Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 ·...
Transcript of Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 ·...
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Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
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Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Marie-Francoise Daza,
Laurent Ariza:
El amor
en los tiempos del coronavirus
LUCIO BOCCARDO
(Sapienza Universita di Roma - Istituto Lombardo)
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Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Singular Problems Associated to Quasilinear Equations
Singular ProblemsAssociated to QuasilinearEquationsA WORKSHOP IN CELEBRATION OF MARIE-FRANÇOISE BIDAUT-VÉRONAND LAURENT VÉRON’S 70TH BIRTHDAY.
June 1-3, 2020
The workshop will take place
over Zoom.
Organizers:
Quoc-Hung Nguyen, ShanghaiTech University
Phuoc-Tai Nguyen, Masaryk University
Speakers
The workshop is co-hosted by Institute of
Mathematical Sciences, ShanghaiTech University
and Department of Mathematics and Statistics,
Masaryk University.
Lucio Boccardo, UNIROMA1, Italy.
Huyuan Chen, JXNU, China.
Julián López Gómez, UCM, Spain.
Manuel Del Pino, Univ. of Bath, UK
Jesús Ildefonso Díaz, UCM, Spain.
Marta García-Huidobro, UC, Chile.
Moshe Marcus, Technion, Israel.
Giuseppe Mingione, UNIPR, Italy.
Vitaly Moroz, Swansea University, UK.
Nguyen Cong Phuc, LSU, USA.
Van Tien Nguyen, NYU, Abu Dhabi.
Alessio Porretta, UNIROMA2, Italy.
Patrizia Pucci, UNIPG, Italy.
Philippe Souplet, LAGA, France
Igor Verbitsky, Univ. of Missouri, USA.
Juan Luis Vázquez, UAM, Spain.
Feng Zhou, ECNU, China.
![Page 4: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/4.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Singular Problems Associated to Quasilinear Equations
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
Bonjour
Buon giorno
Good morning
+thanks to the organizers
![Page 5: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/5.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Singular Problems Associated to Quasilinear Equations
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
Bonjour
Buon giorno
Good morning
+thanks to the organizers
![Page 6: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/6.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Singular Problems Associated to Quasilinear Equations
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
Bonjour
Buon giorno
Good morning
+thanks to the organizers
![Page 7: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/7.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
El amor en los tiempos del coronavirus
![Page 8: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/8.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
El amor en los tiempos del coronavirus
![Page 9: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/9.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
El amor en los tiempos del coronavirus
G.G.M.
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Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
L’amour aux temps du corona
![Page 11: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/11.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
L’amour aux temps du corona
![Page 12: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/12.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
![Page 13: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/13.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
![Page 14: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/14.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
1 . 6 . 2020
Bon anniversaire
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Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
1 . 6 . 2020
Bon anniversaire
![Page 16: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/16.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
1 . 6 . 2020
Bon anniversaire
![Page 17: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/17.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
A WORKSHOP IN CELEBRATION OF MARIE-FRANCOISEBIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.
1 . 6 . 2020
Bon anniversaire
![Page 18: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/18.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
Maximum principle
![Page 19: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/19.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
Maximum principle
Two maximum principles for two friends
![Page 20: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/20.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
Maximum principle
Two maximum principles for two friends
![Page 21: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/21.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
Maximum principle
Two maximum principles for two friends
![Page 22: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/22.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
Maximum principle
Two maximum principles for two friends
![Page 23: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/23.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Les mathematiques aux temps du corona
Maximum principle
Two maximum principles for two friends
![Page 24: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/24.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
![Page 25: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/25.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
![Page 26: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/26.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0sketch:
Tk(s)
−k
−k
k
k
−kk
Gk(s)
−k
k
Th(Gk(s))/h
−k − h
-1
k + h
1
1
![Page 27: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/27.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0sketch:
Tk(s)
−k
−k
k
k
−kk
Gk(s)
−k
k
Th(Gk(s))/h
−k − h
-1
k + h
1
1
![Page 28: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/28.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0sketch:
Tk(s)
