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Marie-Fran coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Marie-Fran coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Marie-Fran coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus

LUCIO BOCCARDO (Sapienza Universit` a di Roma - Istituto Lombardo)

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Singular Problems Associated to Quasilinear Equations

Singular Problems Associated to Quasilinear Equations

A WORKSHOP IN CELEBRATION OF MARIE-FRANÇOISE BIDAUT-VÉRON AND LAURENT VÉRON’S 70TH BIRTHDAY.

June 1-3, 2020 The workshop will take place

  • ver 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.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Singular Problems Associated to Quasilinear Equations

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Singular Problems Associated to Quasilinear Equations

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

Bonjour Buon giorno Good morning

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Singular Problems Associated to Quasilinear Equations

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

Bonjour Buon giorno Good morning + thanks to the organizers

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus El amor en los tiempos del coronavirus

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus El amor en los tiempos del coronavirus

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Marie-Fran¸ coise 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-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus L’amour aux temps du corona

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus L’amour aux temps du corona

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

1 . 6 . 2020

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

1 . 6 . 2020 Bon anniversaire

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona

A WORKSHOP IN CELEBRATION OF MARIE-FRANC ¸OISE BIDAUT-VERON AND LAURENT VERON’S 70TH BIRTHDAY.

1 . 6 . 2020 Bon anniversaire

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona Maximum principle

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona Maximum principle

Two maximum principles for two friends

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona Maximum principle

Two maximum principles for two friends

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona Maximum principle

Two maximum principles for two friends

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona Maximum principle

Two maximum principles for two friends

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Les mathematiques aux temps du corona Maximum principle

Two maximum principles for two friends

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, by David A. and Lucio B. / following [AB,JFA 2014]

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, by David A. and Lucio B. / following [AB,JFA 2014]

Ω bounded open subset of RN

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, by David A. and Lucio B. / following [AB,JFA 2014]

Ω bounded open subset of RN M(x) bounded elliptic matrix

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω)

−div(M(x)∇u)+a(x)u|u|γ−1 =f , γ > 0

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω)

−div(M(x)∇u)+a(x)u|u|γ−1 =f , γ > 0 sketch:

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω)

−div(M(x)∇u)+a(x)u|u|γ−1 =f , γ > 0 sketch:

Tk(s) −k −k k k −k k Gk(s) T (G (s))/h

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω)

−div(M(x)∇u)+a(x)u|u|γ−1 =f , γ > 0 sketch :

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (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)|

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (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)| ⇒

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (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| ≤ Q

1 γ ...⇒... ∃ |u| ≤ Q 1 γ

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−div(M(x)∇u)+a(x)u|u|γ−1 =f , γ > 0

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−div(M(x)∇u)+a(x)u|u|γ−1 =f , γ > 0 f=Q a:

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−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

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−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] ≥ T1{a(x)}[Q − uγ] ≥ 0

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−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] ≥ T1{a(x)}[Q − uγ] ≥ 0⇒ ⇒

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−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] ≥ T1{a(x)}[Q − uγ] ≥ 0⇒ ⇒ u satisfies Strong Maximum Principle,

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Maximum principle A remark dedicated to Marie-Fran¸ coise, 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 1,2

0 (Ω) ∩ L∞(Ω), |u| ≤ Q

1 γ

−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] ≥ T1{a(x)}[Q − uγ] ≥ 0⇒ ⇒ u satisfies Strong Maximum Principle, even if 0 < γ < 1.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Lower order term: Convection/Drift terms

We discuss the existence properties and of distributional solutions for the boundary value problems (the first with a convection term, the second with a drift term)

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Lower order term: Convection/Drift terms

We discuss the existence properties and of distributional solutions for the boundary value problems (the first with a convection term, the second with a drift term)

  • −div(M(x)∇u) = −div(u E(x)) + f (x)

in Ω, u = 0

  • n ∂Ω,
  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,
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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Lower order term: Convection/Drift terms

We discuss the existence properties and of distributional solutions for the boundary value problems (the first with a convection term, the second with a drift term)

  • −div(M(x)∇u) = −div(u E(x)) + f (x)

in Ω, u = 0

  • n ∂Ω,
  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,

We note that

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Lower order term: Convection/Drift terms

We discuss the existence properties and of distributional solutions for the boundary value problems (the first with a convection term, the second with a drift term)

