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A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Lucio Boccardo Dipartimento di Matematica - Universit` a di Roma 1

boccardo@mat.uniroma1.it

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Marrakech, 17.4.2018

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

SABAHU AL-KHAIR

Bonjour Buon giorno

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Shukran

Merci Grazie

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

PREMIER CONGRES FRANCO-MAROCAIN DE MATHEMATIQUES APPLIQUEES

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Ω open ⊂ RN, N > 2,

1today no p=1, no r.h.s. measures

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Ω open ⊂ RN, N > 2, bounded;

1today no p=1, no r.h.s. measures

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Ω open ⊂ RN, N > 2, bounded; r.h.s. f (x) ∈ Lm(Ω), m > 1 1;

1today no p=1, no r.h.s. measures

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Ω open ⊂ RN, N > 2, bounded; r.h.s. f (x) ∈ Lm(Ω), m > 1 1; M(x) elliptic, bounded, measurable matrix: α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, a.e. x ∈ Ω, ∀ ξ ∈ RN, β, α > 0.

1today no p=1, no r.h.s. measures

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions

Ω open ⊂ RN, N > 2, bounded; r.h.s. f (x) ∈ Lm(Ω), m > 1 1; M(x) elliptic, bounded, measurable matrix: α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, a.e. x ∈ Ω, ∀ ξ ∈ RN, β, α > 0. Recall that: m∗ = mN N − m, if 1 < m < N; m∗∗ = mN N − 2m, if 1 < m < N 2 .

1today no p=1, no r.h.s. measures

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems

Linear Dirichlet problems

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Classical results (Stampacchia) and nonclassical res. (B-Gallouet)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Classical results (Stampacchia) and nonclassical res. (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Classical results (Stampacchia) and nonclassical res. (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

we study linear and nonlinear problems, but

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Classical results (Stampacchia) and nonclassical res. (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

we study linear and nonlinear problems, but               

• nly in RN,

no p(x) − operators, no fully nonlinear, no fractional,

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Classical results (Stampacchia) and nonclassical res. (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

we study linear and nonlinear problems, but               

• nly in RN,

no p(x) − operators, no fully nonlinear, no fractional, no smoothness assumptions (even weak) on M(x).

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Classical results (Stampacchia) and nonclassical res. (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

we study linear and nonlinear problems, but               

• nly in RN,

no p(x) − operators, no fully nonlinear, no fractional, no smoothness assumptions (even weak) on M(x). That is: classical and easy framework.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω), weak sol.

2and more [De Giorgi]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω), weak sol.

and   

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω)2.

2and more [De Giorgi]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω).

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

The above red existence results can be seen as an improvement of the

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

The above red existence results can be seen as an improvement of the Calderon-Zygmund theory for linear elliptic operators.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

Note the above th. say that,

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

Note the above th. say that, non smooth M(x),

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

Note the above th. say that, non smooth M(x),          in the blue case, more summability on f ⇒

• nly more summability on u;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

−div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

Recall f (x) ∈ Lm(Ω), m ≥ 1 ⇒ m ≥

2N N+2

u ∈ W 1,2

0 (Ω) weak sol.

  

2N N+2 < m < N 2 ,

u ∈ Lm∗∗(Ω); m = N/2, u exp. summ. m > N/2, u ∈ L∞(Ω). 1 ≤ m <

2N N+2

u ∈ W 1,2

0 (Ω)

• 1 < m <

2N N+2,

u ∈ W 1,m∗ (Ω) distr. sol.; m = 1, u ∈ W 1,q

0 (Ω), q < N N−1.

Note the above th. say that, non smooth M(x),          in the blue case, more summability on f ⇒

• nly more summability on u;

in the red case, more summability on f ⇒ more summability on u and also more summability on ∇u .