−k
−k
k
k
−kk
Gk(s)
−k
k
Th(Gk(s))/h
−k − h
-1
k + h
1
1
![Page 29: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/29.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
sketch:
Tk(s)
−k
−k
k
k
−kk
Gk(s)
−k
k
Th(Gk(s))/h
−k − h
-1
k + h
1
1
![Page 30: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/30.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0sketch:
Tk(s)
−k
−k
k
k
−kk
Gk(s)
−k
k
Th(Gk(s))/h
−k − h
-1
k + h
1
1
![Page 31: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/31.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0sketch:
Tk(s)
−k
−k
k
k
−kk
Gk(s)
−k
k
Th(Gk(s))/h
−k − h
-1
k + h
1
1
![Page 32: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/32.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
sketch :
α
∫Ω
|∇Gk(un)|2 +
∫Ω
an(x)|un|γ|Gk(un)|
≤∫
Ω
|fn||Gk(un)| ≤∫
Ω
Q an(x)|Gk(un)|
⇒posit. +
∫Ω
an(x)[|un|γ − Q]|Gk(un)| ≤ 0
⇒|un| ≤ Q1γ ...⇒... ∃ |u| ≤ Q
1γ
![Page 33: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/33.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
sketch : α
∫Ω
|∇Gk(un)|2 +
∫Ω
an(x)|un|γ|Gk(un)|
≤∫
Ω
|fn||Gk(un)| ≤∫
Ω
Q an(x)|Gk(un)|
⇒posit. +
∫Ω
an(x)[|un|γ − Q]|Gk(un)| ≤ 0
⇒|un| ≤ Q1γ ...⇒... ∃ |u| ≤ Q
1γ
![Page 34: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/34.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
sketch : α
∫Ω
|∇Gk(un)|2 +
∫Ω
an(x)|un|γ|Gk(un)|
≤∫
Ω
|fn||Gk(un)| ≤∫
Ω
Q an(x)|Gk(un)|
⇒
posit. +
∫Ω
an(x)[|un|γ − Q]|Gk(un)| ≤ 0
⇒|un| ≤ Q1γ ...⇒... ∃ |u| ≤ Q
1γ
![Page 35: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/35.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
u ∈ W 1,20 (Ω) ∩ L∞(Ω)
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
sketch : α
∫Ω
|∇Gk(un)|2 +
∫Ω
an(x)|un|γ|Gk(un)|
≤∫
Ω
|fn||Gk(un)| ≤∫
Ω
Q an(x)|Gk(un)|
⇒posit. +
∫Ω
an(x)[|un|γ − Q]|Gk(un)| ≤ 0
⇒|un| ≤ Q1γ ...⇒... ∃ |u| ≤ Q
1γ
![Page 36: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/36.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a: −div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒u satisfies Strong Maximum Principle,even if 0 < γ < 1.
![Page 37: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/37.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a:
−div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒u satisfies Strong Maximum Principle,even if 0 < γ < 1.
![Page 38: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/38.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a: −div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒u satisfies Strong Maximum Principle,even if 0 < γ < 1.
![Page 39: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/39.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a: −div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0
⇒⇒u satisfies Strong Maximum Principle,even if 0 < γ < 1.
![Page 40: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/40.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a: −div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒
u satisfies Strong Maximum Principle,even if 0 < γ < 1.
![Page 41: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/41.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a: −div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒u satisfies Strong Maximum Principle,
even if 0 < γ < 1.
![Page 42: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/42.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Maximum principle
A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]
Ω bounded open subset of RN
M(x) bounded elliptic matrix
|f (x)| ≤ Q a(x) ∈ L1(Ω), Q > 0
∃ : u ∈ W 1,20 (Ω) ∩ L∞(Ω), |u| ≤ Q
1γ
−div(M(x)∇u)+a(x)u|u|γ−1 = f , γ > 0
f=Q a: −div(M(x)∇u)+a(x)uγ =Q a(x), γ > 0
−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒u satisfies Strong Maximum Principle,even if 0 < γ < 1.
![Page 43: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/43.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
![Page 44: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/44.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Lower order term: Convection/Drift terms
We discuss the existence properties and of distributionalsolutions for the boundary value problems (the firstwith a convection term, the second with a drift term)
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
at least formally, if M(x) is symmetric, the twoabove linear problems are in duality.
![Page 45: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/45.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Lower order term: Convection/Drift terms
We discuss the existence properties and of distributionalsolutions for the boundary value problems (the firstwith a convection term, the second with a drift term)
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
at least formally, if M(x) is symmetric, the twoabove linear problems are in duality.
![Page 46: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/46.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Lower order term: Convection/Drift terms
We discuss the existence properties and of distributionalsolutions for the boundary value problems (the firstwith a convection term, the second with a drift term)
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
at least formally, if M(x) is symmetric, the twoabove linear problems are in duality.
![Page 47: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/47.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Lower order term: Convection/Drift terms
We discuss the existence properties and of distributionalsolutions for the boundary value problems (the firstwith a convection term, the second with a drift term)
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
at least formally, if M(x) is symmetric, the twoabove linear problems are in duality.
![Page 48: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/48.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
the differential operators may be not coercive,unless one assumes that either the norm of |E | inLN(Ω) is small, or that div(‖E‖
N) = 0: ...
