Marie-Fran coise Daza, Laurent Ariza: El amor en los tiempos del - PowerPoint PPT Presentation

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 � � |∇ G k ( u n ) | 2 + a n ( x ) | u n | γ | G k ( u n ) | sketch : α Ω Ω � � ≤ | f n || G k ( u n ) | ≤ Q a n ( x ) | G k ( u n ) | Ω Ω � a n ( x )[ | u n | γ − Q ] | G k ( u n ) | ≤ 0 ⇒ posit . + Ω 1 1 γ ... ⇒ ... ∃ | u | ≤ Q ⇒| u n | ≤ Q γ

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 f=Q a:

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 f=Q a: − div ( M ( x ) ∇ u )+ a ( x ) u γ = Q a ( x ) , γ > 0

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 f=Q a: − div ( M ( x ) ∇ u )+ a ( x ) u γ = Q a ( x ) , γ > 0 − div ( M ( x ) ∇ u )= a ( x )[ Q − u γ ] ≥ T 1 { a ( x ) } [ Q − u γ ] ≥ 0

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 f=Q a: − div ( M ( x ) ∇ u )+ a ( x ) u γ = Q a ( x ) , γ > 0 − div ( M ( x ) ∇ u )= a ( x )[ Q − u γ ] ≥ T 1 { a ( x ) } [ Q − u γ ] ≥ 0 ⇒ ⇒

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 f=Q a: − div ( M ( x ) ∇ u )+ a ( x ) u γ = Q a ( x ) , γ > 0 − div ( M ( x ) ∇ u )= a ( x )[ Q − u γ ] ≥ T 1 { a ( x ) } [ Q − u γ ] ≥ 0 ⇒ ⇒ u satisfies Strong Maximum Principle,

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 R N M ( x ) bounded elliptic matrix | f ( x ) | ≤ Q a ( x ) ∈ L 1 (Ω) , Q > 0 1 ∃ : u ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) , | u | ≤ Q γ − div ( M ( x ) ∇ u )+ a ( x ) u | u | γ − 1 = f , γ > 0 f=Q a: − div ( M ( x ) ∇ u )+ a ( x ) u γ = Q a ( x ) , γ > 0 − div ( M ( x ) ∇ u )= a ( x )[ Q − u γ ] ≥ T 1 { a ( x ) } [ Q − u γ ] ≥ 0 ⇒ ⇒ u satisfies Strong Maximum Principle, even if 0 < γ < 1 .

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

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)

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

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 on ∂ Ω , � − div ( M ( x ) ∇ ψ ) = E ( x ) ∇ ψ + g ( x ) in Ω , ψ = 0 on ∂ Ω , We note that

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 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 two above linear problems are in duality.

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

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 on ∂ Ω , � − div ( M ( x ) ∇ ψ ) = E ( x ) ∇ ψ + g ( x ) in Ω , ψ = 0 on ∂ Ω , We note that

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 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 | in L N (Ω) is small, or that div ( � E � N ) = 0 : ...

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.

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 R N , 1 1: dependence w.r.t. x / 2: nonsmooth dependence / Mingione

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 R N , ellipticity: 1 1: dependence w.r.t. x / 2: nonsmooth dependence / Mingione

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 R N , ellipticity: 0 < α, α | ξ | 2 ≤ M ( x ) ξξ, | M ( x ) | 1 ≤ β , 1 1: dependence w.r.t. x / 2: nonsmooth dependence / Mingione

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 R N , ellipticity: 0 < α, α | ξ | 2 ≤ M ( x ) ξξ, | M ( x ) | 1 ≤ β , f ∈ L m (Ω) , 1 ≤ m ≤ ∞ , E ∈ ( L N (Ω)) N 1 1: dependence w.r.t. x / 2: nonsmooth dependence / Mingione

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

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

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

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) : M ( x ) ∇ u ∇ v = u E ( x ) ∇ v + f ( x ) v , Ω Ω Ω ∀ v ∈ W 1 , 2 0 (Ω)

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) : M ( x ) ∇ u ∇ v = u E ( x ) ∇ v + f ( x ) v , Ω Ω Ω ∀ v ∈ W 1 , 2 0 (Ω) / smooth

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 ))

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 )) � � M ( x ) ∇ v ∇ v ± v E ( x ) ∇ v Ω Ω

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 )) � � M ( x ) ∇ v ∇ v ± v E ( x ) ∇ v Ω Ω | v | 2 ∗ � 1 | E ( x ) | N � 1 |∇ v | 2 � 1 � � � 2 ∗ � � N � � |∇ v | 2 − 2 ≥ α Ω Ω Ω Ω