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Right hand side div(F), instead of f(x)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Nonuniqueness of the distributional solutions

Good definition of solution in order to have uniqueness.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Linear Dirichlet problems Classical results (Stampacchia) and nonclassical results (B-Gallouet)

Variational interpretation

T-minima

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix −div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

∃

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix −div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

∃ Ω, M, f (x) ∈ Lm(Ω), m > N

2

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix −div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

∃ Ω, M, f (x) ∈ Lm(Ω), m > N

2

u ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix −div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

∃ Ω, M, f (x) ∈ Lm(Ω), m > N

2

u ∈ W 1,2

0 (Ω) ∩ L∞(Ω), but

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix −div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

∃ Ω, M, f (x) ∈ Lm(Ω), m > N

2

u ∈ W 1,2

0 (Ω) ∩ L∞(Ω), but

u ∈ W 1,m∗ (Ω) 4

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

f ∈ Lm(Ω), 1 < m < ∞, ? ⇒ ? u ∈ W 1,m∗ (Ω)

M(x) elliptic, 3bounded, measurable matrix −div

• M(x) ∇u
• = f ,

in Ω; u = 0,

• n ∂Ω;

∃ Ω, M, f (x) ∈ Lm(Ω), m > N

2

u ∈ W 1,2

0 (Ω) ∩ L∞(Ω), but

u ∈ W 1,m∗ (Ω) 4 2 + δ < m ≤ N

2 work in progress ...

3“Calderon-Zygmund”=M smooth enough: recall a papers by

Brezis and Mingione

4[LB2014,70Haim]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions No Calderon-Zygmund

Summary

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Classic framework

Principal part nonlinear w.r.t. the gradient

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Classic framework

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Classic framework

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]:

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Classic framework

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

weak solution

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Classic framework

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

weak solution +Stampacchia-type-summability:

pN (p−1)N+p ≤ m < N p (p < N)

⇒ u ∈ L[(p−1)m∗]∗(Ω) [B-Gia.] m > N

p

⇒ u ∈ L∞(Ω) [Stampacchia]

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

+Stampacchia-type-summability: u ∈ L[(p−1)m∗]∗(Ω) [B-Gia.]

5parabolic pb. / general Leray-Lions operators

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

+Stampacchia-type-summability: u ∈ L[(p−1)m∗]∗(Ω) [B-Gia.] Nonlinear CZ

5parabolic pb. / general Leray-Lions operators

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

+Stampacchia-type-summability: u ∈ L[(p−1)m∗]∗(Ω) [B-Gia.] Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ u ∈ W1,(p−1)m⋆

(Ω), [BG, 20century] m =

N (p−1)N+1,1 < p < 2 − 1 N ⇒ u ∈ W1,1 0 (Ω), [BG, 2012]5

5parabolic pb. / general Leray-Lions operators

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

+Stampacchia-type-summability: u ∈ L[(p−1)m∗]∗(Ω) [B-Gia.] Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ u ∈ W1,(p−1)m⋆

(Ω), [BG, 20century] m =

N (p−1)N+1,1 < p < 2 − 1 N ⇒ u ∈ W1,1 0 (Ω), [BG, 2012]5

Note that (p − 1)m⋆ > 1 ⇐ ⇒

N (p−1)N+1 < m

5parabolic pb. / general Leray-Lions operators

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

1 ≤ m <

N (p−1)N+1,1 < p < 2 − 1 N :

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

1 ≤ m <

N (p−1)N+1,1 < p < 2 − 1 N : meaning of solution

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

1 ≤ m <

N (p−1)N+1,1 < p < 2 − 1 N : meaning of solution

[BBG...]

• Ω

a(x)|∇u|p−2∇u ∇Tk[u − ϕ] ≤

• Ω

f (x) Tk[u − ϕ] ∀ ϕ ∈ W 1,p

0 (Ω) ∩ L∞(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ

0 < α ≤ a(x) ≤ β, a(x) bounded and measurable −div

• a(x)|∇u|p−2∇u
• = f (x) ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

[Leray-Lions]: m ≥ (p∗)′ =

pN (p−1)N+p ⇒ u ∈ W 1,p 0 (Ω)

+Stampacchia-type-summability: u ∈ L[(p−1)m∗]∗(Ω) [B-Gia.] Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ u ∈ W1,(p−1)m⋆

(Ω), [BG, 20century] m =

N (p−1)N+1,1 < p < 2 − 1 N ⇒ u ∈ W1,1 0 (Ω), [BG, 2012]6

1 ≤ m <

N (p−1)N+1,1 < p < 2 − 1 N : meaning of solution

[BBG...]

• Ω

a(x)|∇u|p−2∇u ∇Tk[u − ϕ] ≤

• Ω

f (x) Tk[u − ϕ] ∀ ϕ ∈ W 1,p

0 (Ω) ∩ L∞(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Nonlinear CZ and p=2: linear problems

last results if p=2

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω) m =

N (p−1)N+1, 1 < p < 2 − 1 N ⇒

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω) m =

N (p−1)N+1, 1 < p < 2 − 1 N ⇒ {un} bdd in

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω) m =

N (p−1)N+1, 1 < p < 2 − 1 N ⇒ {un} bdd in W1,1 0 (Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω) m =

N (p−1)N+1, 1 < p < 2 − 1 N ⇒ {un} bdd in W1,1 0 (Ω)

• W 1,1 ?