![Page 49: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/49.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
the differential operators may be not coercive,unless one assumes that either the norm of |E | inLN(Ω) is small, or that div(‖E‖
N) = 0: ...
![Page 50: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/50.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
We note that
the differential operators may be not coercive,unless one assumes that either the norm of |E | inLN(Ω) is small, or that div(‖E‖
N) = 0: ...
![Page 51: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/51.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Papers concerned with this part of the talk
L. Boccardo: Some developments on Dirichlet problemswith discontinuous coefficients; Boll. Unione Mat. Ital, 2(2009) 285–297.(invited paper in memory of 30-death Stampacchia)
L. Boccardo: Dirichlet problems with singularconvection terms and applications; J. DifferentialEquations, 258 (2015) 2290–2314.
L. Boccardo: Stampacchia-Calderon-Zygmund theoryfor linear elliptic equations with discontinuouscoefficients and singular drift; ESAIM, Control,Optimization and Calculus of Variations, 25 (2019), Art.47, 13 pp.
![Page 52: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/52.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Assumptions
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN ,
ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
1 1: dependence w.r.t. x / 2: nonsmooth dependence /Mingione
![Page 53: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/53.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Assumptions
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN ,
ellipticity:
0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
1 1: dependence w.r.t. x / 2: nonsmooth dependence /Mingione
![Page 54: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/54.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Assumptions
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN ,
ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
1 1: dependence w.r.t. x / 2: nonsmooth dependence /Mingione
![Page 55: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/55.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Assumptions
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN ,
ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
1 1: dependence w.r.t. x / 2: nonsmooth dependence /Mingione
![Page 56: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/56.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Boundary value problem and Lax-Milgram
u weak /distributional solution of the boundary valueproblem
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,u = 0 : ∂Ω.
means
u ∈ W 1,20 (Ω) :
∫Ω
M(x)∇u∇v =
∫Ω
u E (x)∇v +
∫Ω
f (x)v ,
∀ v ∈ W 1,20 (Ω)/smooth
![Page 57: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/57.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Boundary value problem and Lax-Milgram
u weak
/distributional solution of the boundary valueproblem
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,u = 0 : ∂Ω.
means
u ∈ W 1,20 (Ω) :
∫Ω
M(x)∇u∇v =
∫Ω
u E (x)∇v +
∫Ω
f (x)v ,
∀ v ∈ W 1,20 (Ω)/smooth
![Page 58: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/58.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Boundary value problem and Lax-Milgram
u weak /distributional
solution of the boundary valueproblem
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,u = 0 : ∂Ω.
means
u ∈ W 1,20 (Ω) :
∫Ω
M(x)∇u∇v =
∫Ω
u E (x)∇v +
∫Ω
f (x)v ,
∀ v ∈ W 1,20 (Ω)/smooth
![Page 59: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/59.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Boundary value problem and Lax-Milgram
u weak /distributional solution of the boundary valueproblem
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,u = 0 : ∂Ω.
means
u ∈ W 1,20 (Ω) :
∫Ω
M(x)∇u∇v =
∫Ω
u E (x)∇v +
∫Ω
f (x)v ,
∀ v ∈ W 1,20 (Ω)
/smooth
![Page 60: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/60.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Boundary value problem and Lax-Milgram
u weak /distributional solution of the boundary valueproblem
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,u = 0 : ∂Ω.