  • −div(M(x)∇u) = −div(u E(x)) + f (x)

in Ω, u = 0

  • n ∂Ω,
  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,

We note that at least formally, if M(x) is symmetric, the two above linear problems are in duality.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

  • −div(M(x)∇u) = −div(u E(x)) + f (x)

in Ω, u = 0

  • n ∂Ω,
  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,
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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

  • −div(M(x)∇u) = −div(u E(x)) + f (x)

in Ω, u = 0

  • n ∂Ω,
  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,

We note that

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

  • −div(M(x)∇u) = −div(u E(x)) + f (x)

in Ω, u = 0

  • n ∂Ω,
  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,

We note that the differential operators may be not coercive, unless one assumes that either the norm of |E| in LN(Ω) is small, or that div(EN) = 0: ...

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Papers concerned with this part of the talk

  • L. Boccardo: Some developments on Dirichlet problems

with discontinuous coefficients; Boll. Unione Mat. Ital, 2 (2009) 285–297. (invited paper in memory of 30-death Stampacchia)

  • L. Boccardo: Dirichlet problems with singular

convection terms and applications; J. Differential Equations, 258 (2015) 2290–2314.

  • L. Boccardo: Stampacchia-Calderon-Zygmund theory

for linear elliptic equations with discontinuous coefficients and singular drift; ESAIM, Control, Optimization and Calculus of Variations, 25 (2019), Art. 47, 13 pp.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Assumptions

−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω. Ω bounded subset of RN,

1 1: dependence w.r.t. x / 2: nonsmooth dependence /

Mingione

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Assumptions

−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω. Ω bounded subset of RN, ellipticity:

1 1: dependence w.r.t. x / 2: nonsmooth dependence /

Mingione

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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,

1 1: dependence w.r.t. x / 2: nonsmooth dependence /

Mingione

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Boundary value problem and Lax-Milgram

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Boundary value problem and Lax-Milgram

u weak

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SLIDE 58

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Boundary value problem and Lax-Milgram

u weak /distributional

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Boundary value problem and Lax-Milgram

u weak /distributional solution of the boundary value problem −div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω. means u ∈ W 1,2

0 (Ω) :

  • Ω

M(x)∇u∇v =

  • Ω

u E(x)∇v +

  • Ω

f (x)v, ∀ v ∈ W 1,2

0 (Ω)

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SLIDE 60

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Boundary value problem and Lax-Milgram

u weak /distributional solution of the boundary value problem −div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω. means u ∈ W 1,2

0 (Ω) :

  • Ω

M(x)∇u∇v =

  • Ω

u E(x)∇v +

  • Ω

f (x)v, ∀ v ∈ W 1,2

0 (Ω)/smooth

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Coercivity of −div(M(x)∇u) + div(u E(x))

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Coercivity of −div(M(x)∇u) + div(u E(x))

  • Ω

M(x)∇v∇v ±

  • Ω

v E(x)∇v

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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 ELN not too large

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Coercivity

Our approach hinges on test function method

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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 EN small

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

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SLIDE 70

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

Stampacchia-Calderon-Zygmund for the two problems

2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

2

  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂Ω,

3

2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

2

  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂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

2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

2

  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂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 same Stampacchia-Calderon-Zygmund results of the case

2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

2

  • −div(M(x)∇ψ) = E(x) ∇ψ + g(x)

in Ω, ψ = 0

  • n ∂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 same Stampacchia-Calderon-Zygmund results of the case E = 0 .

2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

|E| ∈ LN(Ω)

as E = 0

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

|E| ∈ LN(Ω)

as E = 0

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

|E| ∈ LN(Ω)

as E = 0

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

2N N+2 ≤ m < N 2 ⇒ u ∈ W 1,2 0 (Ω) ∩ Lm∗∗(Ω);

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

2N N+2 ≤ m < N 2 ⇒ u ∈ W 1,2 0 (Ω) ∩ Lm∗∗(Ω);

2 1 < m <

2N N+2 ⇒ u ∈ W 1,m∗

(Ω);

3 m = 1

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

2N N+2 ≤ m < N 2 ⇒ u ∈ W 1,2 0 (Ω) ∩ Lm∗∗(Ω);

2 1 < m <

2N N+2 ⇒ u ∈ W 1,m∗

(Ω);

3 m = 1 ⇒ u ∈ W 1,q

0 (Ω), q < N N−1 .

u :

  • Ω

M∇u∇φ =

  • Ω

u E∇φ +

  • Ω

f φ, ∀ φ ∈ D.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

2N N+2 ≤ m < N 2 ⇒ u ∈ W 1,2 0 (Ω) ∩ Lm∗∗(Ω);