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω) m =

N (p−1)N+1, 1 < p < 2 − 1 N ⇒ {un} bdd in W1,1 0 (Ω)

• W 1,1 ?

∇un(x) converges a.e. to ∇u(x)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Sketch of the existence proof

0 < α ≤ a(x) ≤ β, a(x) measurable, f (x) ∈ Lm(Ω)    −div

• a(x)|∇un|p−2∇un
• =

f (x) 1+ 1

n |f|,

in Ω; un = 0,

• n ∂Ω;

∃ un, and un ∈ L∞(Ω) Nonlinear CZ

• N

(p−1)N+1 < m < pN (p−1)N+p ⇒ {un} bounded in W1,(p−1)m⋆

(Ω) m =

N (p−1)N+1, 1 < p < 2 − 1 N ⇒ {un} bdd in W1,1 0 (Ω)

• W 1,1 ?

∇un(x) converges a.e. to ∇u(x) third step (pass to the limit n → ∞) less difficult

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Principal part nonlinear w.r.t. the gradient Tools in the existence proofs

Summary

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Right hand side f (x) belonging to Marcinkiewicz spaces

C-Z-Stampacchia theory for f (x) belonging to Marcinkiewicz spaces

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Unilateral problems

Unilateral problems and Lewy-Stampacchia inequality

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Recent results and work in progress

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Recent results and work in progress Impact of terms of order 1

(Terms of order 1 do not help coercivity)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Recent results and work in progress Impact of terms of order 1

(Terms of order 1 do not help coercivity)

−div

• M(x) ∇u
• − div(u E(x)) = f ,

in Ω; u = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Recent results and work in progress Impact of terms of order 1

(Terms of order 1 do not help coercivity)

−div

• M(x) ∇u
• − div(u E(x)) = f ,

in Ω; u = 0,

• n ∂Ω;

−div

• M(x) ∇u
• + D(x) · ∇u = f ,

in Ω; u = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m< 2N

N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m< 2N

N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

It is possible to prove the existence of weak solutions, even beyond the natural duality pairing; that is: there exists a weak solution

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m< 2N

N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

It is possible to prove the existence of weak solutions, even beyond the natural duality pairing; that is: there exists a weak solution u ∈ W1,2

0 (Ω) ∩ L(r−1)m(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points (formally)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points (formally) u|u|r−2Lm(Ω) ≤ f Lm(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points (formally) u|u|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ u ∈ L(r−1)m(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points (formally) u|u|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ u ∈ L(r−1)m(Ω) We can use u as test function if

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points (formally) u|u|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ u ∈ L(r−1)m(Ω) We can use u as test function if 1 (r − 1)m + 1 m ≤ 1

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

−div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

Main points (formally) u|u|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ u ∈ L(r−1)m(Ω) We can use u as test function if 1 (r − 1)m + 1 m ≤ 1 1 (r − 1)m + 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ Then

• Ω

M(x) ∇u∇u +

• Ω

|u|r =

• Ω

f (x) u(x) < ∞

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

un|un|r−2Lm(Ω) ≤ f Lm(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

un|un|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ {un} bdd in L(r−1)m

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

un|un|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ {un} bdd in L(r−1)m We can use un as test function if 1 (r − 1)m + 1 m ≤ 1 ⇐ ⇒ m ≥ r ′

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

un|un|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ {un} bdd in L(r−1)m We can use un as test function if 1 (r − 1)m + 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ ⇒

• Ω

M(x) ∇un∇un +

• Ω

|un|r ≤ f Lm(Ω)unLm′(Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

un|un|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ {un} bdd in L(r−1)m We can use un as test function if 1 (r − 1)m + 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ ⇒

• Ω

M(x) ∇un∇un +

• Ω

|un|r ≤ f Lm(Ω)unLm′(Ω) {un} bdd in W 1,2

0 (Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

[R. Cirmi]

r ′ ≤ m <

2N N+2, (r > 2∗)

un ∈ W 1,2

0 (Ω)∩L∞(Ω) : −div

• M(x) ∇un
• +un|un|r−2 =

f 1 + 1

n|f |

un|un|r−2Lm(Ω) ≤ f Lm(Ω) ⇒ {un} bdd in L(r−1)m We can use un as test function if 1 (r − 1)m + 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ ⇒