means
u ∈ W 1,20 (Ω) :
∫Ω
M(x)∇u∇v =
∫Ω
u E (x)∇v +
∫Ω
f (x)v ,
∀ v ∈ W 1,20 (Ω)/smooth
![Page 61: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/61.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Coercivity of −div(M(x)∇u) + div(u E(x))
∫Ω
M(x)∇v∇v ±∫
Ω
v E (x)∇v
≥ α
∫Ω
|∇v |2 −[ ∫
Ω
|v |2∗] 1
2∗[ ∫
Ω
|E (x)|N] 1
N[ ∫
Ω
|∇v |2] 1
2
≥α− 1
S
[ ∫Ω
|E (x)|N] 1
N∫
Ω
|∇v |2
E ∈ LN
‖E‖LN
not too large
![Page 62: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/62.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Coercivity of −div(M(x)∇u) + div(u E(x))
∫Ω
M(x)∇v∇v ±∫
Ω
v E (x)∇v
≥ α
∫Ω
|∇v |2 −[ ∫
Ω
|v |2∗] 1
2∗[ ∫
Ω
|E (x)|N] 1
N[ ∫
Ω
|∇v |2] 1
2
≥α− 1
S
[ ∫Ω
|E (x)|N] 1
N∫
Ω
|∇v |2
E ∈ LN
‖E‖LN
not too large
![Page 63: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/63.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Coercivity of −div(M(x)∇u) + div(u E(x))
∫Ω
M(x)∇v∇v ±∫
Ω
v E (x)∇v
≥ α
∫Ω
|∇v |2 −[ ∫
Ω
|v |2∗] 1
2∗[ ∫
Ω
|E (x)|N] 1
N[ ∫
Ω
|∇v |2] 1
2
≥α− 1
S
[ ∫Ω
|E (x)|N] 1
N∫
Ω
|∇v |2
E ∈ LN
‖E‖LN
not too large
![Page 64: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/64.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Coercivity of −div(M(x)∇u) + div(u E(x))
∫Ω
M(x)∇v∇v ±∫
Ω
v E (x)∇v
≥ α
∫Ω
|∇v |2 −[ ∫
Ω
|v |2∗] 1
2∗[ ∫
Ω
|E (x)|N] 1
N[ ∫
Ω
|∇v |2] 1
2
≥α− 1
S
[ ∫Ω
|E (x)|N] 1
N∫
Ω
|∇v |2
E ∈ LN
‖E‖LN
not too large
![Page 65: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/65.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Coercivity of −div(M(x)∇u) + div(u E(x))
∫Ω
M(x)∇v∇v ±∫
Ω
v E (x)∇v
≥ α
∫Ω
|∇v |2 −[ ∫
Ω
|v |2∗] 1
2∗[ ∫
Ω
|E (x)|N] 1
N[ ∫
Ω
|∇v |2] 1
2
≥α− 1
S
[ ∫Ω
|E (x)|N] 1
N∫
Ω
|∇v |2
E ∈ LN
‖E‖LN
not too large
![Page 66: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/66.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Coercivity of −div(M(x)∇u) + div(u E(x))
∫Ω
M(x)∇v∇v ±∫
Ω
v E (x)∇v
≥ α
∫Ω
|∇v |2 −[ ∫
Ω
|v |2∗] 1
2∗[ ∫
Ω
|E (x)|N] 1
N[ ∫
Ω
|∇v |2] 1
2
≥α− 1
S
[ ∫Ω
|E (x)|N] 1
N∫
Ω
|∇v |2
E ∈ LN
‖E‖LN
not too large
![Page 67: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/67.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Our approach hinges on test function method
The proofs of all the results are very easy
if we assume div(E ) = 0
if we assume ‖E‖N
small
![Page 68: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/68.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Coercivity
Our approach hinges on test function method
The proofs of all the results are very easy
if we assume div(E ) = 0
if we assume ‖E‖N
small
![Page 69: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/69.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
![Page 70: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/70.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems
−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,2
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,3
Ω bounded subset of RN ,ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β,f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
and we prove for both the b.v.p. the sameStampacchia-Calderon-Zygmund results of the caseE = 0 .
2 paper invitation U.M.I. in memory of 30-Stampacchia3 ESAIM-COCV 2019
![Page 71: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/71.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,2
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,3
Ω bounded subset of RN ,ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β,f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
and we prove for both the b.v.p. the sameStampacchia-Calderon-Zygmund results of the caseE = 0 .
2 paper invitation U.M.I. in memory of 30-Stampacchia3 ESAIM-COCV 2019
![Page 72: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/72.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,2
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,3
Ω bounded subset of RN ,ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β,f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
and we prove for both the b.v.p. the sameStampacchia-Calderon-Zygmund results of the caseE = 0 .
2 paper invitation U.M.I. in memory of 30-Stampacchia3 ESAIM-COCV 2019
![Page 73: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/73.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,2
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,3
Ω bounded subset of RN ,ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β,f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
and we prove for both the b.v.p. the sameStampacchia-Calderon-Zygmund results of the case
E = 0 .
2 paper invitation U.M.I. in memory of 30-Stampacchia3 ESAIM-COCV 2019
![Page 74: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/74.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,
u = 0 on ∂Ω,2
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,3
Ω bounded subset of RN ,ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β,f ∈ Lm(Ω), 1 ≤ m ≤ ∞,
E ∈ (LN(Ω))N
and we prove for both the b.v.p. the sameStampacchia-Calderon-Zygmund results of the caseE = 0 .
2 paper invitation U.M.I. in memory of 30-Stampacchia3 ESAIM-COCV 2019
![Page 75: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/75.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0
![Page 76: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/76.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0
![Page 77: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/77.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
1 2NN+2≤ m < N
2⇒ u ∈ W 1,2
0 (Ω) ∩ Lm∗∗(Ω);2 1 < m < 2N
N+2⇒ u ∈ W 1,m∗
0 (Ω);3 m = 1 ⇒ u ∈ W 1,q
0 (Ω), q < NN−1
.
u :
∫Ω
M∇u∇φ =
∫Ω
u E∇φ +
∫Ω
f φ, ∀φ ∈ D.