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 )) � � M ( x ) ∇ v ∇ v ± v E ( x ) ∇ v Ω Ω | v | 2 ∗ � 1 | E ( x ) | N � 1 |∇ v | 2 � 1 � � � 2 ∗ � � N � � |∇ v | 2 − 2 ≥ α Ω Ω Ω Ω | E ( x ) | N � 1 α − 1 � � N � � � |∇ v | 2 ≥ S Ω Ω

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 )) � � M ( x ) ∇ v ∇ v ± v E ( x ) ∇ v Ω Ω | v | 2 ∗ � 1 | E ( x ) | N � 1 |∇ v | 2 � 1 � � � 2 ∗ � � N � � |∇ v | 2 − 2 ≥ α Ω Ω Ω Ω | E ( x ) | N � 1 α − 1 � � N � � � |∇ v | 2 ≥ S Ω Ω E ∈ L N

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 )) � � M ( x ) ∇ v ∇ v ± v E ( x ) ∇ v Ω Ω | v | 2 ∗ � 1 | E ( x ) | N � 1 |∇ v | 2 � 1 � � � 2 ∗ � � N � � |∇ v | 2 − 2 ≥ α Ω Ω Ω Ω | E ( x ) | N � 1 α − 1 � � N � � � |∇ v | 2 ≥ S Ω Ω E ∈ L N � E � L N not too large

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

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 � E � N small

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

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

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 , 2 u = 0 on ∂ Ω , � − div ( M ( x ) ∇ ψ ) = E ( x ) ∇ ψ + g ( x ) in Ω , 3 ψ = 0 on ∂ Ω , 2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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 , 2 u = 0 on ∂ Ω , � − div ( M ( x ) ∇ ψ ) = E ( x ) ∇ ψ + g ( x ) in Ω , 3 ψ = 0 on ∂ Ω , Ω bounded subset of R N , ellipticity: 0 < α, α | ξ | 2 ≤ M ( x ) ξξ, | M ( x ) | ≤ β , f ∈ L m (Ω) , 1 ≤ m ≤ ∞ , E ∈ ( L N (Ω)) N 2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019

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 , 2 u = 0 on ∂ Ω , � − div ( M ( x ) ∇ ψ ) = E ( x ) ∇ ψ + g ( x ) in Ω , 3 ψ = 0 on ∂ Ω , Ω bounded subset of R N , ellipticity: 0 < α, α | ξ | 2 ≤ M ( x ) ξξ, | M ( x ) | ≤ β , f ∈ L m (Ω) , 1 ≤ m ≤ ∞ , E ∈ ( L N (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

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 , 2 u = 0 on ∂ Ω , � − div ( M ( x ) ∇ ψ ) = E ( x ) ∇ ψ + g ( x ) in Ω , 3 ψ = 0 on ∂ Ω , Ω bounded subset of R N , ellipticity: 0 < α, α | ξ | 2 ≤ M ( x ) ξξ, | M ( x ) | ≤ β , f ∈ L m (Ω) , 1 ≤ m ≤ ∞ , E ∈ ( L N (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

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 | ∈ L N (Ω) as E = 0

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 | ∈ L N (Ω) as E = 0

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 | ∈ L N (Ω) as E = 0

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 | ∈ L N (Ω) as E = 0 � − div ( M ( x ) ∇ u )) = − div ( u E ( x )) + f ( x ) : Ω , u = 0 : ∂ Ω 2 ⇒ u ∈ W 1 , 2 N +2 ≤ m < N 2 N 0 (Ω) ∩ L m ∗∗ (Ω) ; 1

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 | ∈ L N (Ω) as E = 0 � − div ( M ( x ) ∇ u )) = − div ( u E ( x )) + f ( x ) : Ω , u = 0 : ∂ Ω 2 ⇒ u ∈ W 1 , 2 N +2 ≤ m < N 2 N 0 (Ω) ∩ L m ∗∗ (Ω) ; 1 N +2 ⇒ u ∈ W 1 , m ∗ 2 1 < m < 2 N (Ω) ; 0 3 m = 1

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 | ∈ L N (Ω) as E = 0 � − div ( M ( x ) ∇ u )) = − div ( u E ( x )) + f ( x ) : Ω , u = 0 : ∂ Ω 2 ⇒ u ∈ W 1 , 2 N +2 ≤ m < N 2 N 0 (Ω) ∩ L m ∗∗ (Ω) ; 1 N +2 ⇒ u ∈ W 1 , m ∗ 2 1 < m < 2 N (Ω) ; 0 3 m = 1 ⇒ u ∈ W 1 , q N 0 (Ω) , q < N − 1 . � � � u : M ∇ u ∇ φ = u E ∇ φ + f φ, ∀ φ ∈ D . Ω Ω Ω