• Ω

M(x) ∇un∇un +

• Ω

|un|r ≤ f Lm(Ω)unLm′(Ω) {un} bdd in W 1,2

0 (Ω)

Pass to the limit

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

Regularizing effect

r ′ ≤ m< 2N

N+2

Regularizing effect:

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

Regularizing effect

r ′ ≤ m< 2N

N+2

Regularizing effect: the solution u of −div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

is more regular than w

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

Regularizing effect

r ′ ≤ m< 2N

N+2

Regularizing effect: the solution u of −div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

is more regular than w solution of −div

• M(x)∇w
• = f ∈ Lm(Ω),

in Ω; w = 0,

• n ∂Ω;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

Regularizing effect

r ′ ≤ m< 2N

N+2

Regularizing effect: the solution u of −div

• M(x) ∇u
• + u|u|r−2 = f ∈ Lm(Ω),

in Ω; u = 0,

• n ∂Ω;

is more regular than w solution of −div

• M(x)∇w
• = f ∈ Lm(Ω),

in Ω; w = 0,

• n ∂Ω;

since u ∈ W 1,2

0 (Ω), but w ∈ W 1,2 0 (Ω):

w only belongs to W 1,m∗ (Ω) and m <

2N N+2 ⇒ m∗ < 2.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

0 ≤ f ∈ L1(Ω)

In the previous semilinear problem, we see that the summability of the solution u increases (recall that Ω is bounded), if the growth of the lower order term increases, even up “infinity growth”:

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

0 ≤ f ∈ L1(Ω)

In the previous semilinear problem, we see that the summability of the solution u increases (recall that Ω is bounded), if the growth of the lower order term increases, even up “infinity growth”: In [B-Orsina] is proved the existence of 0 ≤ u ∈ W 1,2

0 (Ω), solution in W 1,2 0 (Ω) of

−div

• M(x) ∇u
• = f (x)

u(x),

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

0 ≤ f ∈ L1(Ω)

In the previous semilinear problem, we see that the summability of the solution u increases (recall that Ω is bounded), if the growth of the lower order term increases, even up “infinity growth”: In [B-Orsina] is proved the existence of 0 ≤ u ∈ W 1,2

0 (Ω), solution in W 1,2 0 (Ω) of

−div

• M(x) ∇u
• = f (x)

u(x), M ∈ R+. In [B-to-Benilan] is proved the existence

• f a weak solution 0 ≤ u ≤ M, with

µ({u(x) = M}) = 0, of u ∈ W 1,2

0 (Ω) ∩ L∞(Ω) : −div

• M(x) ∇u
• +

u u − M = f (x). No right hand side Dirac mass.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x).

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q ⇒ u ∈ W 1,2

0 (Ω)

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q ⇒ u ∈ W 1,2

0 (Ω)

Note: radial ex.

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q ⇒ u ∈ W 1,2

0 (Ω)

Note: radial ex. u ∈ L∞(Ω) proved,

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q ⇒ u ∈ W 1,2

0 (Ω)

Note: radial ex. u ∈ L∞(Ω) proved, u ∈ C 0,α false;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q ⇒ u ∈ W 1,2

0 (Ω)

Note: radial ex. u ∈ L∞(Ω) proved, u ∈ C 0,α false; data L1 ok,

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

  

• Ω

M(x)∇u∇ϕ +

• Ω

a(x) u ϕ =

• Ω

f (x)ϕ, ∀ ϕ ∈ W 1,2

0 (Ω) ∩ L∞(Ω),

   0 ≤ a(x) ∈ L1(Ω), ∃ Q > 0 such that |f (x)|

∈L1(Ω)

≤ Q a(x). ⇒ |u| ≤ Q ⇒ u ∈ W 1,2

0 (Ω)

Note: radial ex. u ∈ L∞(Ω) proved, u ∈ C 0,α false; data L1 ok, no-measures;

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Regularizing effect of lower order terms of order zero

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Lower order terms having natural growth with respect to the gradient

A survey on nonlinear Dirichlet pb: classical results and Calderon-Zygmund theory for infinite energy solutions Regularizing effect of some lower order terms Lower order terms having natural growth with respect to the gradient

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