Theorem (70-Brezis)
E = 0, m > N2
, it is false that u ∈ W 1,m∗
0 (Ω)
Remark
E = 0, 2NN+2
+ δMeyers < m < N2
, u ∈?
![Page 78: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/78.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
1 2NN+2≤ m < N
2⇒ u ∈ W 1,2
0 (Ω) ∩ Lm∗∗(Ω);
2 1 < m < 2NN+2⇒ u ∈ W 1,m∗
0 (Ω);3 m = 1 ⇒ u ∈ W 1,q
0 (Ω), q < NN−1
.
u :
∫Ω
M∇u∇φ =
∫Ω
u E∇φ +
∫Ω
f φ, ∀φ ∈ D.
Theorem (70-Brezis)
E = 0, m > N2
, it is false that u ∈ W 1,m∗
0 (Ω)
Remark
E = 0, 2NN+2
+ δMeyers < m < N2
, u ∈?
![Page 79: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/79.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
1 2NN+2≤ m < N
2⇒ u ∈ W 1,2
0 (Ω) ∩ Lm∗∗(Ω);2 1 < m < 2N
N+2⇒ u ∈ W 1,m∗
0 (Ω);3 m = 1
⇒ u ∈ W 1,q0 (Ω), q < N
N−1.
u :
∫Ω
M∇u∇φ =
∫Ω
u E∇φ +
∫Ω
f φ, ∀φ ∈ D.
Theorem (70-Brezis)
E = 0, m > N2
, it is false that u ∈ W 1,m∗
0 (Ω)
Remark
E = 0, 2NN+2
+ δMeyers < m < N2
, u ∈?
![Page 80: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/80.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
1 2NN+2≤ m < N
2⇒ u ∈ W 1,2
0 (Ω) ∩ Lm∗∗(Ω);2 1 < m < 2N
N+2⇒ u ∈ W 1,m∗
0 (Ω);3 m = 1 ⇒ u ∈ W 1,q
0 (Ω), q < NN−1
.
u :
∫Ω
M∇u∇φ =
∫Ω
u E∇φ +
∫Ω
f φ, ∀φ ∈ D.
Theorem (70-Brezis)
E = 0, m > N2
, it is false that u ∈ W 1,m∗
0 (Ω)
Remark
E = 0, 2NN+2
+ δMeyers < m < N2
, u ∈?
![Page 81: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/81.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
1 2NN+2≤ m < N
2⇒ u ∈ W 1,2
0 (Ω) ∩ Lm∗∗(Ω);2 1 < m < 2N
N+2⇒ u ∈ W 1,m∗
0 (Ω);3 m = 1 ⇒ u ∈ W 1,q
0 (Ω), q < NN−1
.
u :
∫Ω
M∇u∇φ =
∫Ω
u E∇φ +
∫Ω
f φ, ∀φ ∈ D.
Theorem (70-Brezis)
E = 0, m > N2
, it is false that u ∈ W 1,m∗
0 (Ω)
Remark
E = 0, 2NN+2
+ δMeyers < m < N2
, u ∈?
![Page 82: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/82.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
|E | ∈ LN(Ω)
as E = 0−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
1 2NN+2≤ m < N
2⇒ u ∈ W 1,2
0 (Ω) ∩ Lm∗∗(Ω);2 1 < m < 2N
N+2⇒ u ∈ W 1,m∗
0 (Ω);3 m = 1 ⇒ u ∈ W 1,q
0 (Ω), q < NN−1
.
u :
∫Ω
M∇u∇φ =
∫Ω
u E∇φ +
∫Ω
f φ, ∀φ ∈ D.
Theorem (70-Brezis)
E = 0, m > N2
, it is false that u ∈ W 1,m∗
0 (Ω)
Remark
E = 0, 2NN+2
+ δMeyers < m < N2
, u ∈?
![Page 83: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/83.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Same results for the drift problem
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,4
4 ESAIM-COCV 2019
![Page 84: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/84.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Same results for the drift problem
−div(M(x)∇ψ) = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,4
4 ESAIM-COCV 2019
![Page 85: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/85.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
“Nonlinear”
approach to a linear problem
−div(M(x)∇un) = −div( un
1 + 1n|un|
E (x))
+ f (x)
![Page 86: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/86.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
“Nonlinear”
approach to a linear problem
−div(M(x)∇un) = −div( un
1 + 1n|un|
E (x))
+ f (x)
![Page 87: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/87.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
“Nonlinear” approach to a linear problem
−div(M(x)∇un) = −div( un
1 + 1n|un|
E (x))
+ f (x)
![Page 88: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/88.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Existence of weak/distributional solutions. Summability properties of solutions
Other recent papers
L. Boccardo, S. Buccheri, G.R. Cirmi: Two linear noncoerciveDirichlet problems in duality; Milan J. Math. 86 (2018),97–104.