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 | ∈ L N (Ω) as E = 0 � − div ( M ( x ) ∇ u )) = − div ( u E ( x )) + f ( x ) : Ω , u = 0 : ∂ Ω 2 ⇒ u ∈ W 1 , 2 N +2 ≤ m < N 2 N 0 (Ω) ∩ L m ∗∗ (Ω) ; 1 N +2 ⇒ u ∈ W 1 , m ∗ 2 1 < m < 2 N (Ω) ; 0 3 m = 1 ⇒ u ∈ W 1 , q N 0 (Ω) , q < N − 1 . � � � u : M ∇ u ∇ φ = u E ∇ φ + f φ, ∀ φ ∈ D . Ω Ω Ω Theorem (70-Brezis) 2 , it is false that u ∈ W 1 , m ∗ E = 0 , m > N (Ω) 0

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 | ∈ L N (Ω) as E = 0 � − div ( M ( x ) ∇ u )) = − div ( u E ( x )) + f ( x ) : Ω , u = 0 : ∂ Ω 2 ⇒ u ∈ W 1 , 2 N +2 ≤ m < N 2 N 0 (Ω) ∩ L m ∗∗ (Ω) ; 1 N +2 ⇒ u ∈ W 1 , m ∗ 2 1 < m < 2 N (Ω) ; 0 3 m = 1 ⇒ u ∈ W 1 , q N 0 (Ω) , q < N − 1 . � � � u : M ∇ u ∇ φ = u E ∇ φ + f φ, ∀ φ ∈ D . Ω Ω Ω Theorem (70-Brezis) 2 , it is false that u ∈ W 1 , m ∗ E = 0 , m > N (Ω) 0 Remark N +2 + δ Meyers < m < N 2 N E = 0 , 2 , u ∈ ?

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

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 , 4 ψ = 0 on ∂ Ω , 4 ESAIM-COCV 2019

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”

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”

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 u n � � − div ( M ( x ) ∇ u n ) = − div n | u n | E ( x ) + f ( x ) 1 + 1

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.

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭✭✭✭ E ∈ ( L N (Ω)) N ✭

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If E �∈ ( L N (Ω)) N , even for nothing, as in | E | ≤ | A | | x | , A ∈ R , 0 ∈ Ω , 5 JDE 2015; +Orsina, Nonlin.Anal. 2019

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If E �∈ ( L N (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 5 JDE 2015; +Orsina, Nonlin.Anal. 2019

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If E �∈ ( L N (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 − 2 m ) 2 N N +2 ≤ m < N , and 2 , then m u ∈ W 1 , 2 0 (Ω) ∩ L m ∗∗ (Ω) ; 2) if | A | < α ( N − 2 m ) 2 N , and 1 < m < N +2 , then m u ∈ W 1 , m ∗ (Ω) ; 0 N N − 1 (Ω)) N 3) if | A | < α ( N − 2) , and m = 1 , then ∇ u ∈ ( M and u ∈ W 1 , q N 0 (Ω) , for every q < N − 1 ; 4) if α ( N − 2) ≤ | A | < α ( N − 1) , then u ∈ W 1 , q 0 (Ω) , for N α every q < | A | + α 5 JDE 2015; +Orsina, Nonlin.Anal. 2019

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ Radial ex.

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ E ∈ ( L 2 (Ω)) N 6 JDE 2015

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ E ∈ ( L 2 (Ω)) N � definition of solution ; 6 existence of solution . 6 JDE 2015

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If we add the zero order term “ + u ”, the framework changes completely.

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If we add the zero order term “ + u ”, the framework changes completely. A , u n ∈ W 1 , 2 0 (Ω) : u n � � − div ( M ( x ) ∇ u n ) + A u n = − div n | u n | E ( x ) + f ( x ) 1 + 1

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If we add the zero order term “ + u ”, the framework changes completely. A , u n ∈ W 1 , 2 0 (Ω) : u n � � − div ( M ( x ) ∇ u n ) + A u n = − div n | u n | E ( x ) + f ( x ) 1 + 1 Simpler proofs PhD

Marie-Fran¸ coise Daza, Laurent Ariza: El amor en los tiempos del coronavirus First Maximum Principle, dedicated to Marie-Fran¸ coise ✭ ✭✭✭✭✭✭✭ ✭✭✭✭ E ∈ ( L N (Ω)) N ✭ If we add the zero order term “ + u ”, the framework changes completely. A , u n ∈ W 1 , 2 0 (Ω) : u n � � − div ( M ( x ) ∇ u n ) + A u n = − div n | u n | E ( x ) + f ( x ) 1 + 1 Simpler proofs PhD course, UCM, November 2019