A survey on nonlinear Dirichlet pb: classical results and - PowerPoint PPT Presentation

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 , on ∂ Ω ; Recall f ( x ) ∈ L m (Ω) , m ≥ 1 ⇒  2 N N +2 < m < N u ∈ L m ∗∗ (Ω); 2 , 2 N m ≥  N +2 m = N / 2 , u exp. summ. u ∈ W 1 , 2 0 (Ω) weak sol .  m > N / 2 , u ∈ L ∞ (Ω) . � 2 N 1 ≤ m < u ∈ W 1 , m ∗ 2 N 1 < m < N +2 , (Ω) distr . sol . ; N +2 0 u ∈ W 1 , q N m = 1 , 0 (Ω) , q < N − 1 . u �∈ W 1 , 2 0 (Ω) 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 , on ∂ Ω ; Recall f ( x ) ∈ L m (Ω) , m ≥ 1 ⇒  2 N u ∈ L m ∗∗ (Ω); N +2 < m < N 2 , 2 N m ≥  N +2 m = N / 2 , u exp. summ. u ∈ W 1 , 2 0 (Ω) weak sol .  m > N / 2 , u ∈ L ∞ (Ω) . � 2 N 1 ≤ m < u ∈ W 1 , m ∗ 2 N 1 < m < N +2 , (Ω) distr . sol . ; N +2 0 u ∈ W 1 , q N m = 1 , 0 (Ω) , q < N − 1 . 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 , on ∂ Ω ; Recall f ( x ) ∈ L m (Ω) , m ≥ 1 ⇒  2 N u ∈ L m ∗∗ (Ω); N +2 < m < N 2 , 2 N m ≥  N +2 m = N / 2 , u exp. summ. u ∈ W 1 , 2 0 (Ω) weak sol .  m > N / 2 , u ∈ L ∞ (Ω) . � 2 N 1 ≤ m < u ∈ W 1 , m ∗ 2 N 1 < m < N +2 , (Ω) distr . sol . ; N +2 0 u ∈ W 1 , q N m = 1 , 0 (Ω) , q < N − 1 . u �∈ W 1 , 2 0 (Ω) 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 , on ∂ Ω ; Recall f ( x ) ∈ L m (Ω) , m ≥ 1 ⇒  2 N u ∈ L m ∗∗ (Ω); N +2 < m < N 2 , 2 N m ≥  N +2 m = N / 2 , u exp. summ. u ∈ W 1 , 2 0 (Ω) weak sol .  m > N / 2 , u ∈ L ∞ (Ω) . � 2 N 1 ≤ m < u ∈ W 1 , m ∗ 2 N 1 < m < N +2 , (Ω) distr . sol . ; N +2 0 u ∈ W 1 , q N m = 1 , 0 (Ω) , q < N − 1 . u �∈ W 1 , 2 0 (Ω) 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 , on ∂ Ω ; Recall f ( x ) ∈ L m (Ω) , m ≥ 1 ⇒  2 N u ∈ L m ∗∗ (Ω); N +2 < m < N 2 , 2 N m ≥  N +2 m = N / 2 , u exp. summ. u ∈ W 1 , 2 0 (Ω) weak sol .  m > N / 2 , u ∈ L ∞ (Ω) . � 2 N 1 ≤ m < u ∈ W 1 , m ∗ 2 N 1 < m < N +2 , (Ω) distr . sol . ; N +2 0 u ∈ W 1 , q N m = 1 , 0 (Ω) , q < N − 1 . u �∈ W 1 , 2 0 (Ω) Note the above th. say that, non smooth M(x),  in the blue case, more summability on f ⇒     only 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 , on ∂ Ω ; Recall f ( x ) ∈ L m (Ω) , m ≥ 1 ⇒  2 N u ∈ L m ∗∗ (Ω); N +2 < m < N 2 , 2 N m ≥  N +2 m = N / 2 , u exp. summ. u ∈ W 1 , 2 0 (Ω) weak sol .  m > N / 2 , u ∈ L ∞ (Ω) . � 2 N 1 ≤ m < u ∈ W 1 , m ∗ 2 N 1 < m < N +2 , (Ω) distr . sol . ; N +2 0 u ∈ W 1 , q N m = 1 , 0 (Ω) , q < N − 1 . u �∈ W 1 , 2 0 (Ω) Note the above th. say that, non smooth M(x),  in the blue case, more summability on f ⇒     only 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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, 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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, measurable matrix � − div � � M ( x ) ∇ u = f , in Ω ; u = 0 , on ∂ 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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, measurable matrix � − div � � M ( x ) ∇ u = f , in Ω ; u = 0 , on ∂ Ω ; ∃ Ω , M , f ( x ) ∈ L m (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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, measurable matrix � − div � � M ( x ) ∇ u = f , in Ω ; u = 0 , on ∂ Ω ; ∃ Ω , M , f ( x ) ∈ L m (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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, measurable matrix � − div � � M ( x ) ∇ u = f , in Ω ; u = 0 , on ∂ Ω ; ∃ Ω , M , f ( x ) ∈ L m (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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, measurable matrix � − div � � M ( x ) ∇ u = f , in Ω ; u = 0 , on ∂ Ω ; ∃ Ω , M , f ( x ) ∈ L m (Ω) , m > N 2 u ∈ W 1 , 2 u �∈ W 1 , m ∗ (Ω) 4 0 (Ω) ∩ L ∞ (Ω) , but 0 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 ∈ L m (Ω) , 1 < m < ∞ , ? ⇒ ? u ∈ W 1 , m ∗ (Ω) 0 M ( x ) elliptic, 3 bounded, measurable