L. Boccardo, S. Buccheri, R.G. Cirmi: Calderon-Zygmundtheory for infinite energy solutions of nonlinear ellipticequations with singular drift; NODEA, to appear.
L. Boccardo, S. Buccheri: A nonlinear homotopy between twolinear Dirichlet problems; Rev. Mat. Complutense, to appear.
L. Boccardo: Two semilinear Dirichlet problems “almost” induality; Boll. Unione Mat. Ital. 12 (2019), 349–356.
L. Boccardo, L. Orsina, A. Porretta: Some noncoerciveparabolic equations with lower order terms in divergence form.Dedicated to Philippe Benilan. J. Evol. Equ. 3 (2003),407–418.
![Page 89: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/89.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((E ∈ (LN (Ω))N
![Page 90: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/90.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((E ∈ (LN (Ω))N
If E 6∈ (LN(Ω))N , even for nothing, as in
|E | ≤ |A||x | , A ∈ R , 0 ∈ Ω,
the framework changes completely:u ∈W 1,2
0 (Ω) or u ∈W 1,q0 (Ω) depends on the size of A . 5
1) if |A| < α(N−2m)m
, and 2NN+2≤ m < N
2, then
u ∈ W 1,20 (Ω) ∩ Lm∗∗(Ω);
2) if |A| < α(N−2m)m
, and 1 < m < 2NN+2
, then
u ∈ W 1,m∗
0 (Ω);
3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (MN
N−1 (Ω))N
and u ∈ W 1,q0 (Ω), for every q < N
N−1;
4) if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W 1,q0 (Ω), for
every q < Nα|A|+α
5JDE 2015; +Orsina, Nonlin.Anal. 2019
![Page 91: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/91.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((E ∈ (LN (Ω))N
If E 6∈ (LN(Ω))N , even for nothing, as in
|E | ≤ |A||x | , A ∈ R , 0 ∈ Ω,
the framework changes completely:u ∈W 1,2
0 (Ω) or u ∈W 1,q0 (Ω) depends on the size of A . 5
1) if |A| < α(N−2m)m
, and 2NN+2≤ m < N
2, then
u ∈ W 1,20 (Ω) ∩ Lm∗∗(Ω);
2) if |A| < α(N−2m)m
, and 1 < m < 2NN+2
, then
u ∈ W 1,m∗
0 (Ω);
3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (MN
N−1 (Ω))N
and u ∈ W 1,q0 (Ω), for every q < N
N−1;
4) if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W 1,q0 (Ω), for
every q < Nα|A|+α
5JDE 2015; +Orsina, Nonlin.Anal. 2019
![Page 92: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/92.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((E ∈ (LN (Ω))N
If E 6∈ (LN(Ω))N , even for nothing, as in
|E | ≤ |A||x | , A ∈ R , 0 ∈ Ω,
the framework changes completely:u ∈W 1,2
0 (Ω) or u ∈W 1,q0 (Ω) depends on the size of A . 5
1) if |A| < α(N−2m)m
, and 2NN+2≤ m < N
2, then
u ∈ W 1,20 (Ω) ∩ Lm∗∗(Ω);
2) if |A| < α(N−2m)m
, and 1 < m < 2NN+2
, then
u ∈ W 1,m∗
0 (Ω);
3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (MN
N−1 (Ω))N
and u ∈ W 1,q0 (Ω), for every q < N
N−1;
4) if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W 1,q0 (Ω), for
every q < Nα|A|+α
5JDE 2015; +Orsina, Nonlin.Anal. 2019
![Page 93: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/93.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((E ∈ (LN (Ω))N
Radial ex.
