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.