Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 ·...

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)


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.


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


El amor en los tiempos del coronavirus


El amor en los tiempos del coronavirus

G.G.M.


L’amour aux temps du corona


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



Maximum principle



Maximum principle

Two maximum principles for two friends


Maximum principle


Maximum principle

A remark dedicated to Marie-Francoise, by David A. and Lucio B. / following [AB,JFA 2014]


Maximum principle


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


Maximum principle




|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


Maximum principle




|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


Maximum principle




|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


Maximum principle




|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


Maximum principle




|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


Maximum principle




|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


Maximum principle




|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.


Maximum principle




|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



Maximum principle




|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




Maximum principle




|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


−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0

⇒⇒u satisfies Strong Maximum Principle,even if 0 < γ < 1.


Maximum principle




|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


−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒

u satisfies Strong Maximum Principle,even if 0 < γ < 1.


Maximum principle




|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


−div(M(x)∇u)=a(x)[Q − uγ] ≥ T1a(x)[Q − uγ] ≥ 0⇒⇒u satisfies Strong Maximum Principle,

even if 0 < γ < 1.


Maximum principle




|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




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.



Coercivity


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: ...



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.



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



Coercivity

Assumptions


u = 0 : ∂Ω.


ellipticity:

0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,

f ∈ Lm(Ω), 1 ≤ m ≤ ∞,

E ∈ (LN(Ω))N




Coercivity

Assumptions


u = 0 : ∂Ω.


ellipticity: 0 < α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,

f ∈ Lm(Ω), 1 ≤ m ≤ ∞,

E ∈ (LN(Ω))N




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



Coercivity


u weak

/distributional solution of the boundary valueproblem


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



Coercivity


u weak /distributional

solution of the boundary valueproblem


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



Coercivity




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



Coercivity




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



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



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



Existence of weak/distributional solutions. Summability properties of solutions




Stampacchia-Calderon-Zygmund for the two problems


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




Stampacchia-Calderon-Zygmund for the two problems−div(M(x)∇u) = −div(u E (x)) + f (x) in Ω,

u = 0 on ∂Ω,2


ψ = 0 on ∂Ω,3


E ∈ (LN(Ω))N







u = 0 on ∂Ω,2


ψ = 0 on ∂Ω,3


E ∈ (LN(Ω))N

and we prove for both the b.v.p. the sameStampacchia-Calderon-Zygmund results of the case

E = 0 .






u = 0 on ∂Ω,2


ψ = 0 on ∂Ω,3


E ∈ (LN(Ω))N






|E | ∈ LN(Ω)

as E = 0




|E | ∈ LN(Ω)

as E = 0


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 ∈?




|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


0 (Ω)

Remark

E = 0, 2NN+2

+ δMeyers < m < N2

, u ∈?




|E | ∈ LN(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


0 (Ω)

Remark

E = 0, 2NN+2

+ δMeyers < m < N2

, u ∈?




|E | ∈ LN(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


0 (Ω)

Remark

E = 0, 2NN+2

+ δMeyers < m < N2

, u ∈?




Same results for the drift problem


ψ = 0 on ∂Ω,4

4 ESAIM-COCV 2019




“Nonlinear”

approach to a linear problem

−div(M(x)∇un) = −div( un

1 + 1n|un|

E (x))

+ f (x)




“Nonlinear” approach to a linear problem

−div(M(x)∇un) = −div( un

1 + 1n|un|

E (x))

+ f (x)




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.



(((((E ∈ (LN (Ω))N



(((((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



(((((E ∈ (LN (Ω))N

Radial ex.



(((((((((((((E ∈ (LN (Ω))N



(((((((((((((E ∈ (LN (Ω))N

E ∈ (L2(Ω))N

definition of solution;

existence of solution.6

6 JDE 2015



(((((((((((((E ∈ (LN (Ω))N

E ∈ (L2(Ω))N definition of solution;

existence of solution.6

6 JDE 2015



(((((((((((((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



(((((((((((((E ∈ (LN (Ω))N


A , un ∈ W 1,20 (Ω) :


1 + 1n|un|

E (x))

+ f (x)

Simpler proofs

PhD

course, UCM, November 2019



(((((((((((((E ∈ (LN (Ω))N


A , un ∈ W 1,20 (Ω) :


1 + 1n|un|

E (x))

+ f (x)

Simpler proofs

PhD course, UCM, November 2019



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




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



Convection [L.B. 2020]





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.





u = 0 : ∂Ω

E , f ∈

?

Theorem


and f (x) ≥ 0 (of



a weak max. pr.

Remark






u = 0 : ∂Ω

E , f ∈ ?

Theorem


and f (x) ≥ 0 (of



a weak max. pr.

Remark




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



a weak max. pr.

Remark



Second Maximum principle, dedicated to Laurent [L.B. - Luigi Orsina, Advanced Nonlinear Sudies, 2020]



with Alberto Farina



books



in a conference in Cortona (organizers Juan Luis and Lucio)



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.



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.




Background




Existence with

regularizing effect [B-Gallouet],







Background











Background






Theorem (Weak Maximum Principle /

easy )





Background






Theorem (Weak Maximum Principle / easy

)





Background











pour Laurent



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.


Next future


Next future

to MARIE-F. and LAURENT


Next future

to MARIE-F. and LAURENT

je vous souhaite tous le bien

without


Merci


Merci

Thanks


Merci

Thanks

Ciao

Marie-Fran˘coise Daza, Laurent Ariza: El amor en los tiempos del … · 2020-06-04 ·...

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