![Page 94: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/94.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
![Page 95: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/95.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
E ∈ (L2(Ω))N
definition of solution;
existence of solution.6
6 JDE 2015
![Page 96: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/96.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
E ∈ (L2(Ω))N definition of solution;
existence of solution.6
6 JDE 2015
![Page 97: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/97.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un ∈ W 1,20 (Ω) :
−div(M(x)∇un) + A un = −div( un
1 + 1n|un|
E (x))
+ f (x)
Simpler proofs
PhD course, UCM, November 2019
![Page 98: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/98.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un ∈ W 1,20 (Ω) :
−div(M(x)∇un) + A un = −div( un
1 + 1n|un|
E (x))
+ f (x)
Simpler proofs
PhD course, UCM, November 2019
![Page 99: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/99.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un ∈ W 1,20 (Ω) :
−div(M(x)∇un) + A un = −div( un
1 + 1n|un|
E (x))
+ f (x)
Simpler proofs
PhD
course, UCM, November 2019
![Page 100: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/100.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
(((((((((((((E ∈ (LN (Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un ∈ W 1,20 (Ω) :
−div(M(x)∇un) + A un = −div( un
1 + 1n|un|
E (x))
+ f (x)
Simpler proofs
PhD course, UCM, November 2019
![Page 101: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/101.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
![Page 102: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/102.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
By duality: problems with very singular drifts
−div(M(x)∇ψ) + ψ = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,7
E ∈ L2 8, f bounded ⇒ u ∈ W 1,20 (Ω), bounded;
application to the existence in some Hamilton-J.eq. with lower order term having q-dependencew.r.t. gradient, q < 2.
7 DIE 20198 only L2
![Page 103: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/103.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
By duality: problems with very singular drifts
−div(M(x)∇ψ) + ψ = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,7
E ∈ L2 8, f bounded ⇒ u ∈ W 1,20 (Ω), bounded;
application to the existence in some Hamilton-J.eq. with lower order term having q-dependencew.r.t. gradient, q < 2.
7 DIE 20198 only L2
![Page 104: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/104.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
By duality: problems with very singular drifts
−div(M(x)∇ψ) + ψ = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,7
E ∈ L2 8, f bounded ⇒ u ∈ W 1,20 (Ω), bounded;
application to the existence in some Hamilton-J.eq. with lower order term having q-dependencew.r.t. gradient, q < 2.
7 DIE 20198 only L2
![Page 105: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/105.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
By duality: problems with very singular drifts
−div(M(x)∇ψ) + ψ = E (x)∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,7
E ∈ L2 8, f bounded ⇒ u ∈ W 1,20 (Ω), bounded;
application to the existence in some Hamilton-J.eq. with lower order term having q-dependencew.r.t. gradient, q < 2.
7 DIE 20198 only L2
![Page 106: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/106.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
An elliptic system connected with the mathematical study of PDE models forchemotaxis
−div(A(x)∇u) + u = −div(u M(x)∇ψ) + f (x) ,−div(M(x)∇ψ) = uθ .
910
9JDE 201510Comm.PDE + L. Orsina
![Page 107: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/107.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Impact of a zero order term
An elliptic system connected with the mathematical study of PDE models forchemotaxis
−div(A(x)∇u) + u = −div(u M(x)∇ψ) + f (x) ,−div(M(x)∇ψ) = uθ .
910
9JDE 201510Comm.PDE + L. Orsina
![Page 108: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/108.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Convection [L.B. 2020]
![Page 109: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/109.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Convection [L.B. 2020]
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
E , f ∈ ?
Theorem
E ∈ (LN(Ω))N , f ∈ Lm(Ω) with m ≥ 2NN+2
and f (x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 110: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/110.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Convection [L.B. 2020]
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
E , f ∈
?
Theorem
E ∈ (LN(Ω))N , f ∈ Lm(Ω) with m ≥ 2NN+2
and f (x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 111: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/111.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Convection [L.B. 2020]
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
E , f ∈ ?
Theorem
E ∈ (LN(Ω))N , f ∈ Lm(Ω) with m ≥ 2NN+2
and f (x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 112: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/112.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Convection [L.B. 2020]
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
E , f ∈ ?
Theorem
E ∈ (LN(Ω))N , f ∈ Lm(Ω) with m ≥ 2NN+2
and f (x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 113: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/113.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Convection [L.B. 2020]
−div(M(x)∇u)) = −div(u E (x)) + f (x) : Ω,
u = 0 : ∂Ω
E , f ∈ ?
Theorem
E ∈ (LN(Ω))N , f ∈ Lm(Ω) with m ≥ 2NN+2
and f (x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 114: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/114.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Drift [L.B. 2020]
−div(M(x)∇ψ)) = −div(ψ E (x)) + g(x) : Ω,
ψ = 0 : ∂Ω
Theorem
E ∈ (LN(Ω))N , g ∈ Lm(Ω) with m ≥ 2NN+2
and g(x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 115: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/115.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
First Maximum Principle, dedicated to Marie-Francoise
Drift [L.B. 2020]
−div(M(x)∇ψ)) = −div(ψ E (x)) + g(x) : Ω,
ψ = 0 : ∂Ω
Theorem
E ∈ (LN(Ω))N , g ∈ Lm(Ω) with m ≥ 2NN+2
and g(x) ≥ 0 (of
course not zero a.e.). Then the solution u ∈ W 1,20 (Ω) is
positive and it is zero at most on a set of zeroLebesgue measure a.
a weak max. pr.