2 1 < m <

2N N+2 ⇒ u ∈ W 1,m∗

(Ω);

3 m = 1 ⇒ u ∈ W 1,q

0 (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 > N

2 , it is false that u ∈ W 1,m∗

(Ω)

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

2N N+2 ≤ m < N 2 ⇒ u ∈ W 1,2 0 (Ω) ∩ Lm∗∗(Ω);

2 1 < m <

2N N+2 ⇒ u ∈ W 1,m∗

(Ω);

3 m = 1 ⇒ u ∈ W 1,q

0 (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 > N

2 , it is false that u ∈ W 1,m∗

(Ω) Remark E = 0,

2N N+2 + δMeyers < m < N 2 , u ∈?

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

Same results for the drift problem

4 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

4

4 ESAIM-COCV 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

“Nonlinear”

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

“Nonlinear”

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

“Nonlinear” approach to a linear problem

−div(M(x)∇un) = −div

  • un

1 + 1

n|un| E(x)

  • + f (x)
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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Existence of weak/distributional solutions. Summability properties of solutions

Other recent papers

  • L. Boccardo, S. Buccheri, G.R. Cirmi: Two linear noncoercive

Dirichlet problems in duality; Milan J. Math. 86 (2018), 97–104.

  • L. Boccardo, S. Buccheri, R.G. Cirmi: Calderon-Zygmund

theory for infinite energy solutions of nonlinear elliptic equations with singular drift; NODEA, to appear.

  • L. Boccardo, S. Buccheri: A nonlinear homotopy between two

linear Dirichlet problems; Rev. Mat. Complutense, to appear.

  • L. Boccardo: Two semilinear Dirichlet problems “almost” in

duality; Boll. Unione Mat. Ital. 12 (2019), 349–356.

  • L. Boccardo, L. Orsina, A. Porretta: Some noncoercive

parabolic equations with lower order terms in divergence form. Dedicated to Philippe B´

  • enilan. J. Evol. Equ. 3 (2003),

407–418.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭ ✭

E ∈ (LN(Ω))N

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭ ✭

E ∈ (LN(Ω))N

If E ∈ (LN(Ω))N, even for nothing, as in |E| ≤ |A|

|x|,

A ∈ R , 0 ∈ Ω,

5JDE 2015; +Orsina, Nonlin.Anal. 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭ ✭

E ∈ (LN(Ω))N

If E ∈ (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,q 0 (Ω) depends on the size of A . 5

5JDE 2015; +Orsina, Nonlin.Anal. 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭ ✭

E ∈ (LN(Ω))N

If E ∈ (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,q 0 (Ω) depends on the size of A . 5

1) if |A| < α(N−2m)

m

, and

2N N+2 ≤ m < N 2 , then

u ∈ W 1,2

0 (Ω) ∩ Lm∗∗(Ω);

2) if |A| < α(N−2m)

m

, and 1 < m <

2N N+2, then

u ∈ W 1,m∗ (Ω); 3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (M

N N−1(Ω))N

and u ∈ W 1,q

0 (Ω), for every q < N N−1;

4) if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W 1,q

0 (Ω), for

every q <

Nα |A|+α

5JDE 2015; +Orsina, Nonlin.Anal. 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭ ✭

E ∈ (LN(Ω))N

Radial ex.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

E ∈ (L2(Ω))N

6 JDE 2015

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

E ∈ (L2(Ω))N

  • definition of solution;

existence of solution.

6

6 JDE 2015

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

If we add the zero order term “+u”, the framework changes completely.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

If we add the zero order term “+u”, the framework changes completely.

A , un ∈ W 1,2

0 (Ω) :

−div(M(x)∇un) + A un = −div

  • un

1 + 1

n|un| E(x)

  • + f (x)
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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

If we add the zero order term “+u”, the framework changes completely.

A , un ∈ W 1,2

0 (Ω) :

−div(M(x)∇un) + A un = −div

  • un

1 + 1

n|un| E(x)

  • + f (x)

Simpler proofs PhD

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise

✭✭✭✭✭✭✭ ✭ ✭✭✭✭ ✭

E ∈ (LN(Ω))N

If we add the zero order term “+u”, the framework changes completely.