matrix � − div � � M ( x ) ∇ u = f , in Ω ; u = 0 , on ∂ Ω ; ∃ Ω , M , f ( x ) ∈ L m (Ω) , m > N 2 u ∈ W 1 , 2 u �∈ W 1 , m ∗ (Ω) 4 0 (Ω) ∩ L ∞ (Ω) , but 0 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 0 (Ω) weak solution +Stampacchia-type-summability: ( p − 1) N + p ≤ m < N pN ⇒ u ∈ L [( p − 1) m ∗ ] ∗ (Ω) [B-Gia.] p ( p < N ) m > N ⇒ u ∈ L ∞ (Ω) [Stampacchia] 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 Nonlinear CZ 0 < α ≤ a ( x ) ≤ β , a(x) bounded and measurable � − div � � a ( x ) |∇ u | p − 2 ∇ u = f ( x ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 0 (Ω) +Stampacchia-type-summability: u ∈ L [( p − 1) m ∗ ] ∗ (Ω) [B-Gia.] 5 parabolic 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 0 (Ω) +Stampacchia-type-summability: u ∈ L [( p − 1) m ∗ ] ∗ (Ω) [B-Gia.] Nonlinear CZ 5 parabolic 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 0 (Ω) +Stampacchia-type-summability: u ∈ L [( p − 1) m ∗ ] ∗ (Ω) [B-Gia.] Nonlinear CZ � ( p − 1) N + p ⇒ u ∈ W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) , [ BG , 20 century ] 0 N ⇒ u ∈ W 1 , 1 ( p − 1) N +1 , 1 < p < 2 − 1 N 0 ( Ω ) , [ BG , 2012] 5 m = 5 parabolic 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 0 (Ω) +Stampacchia-type-summability: u ∈ L [( p − 1) m ∗ ] ∗ (Ω) [B-Gia.] Nonlinear CZ � ( p − 1) N + p ⇒ u ∈ W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) , [ BG , 20 century ] 0 N ⇒ u ∈ W 1 , 1 ( p − 1) N +1 , 1 < p < 2 − 1 N 0 ( Ω ) , [ BG , 2012] 5 m = Note that ( p − 1) m ⋆ > 1 ⇐ N ⇒ ( p − 1) N +1 < m 5 parabolic 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; ( p − 1) N +1 , 1 < p < 2 − 1 N 1 ≤ m < 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; ( p − 1) N +1 , 1 < p < 2 − 1 N 1 ≤ m < 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; ( p − 1) N +1 , 1 < p < 2 − 1 N 1 ≤ m < N : meaning of solution � � a ( x ) |∇ u | p − 2 ∇ u ∇ T k [ u − ϕ ] ≤ [ BBG ... ] f ( x ) T k [ u − ϕ ] Ω Ω ∀ 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 ) ∈ L m ( Ω ) , in Ω ; u = 0 , on ∂ Ω ; [Leray-Lions]: m ≥ ( p ∗ ) ′ = ( p − 1) N + p ⇒ u ∈ W 1 , p pN 0 (Ω) +Stampacchia-type-summability: u ∈ L [( p − 1) m ∗ ] ∗ (Ω) [B-Gia.] Nonlinear CZ � ( p − 1) N + p ⇒ u ∈ W 1 , ( p − 1 ) m ⋆ pN N ( p − 1) N +1 < m < ( Ω ) , [ BG , 20 century ] 0 N ⇒ u ∈ W 1 , 1 ( p − 1) N +1 , 1 < p < 2 − 1 N 0 ( Ω ) , [ BG , 2012] 6 m = ( p − 1) N +1 , 1 < p < 2 − 1 N 1 ≤ m < N : meaning of solution � � a ( x ) |∇ u | p − 2 ∇ u ∇ T k [ u − ϕ ] ≤ [ BBG ... ] f ( x ) T k [ u − ϕ ] Ω Ω ∀ 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 ) ∈ L 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � N pN ( p − 1) N +1 < m < ( 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0

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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0 ( p − 1) N +1 , 1 < p < 2 − 1 N m = 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0 ( p − 1) N +1 , 1 < p < 2 − 1 N m = N ⇒ { u n } 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0 N ⇒ { u n } bdd in W 1 , 1 ( p − 1) N +1 , 1 < p < 2 − 1 N m = 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0 � N ⇒ { u n } bdd in W 1 , 1 W 1 , 1 ? ( p − 1) N +1 , 1 < p < 2 − 1 N m = 0 ( Ω ) 0

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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0 � N ⇒ { u n } bdd in W 1 , 1 W 1 , 1 ? ( p − 1) N +1 , 1 < p < 2 − 1 N m = 0 ( Ω ) 0 ∇ u n ( 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 ) ∈ L m ( Ω )  f ( x ) � � a ( x ) |∇ u n | p − 2 ∇ u n  − div = n | f | , in Ω ; 1+ 1 ∃ u n , and u n ∈ L ∞ (Ω)  u n = 0 , on ∂ Ω ; Nonlinear CZ � ( p − 1) N + p ⇒ { u n } bounded in W 1 , ( p − 1 ) m ⋆ N pN ( p − 1) N +1 < m < ( Ω ) 0 � N ⇒ { u n } bdd in W 1 , 1 W 1 , 1 ? ( p − 1) N +1 , 1 < p < 2 − 1 N m = 0 ( Ω ) 0 ∇ u n ( 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 , on ∂ 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 , on ∂ Ω ; � − div � � M ( x ) ∇ u + D ( x ) · ∇ u = f , in Ω ; u = 0 , on ∂ 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 ∈ W 1 , 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; Main points (formally) � u | u | r − 2 � L m (Ω) ≤ � f � L 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; Main points (formally) � u | u | r − 2 � L m (Ω) ≤ � f � L m (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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; Main points (formally) � u | u | r − 2 � L m (Ω) ≤ � f � L m (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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; Main points (formally) � u | u | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ u ∈ L ( r − 1) m (Ω) ( r − 1) m + 1 1 We can use u as test function if 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 < 2 N N +2 , ( r > 2 ∗ ) � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; Main points (formally) � u | u | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ u ∈ L ( r − 1) m (Ω) ( r − 1) m + 1 1 We can use u as test function if m ≤ 1 ( r − 1) m + 1 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ Then � � � | u | r = M ( x ) ∇ u ∇ u + f ( x ) u ( x ) < ∞ Ω Ω 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 1 + 1 n | f | � u n | u n | r − 2 � L m (Ω) ≤ � f � L 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 1 + 1 n | f | � u n | u n | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ { u n } 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 1 + 1 n | f | � u n | u n | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ { u n } bdd in L ( r − 1) m We can use u n as test function if ( r − 1) m + 1 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 1 + 1 n | f | � u n | u n | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ { u n } bdd in L ( r − 1) m We can use u n as test function if ( r − 1) m + 1 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ � � | u n | r ≤ � f � L m (Ω) � u n � L m ′ (Ω) ⇒ M ( x ) ∇ u n ∇ u n + Ω 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 1 + 1 n | f | � u n | u n | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ { u n } bdd in L ( r − 1) m We can use u n as test function if ( r − 1) m + 1 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ � � | u n | r ≤ � f � L m (Ω) � u n � L m ′ (Ω) ⇒ M ( x ) ∇ u n ∇ u n + Ω Ω { u n } 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 < 2 N N +2 , ( r > 2 ∗ ) f � � + u n | u n | r − 2 = u n ∈ W 1 , 2 0 (Ω) ∩ L ∞ (Ω) : − div M ( x ) ∇ u n 1 + 1 n | f | � u n | u n | r − 2 � L m (Ω) ≤ � f � L m (Ω) ⇒ { u n } bdd in L ( r − 1) m We can use u n as test function if ( r − 1) m + 1 1 m ≤ 1 ⇐ ⇒ m ≥ r ′ � � | u n | r ≤ � f � L m (Ω) � u n � L m ′ (Ω) ⇒ M ( x ) ∇ u n ∇ u n + Ω Ω { u n } 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 < 2 N 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 < 2 N N +2 Regularizing effect: the solution u of � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ 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 < 2 N N +2 Regularizing effect: the solution u of � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; is more regular than w solution of � − div � � = f ∈ L m (Ω) , M ( x ) ∇ w in Ω ; w = 0 , on ∂ 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 < 2 N N +2 Regularizing effect: the solution u of � − div � � + u | u | r − 2 = f ∈ L m (Ω) , M ( x ) ∇ u in Ω ; u = 0 , on ∂ Ω ; is more regular than w solution of � − div � � = f ∈ L m (Ω) , M ( x ) ∇ w in Ω ; w = 0 , on ∂ Ω ; since u ∈ W 1 , 2 0 (Ω) , but w �∈ W 1 , 2 0 (Ω) : N +2 ⇒ m ∗ < 2 . w only belongs to W 1 , m ∗ 2 N (Ω) and m < 0