On the Hölder regularity of all extrema in Hilbert’s 19th Problem

Abstract

Abstract Let Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} be a C 1 {C^{1}} smooth compact domain. Furthermore, let F : Ω × ℝ n ⁢ N → ℝ {F:\Omega\times\mathbb{R}^{nN}\to\mathbb{R}} , F ⁢ ( x , p ) {F(x,p)} , be C 0 {C^{0}} , differentiable with respect to p, and with F p := D p ⁢ F {F_{p}:=D_{p}F} continuous on Ω × ℝ n ⁢ N {\Omega\times\mathbb{R}^{nN}} and F strictly convex in p. Consider an n ⁢ N × n ⁢ N {nN\times nN} matrix A = ( A α ⁢ β i ⁢ j ) ∈ C 0 ⁢ ( Ω ) {A=(A^{{ij}}_{\alpha\beta})\in C^{0}(\Omega)} satisfying (0.1) A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ξ α i ⁢ ξ β j = A β ⁢ α j ⁢ i ⁢ ( x ) ⁢ ξ α i ⁢ ξ β j ≥ λ ⁢ | ξ | 2 , λ > 0 . A^{ij}_{\alpha\beta}(x)\xi^{i}_{\alpha}\xi^{j}_{\beta}=A^{ji}_{\beta\alpha}(x)% \xi^{i}_{\alpha}\xi^{j}_{\beta}\geq\lambda\lvert\xi\rvert^{2},\quad\lambda>0. Suppose that (0.2) lim | p | → ∞ ⁡ 1 | p | ⁢ ( D p ⁢ F ⁢ ( x , p ) - A ⁢ ( x ) ⁢ p ) = 0 , \displaystyle\lim_{\lvert p\rvert\to\infty}\frac{1}{\lvert p\rvert}(D_{p}F(x,p% )-A(x)p)=0, (0.3) - C 0 + c 0 ⁢ | p | 2 ≤ F ⁢ ( x , p ) ≤ C 0 ⁢ ( 1 + | p | 2 ) , \displaystyle{-}C_{0}+c_{0}\lvert p\rvert^{2}\leq F(x,p)\leq C_{0}(1+\lvert p% \rvert^{2}), (0.4) | F p ⁢ ( x , p ) - F p ⁢ ( x , q ) | ≤ C 0 ⁢ | p - q | , \displaystyle\lvert F_{p}(x,p)-F_{p}(x,q)\rvert\leq C_{0}\lvert p-q\rvert, (0.5) 〈 F p ⁢ ( x , p ) - F p ⁢ ( x , q ) , p - q 〉 ≥ c 0 ⁢ | p - q | 2 \displaystyle\langle F_{p}(x,p)-F_{p}(x,q),p-q\rangle\geq c_{0}\lvert p-q% \rvert^{2} uniformly in x and with positive constants c 0 {c_{0}} and C 0 {C_{0}} . Consider the functional (0.6) J ⁢ ( u ) := ∫ Ω F ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x , J(u):=\int_{\Omega}F(x,Du(x))\,dx+\int_{\Omega}G(x,u)\,dx, where G ⁢ ( x , ⋅ ) ∈ C 1 ⁢ ( ℝ N ) {G(x,\cdot\,)\in C^{1}(\mathbb{R}^{N})} for each x ∈ Ω {x\in\Omega} , G ⁢ ( ⋅ , u ) {G(\,\cdot\,,u)} is measurable for each u ∈ ℝ N {u\in\mathbb{R}^{N}} , and (0.7) | G u ⁢ ( x , u ) | ≤ C 0 ⁢ ( 1 + | u | s ) \lvert G_{u}(x,u)\rvert\leq C_{0}(1+\lvert u\rvert^{s}) with s < n + 2 n - 2 {s<\frac{n+2}{n-2}} . Under these conditions, we shall show that if n > 2 {n>2} , then any weak solution u ∈ W 1 , 2 ⁢ ( Ω , ℝ N ) {u\in W^{1,2}(\Omega,\mathbb{R}^{N})} of the Euler equations of J, i.e. ∑ α ∂ ∂ ⁡ x α ⁢ F p α i ⁢ ( x , D ⁢ u ) = G u i ⁢ ( x , u ) , i = 1 , … , N , \sum_{\alpha}\frac{\partial}{\partial x^{\alpha}}F_{p^{i}_{\alpha}}(x,Du)=G_{u% ^{i}}(x,u),\quad i=1,\ldots,N, is Hölder continuous in the interior of Ω and under appropriate boundary conditions also Hölder continuous up to the boundary.