A , un ∈ W 1,2

0 (Ω) :

−div(M(x)∇un) + A un = −div

  • un

1 + 1

n|un| E(x)

  • + f (x)

Simpler proofs PhD course, UCM, November 2019

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Impact of a zero order term

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Impact of a zero order term

By duality: problems with very singular drifts

7 DIE 2019 8 only L2

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

7

7 DIE 2019 8 only L2

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

7

E ∈ L2

8, f bounded ⇒ u ∈ W 1,2 0 (Ω), bounded;

7 DIE 2019 8 only L2

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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

  • n ∂Ω,

7

E ∈ L2

8, f bounded ⇒ u ∈ W 1,2 0 (Ω), bounded;

application to the existence in some Hamilton-J.

  • eq. with lower order term having q-dependence

w.r.t. gradient, q < 2.

7 DIE 2019 8 only L2

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Impact of a zero order term

An elliptic system connected with the mathematical study of PDE models for chemotaxis

9JDE 2015 10Comm.PDE + L. Orsina

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Impact of a zero order term

An elliptic system connected with the mathematical study of PDE models for chemotaxis

−div(A(x)∇u) + u = −div(u M(x)∇ψ) + f (x) , −div(M(x)∇ψ) = uθ .

910

9JDE 2015 10Comm.PDE + L. Orsina

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Convection [L.B. 2020]

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Convection [L.B. 2020]

−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Convection [L.B. 2020]

−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω E, f ∈

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Convection [L.B. 2020]

−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω, u = 0 : ∂Ω E, f ∈ ?

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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 ≥

2N N+2 and f (x) ≥ 0 (of

course not zero a.e.). Then the solution u ∈ W 1,2

0 (Ω) is

positive and it is zero at most on a set of zero Lebesgue measure a.

a weak max. pr.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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 ≥

2N N+2 and f (x) ≥ 0 (of

course not zero a.e.). Then the solution u ∈ W 1,2

0 (Ω) is

positive and it is zero at most on a set of zero Lebesgue measure a.

a weak max. pr.

Remark In the proof we only need E ∈ (L2(Ω))N: blue and red assumptions.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise Drift [L.B. 2020]

−div(M(x)∇ψ)) = −div(ψ E(x)) + g(x) : Ω, ψ = 0 : ∂Ω

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise 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 ≥

2N N+2 and g(x) ≥ 0 (of

course not zero a.e.). Then the solution u ∈ W 1,2

0 (Ω) is

positive and it is zero at most on a set of zero Lebesgue measure a.

a weak max. pr.

Remark In the proof we only need E ∈ (L2(Ω))N: blue and red assumptions.

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Marie-Fran¸ coise 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

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Marie-Fran¸ coise 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

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Marie-Fran¸ coise 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

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Marie-Fran¸ coise 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

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Marie-Fran¸ coise 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)

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Marie-Fran¸ coise 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)

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Marie-Fran¸ coise 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 integral functionals

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Marie-Fran¸ coise 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 integral functionals J(v) = 1 2

  • Ω

A(x, v)|∇v|2 + λ 2

  • Ω

v 2 −

  • Ω

f v λ > 0.

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Marie-Fran¸ coise 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 integral functionals 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

  • n ∂Ω.
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Marie-Fran¸ coise 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

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Marie-Fran¸ coise 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 1,2

0 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)

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Marie-Fran¸ coise 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 1,2

0 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)

Existence [+several friends],

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Marie-Fran¸ coise 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 1,2

0 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)

Existence [+several friends], Existence with

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Marie-Fran¸ coise 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 1,2

0 (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).

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Marie-Fran¸ coise 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 1,2

0 (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 /

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Marie-Fran¸ coise 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 1,2

0 (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

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Marie-Fran¸ coise 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 1,2

0 (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 ≥ 0 almost everywhere in Ω.

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Marie-Fran¸ coise 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 1,2

0 (Ω) : −div([a(x)+|u|q]∇u)+λ u+b(x) u|u|p−1|∇u|2 = f (x)

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Marie-Fran¸ coise 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 1,2

0 (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

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Marie-Fran¸ coise 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 1,2

0 (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 that u(x) ≥ mω almost everywhere in ω.

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Next future

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Next future to MARIE-F. and LAURENT

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Next future to MARIE-F. and LAURENT

je vous souhaite tous le bien

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Next future to MARIE-F. and LAURENT

je vous souhaite tous le bien

without

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Next future to MARIE-F. and LAURENT

je vous souhaite tous le bien

without

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Merci

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Merci Thanks

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Merci Thanks

Ciao

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Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus Merci Thanks

Ciao