Remark
In the proof we only need E ∈ (L2(Ω))N : blue and redassumptions.
![Page 116: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/116.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
![Page 117: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/117.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
with Alberto Farina
![Page 118: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/118.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
with Alberto Farina
![Page 119: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/119.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
books
![Page 120: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/120.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
books
![Page 121: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/121.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
in a conference in Cortona (organizers Juan Luis and Lucio)
![Page 122: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/122.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
in a conference in Cortona (organizers Juan Luis and Lucio)
![Page 123: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/123.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Calculus of Variations (in the study of integral functionals)
Recall this large and important class of integralfunctionals
J(v) =1
2
∫Ω
A(x , v)|∇v |2 +λ
2
∫Ω
v 2 −∫
Ω
f v λ > 0.
The Euler-Lagrange equation for J is (at least formally)the quasilinear elliptic problem−div(A(x , u)∇u) + 1
2A′(x , u)|∇u|2 + λ u = f in Ω,
u = 0 on ∂Ω.
![Page 124: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/124.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Calculus of Variations (in the study of integral functionals)
Recall this large and important class of integralfunctionals
J(v) =1
2
∫Ω
A(x , v)|∇v |2 +λ
2
∫Ω
v 2 −∫
Ω
f v λ > 0.
The Euler-Lagrange equation for J is (at least formally)the quasilinear elliptic problem−div(A(x , u)∇u) + 1
2A′(x , u)|∇u|2 + λ u = f in Ω,
u = 0 on ∂Ω.
![Page 125: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/125.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Calculus of Variations (in the study of integral functionals)
Recall this large and important class of integralfunctionals
J(v) =1
2
∫Ω
A(x , v)|∇v |2 +λ
2
∫Ω
v 2 −∫
Ω
f v λ > 0.
The Euler-Lagrange equation for J is (at least formally)the quasilinear elliptic problem−div(A(x , u)∇u) + 1
2A′(x , u)|∇u|2 + λ u = f in Ω,
u = 0 on ∂Ω.
![Page 126: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/126.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
![Page 127: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/127.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle / easy )
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 128: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/128.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle / easy )
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 129: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/129.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with
regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle / easy )
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 130: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/130.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle / easy )
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 131: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/131.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle /
easy )
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 132: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/132.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle / easy
)
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 133: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/133.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
Background
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
Existence [+several friends],
Existence with regularizing effect [B-Gallouet],
(L.B. dedicated to 60-Laurent).
Theorem (Weak Maximum Principle / easy )
If f ≥ 0, then the weak solution u is such that u ≥ 0almost everywhere in Ω.
![Page 134: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/134.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
pour Laurent
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
even with f(x) very singular
Theorem (Strong Maximum Principle)
If f ≥ 0 (and not almost everywhere equal to zero),then for every set ω ⊂⊂ Ω there exists mω > 0 such thatu(x) ≥ mω almost everywhere in ω.
![Page 135: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/135.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
pour Laurent
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
even with f(x) very singular
Theorem (Strong Maximum Principle)
If f ≥ 0 (and not almost everywhere equal to zero),then for every set ω ⊂⊂ Ω there exists mω > 0 such thatu(x) ≥ mω almost everywhere in ω.
![Page 136: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/136.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]
Quasilinear Dirichlet problems having l.o.t. with natural growth
pour Laurent
Consider the Dirichlet problem
u ∈ W 1,20 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)
even with f(x) very singular
Theorem (Strong Maximum Principle)
If f ≥ 0 (and not almost everywhere equal to zero),then for every set ω ⊂⊂ Ω there exists mω > 0 such thatu(x) ≥ mω almost everywhere in ω.
![Page 137: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/137.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Next future
![Page 138: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/138.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Next future
to MARIE-F. and LAURENT
![Page 139: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/139.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Next future
to MARIE-F. and LAURENT
je vous souhaite tous le bien
without
![Page 140: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/140.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Next future
to MARIE-F. and LAURENT
je vous souhaite tous le bien
without
![Page 141: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/141.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Next future
to MARIE-F. and LAURENT
je vous souhaite tous le bien
without
![Page 142: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/142.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Merci
![Page 143: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/143.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Merci
Thanks
![Page 144: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/144.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Merci
Thanks
Ciao
![Page 145: Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 · Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle](https://reader033.fdocumento.com/reader033/viewer/2022060300/5f0810c67e708231d420290b/html5/thumbnails/145.jpg)
Marie-Francoise Daza, Laurent Ariza: El amor en los tiempos del coronavirus
Merci
Thanks
Ciao