Continuity of Solutions for a problem in the Calculus of Variations - PowerPoint PPT Presentation

Continuity of Solutions for a problem in the Calculus of Variations Pierre Bousquet June 2011, Ancona 1 / 18

A basic problem in the Calculus of Variations � To minimize u �→ L ( ∇ u ( x )) dx Ω u | ∂ Ω = ϕ Standing Assumptions ◮ Ω ⊂ R n bounded open set ◮ ϕ : ∂ Ω → R continuous ◮ L : R n → R strictly convex and superlinear The regularity problem ◮ Is the solution smooth in Ω ? ◮ Is the solution continuous on Ω ? 2 / 18

A basic example � |∇ u ( x ) | 2 dx To minimize u �→ Ω u | ∂ Ω = ϕ Regularity properties ◮ u is analytic on Ω ◮ u is continuous at any regular point γ ∈ ∂ Ω 3 / 18

De Giorgi’s Theorem Theorem Assume ◮ L ∈ C 2 , ∇ 2 L > 0 ◮ the solution u is locally Lipschitz in Ω Then u is locally C 1 ,α in Ω The partial derivatives of u satisfy an elliptic equation of the form div ( A ( x ) ∇ v ) = 0 A ( x ) = ∇ L ( ∇ u ( x )) By Schauder’s Theory ◮ L smooth = ⇒ u smooth By Bernstein’s Theorem ◮ L analytic = ⇒ u analytic 4 / 18

A counterexample (Giaquinta, Marcellini) A nice Lagrangian... n − 1 + 1 L ( ξ ) = ξ 2 1 + · · · + ξ 2 2 ξ 4 n ...a singular minimum x 2 n u ( x 1 , . . . ,x n ) = c n �� n − 1 i =1 x 2 i Two open problems ◮ u locally bounded = ⇒ u continuous ? ◮ ϕ continuous = ⇒ u continuous ? 5 / 18

Lipschitz regularity on uniformly convex sets Theorem (Miranda) Assume ◮ Ω uniformly convex (= enclosing sphere condition) ◮ ϕ is C 2 Then u ∈ W 1 , ∞ (Ω) Theorem (Clarke) Assume ◮ Ω uniformly convex ◮ ϕ is semiconvex Then u ∈ W 1 , ∞ loc (Ω) ∩ C 0 (Ω) Counterexample to global Lipschitzness Ω = B (0 , 1) ⊂ R 2 L ( ξ ) = | ξ | 2 , , ϕ ( x,y ) = | y | 6 / 18

The bounded slope condition (γ,ϕ(γ)) (δ,ϕ(δ)) γ δ 7 / 18

Lipschitz regularity on convex sets Theorem (Miranda) Assume ◮ Ω convex ◮ ϕ bounded slope condition Then u ∈ W 1 , ∞ (Ω) Theorem (Clarke) Assume ◮ Ω convex ◮ ϕ lower bounded slope condition Then u ∈ W 1 , ∞ loc (Ω) 8 / 18

A Lipschitz continuity result on a non convex domain Theorem (Cellina) Assume ◮ Ω exterior sphere condition ◮ L ( ξ ) = l ( | ξ | ) ◮ ϕ constant on each connected components of ∂ Ω Then u ∈ W 1 , ∞ (Ω) 9 / 18

Continuity up to the boundary Theorem (B.) Assume that ϕ is continuous and one of the following ◮ Ω convex ◮ Ω smooth and L ( ξ ) = l ( | ξ | ) Then u is continuous Theorem (Mariconda-Treu) Assume ◮ Ω convex ◮ ϕ Lipschitz continuous ◮ L coercive of order p > 1 p − 1 Then u is H¨ older continuous (of order n + p − 1 ) 10 / 18

A maximum principle : the Rado-Haar Lemma Let x, y ∈ Ω and τ := x − y. Compare the minimum u with u τ ( x ) := u ( x + τ ) Ω Ω−τ τ x y = x−y An estimate on the modulus of continuity | u ( x ′ ) − ϕ ( y ′ ) | | u ( x ) − u ( y ) | ≤ sup x ′ ∈ Ω ,y ′ ∈ ∂ Ω | x ′ − y ′ |≤| x − y | 11 / 18

Lower and upper barriers Definition v : Ω → R is an upper barrier at γ ∈ ∂ Ω if ◮ v ∈ W 1 , 1 (Ω) ∩ C 0 (Ω) ◮ v ( γ ) = ϕ ( γ ) ◮ v ≥ u a.e. on Ω Example: concave functions Rado Haar Lemma + barriers = ⇒ continuity on Ω 12 / 18

Implicit barriers and continuity I Lemma Assume ◮ Ω convex ◮ ϕ Lipschitz continuous Then u continuous Proof: Let u be the solution of the original problem and γ ∈ ∂ Ω. 1 st step prove that u is continuous at γ when Ω is a cube 2 nd step an auxiliary variational problem � ( P 0 ) To minimize v �→ L ( ∇ v ) , v | ∂ Ω 0 = ϕ 0 Ω 0 13 / 18

Implicit barriers and continuity II ϕ γ 0 ϕ Ω Ω 0 ϕ 0 ( x ) = ϕ ( γ ) + K ϕ | x − γ | ≥ ϕ ( x ) ◮ ϕ 0 convex = ⇒ ϕ 0 lower barrier for ( P 0 ) ◮ the solution u 0 for ( P 0 ) ≥ ϕ 0 ≥ ϕ ◮ u 0 is an implicit upper barrier at γ : u 0 ≥ u on Ω 14 / 18

More general Lagrangians � To minimize u �→ ( L ( ∇ u ( x )) + G ( x,u ( x ))) dx Ω u | ∂ Ω = ϕ Standing Assumptions ξ, ξ ′ ∈ R n ◮ L uniformly convex: ∃ α > 0 s.t. ∀ θ ∈ (0 , 1) , θL ( ξ ) + (1 − θ ) L ( ξ ′ ) − L ( θξ + (1 − θ ) ξ ′ ) ≥ α | ξ − ξ ′ | 2 ◮ G measurable in x and locally Lipschitz in u 15 / 18

Lipschitz continuity results Theorem (Stampacchia, B.-Clarke) Assume that Ω is convex and u is bounded. Then ◮ ϕ satisfies the bounded slope condition u ∈ W 1 , ∞ (Ω) = ⇒ ◮ ϕ satisfies the lower bounded slope condition = ⇒ u ∈ W 1 , ∞ loc (Ω) ∩ C 0 (Ω) 16 / 18

Continuity results Theorem (B.) Assume that Ω is smooth, L ( ξ ) = l ( | ξ | ) and u is bounded. Then ◮ ϕ continuous u ∈ C 0 (Ω) = ⇒ 1 u ∈ C 0 , n +1 (Ω) ◮ ϕ Lipschitz continuous = ⇒ 17 / 18

A final counterexample (Esposito-Leonetti-Mingione, Fonseca-Mal´ y-Mingione) � |∇ u ( x ) | p + a ( x ) |∇ u ( x ) | q ) dx To minimize u �→ Ω 1 < p < n < n + 1 < q < + ∞ a ∈ C 1 , Ω = a cube , a ≥ 0 , ϕ linear The set of non-Lebesgue points of the solution has (almost) dimension N − p ! A final open problem : What about autonomous Lagrangians? 18 / 18