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a.e. in G assumption Banach space boundary curve boundary of G boundary value problem closed subregion coefficients compact set compact support consider continuous function converges a.e. cube curl define definition denote Dirichlet integral Dirichlet's principle div f domain G elliptic equality sign holds exists Fatou's theorem finite interval fixed point theorem fn(x following properties Fourier series function of class functions f functions in G Furthermore Hence Holder inequality infinitely differentiable function Laplacian Lebesgue integrals Let f(x Let G linear functional linear manifold locally convex mapping Math minimum problem Minkowski Inequality n-tuples neighborhood non-linear problem norm partial differential equations plane point in G priori estimates proof prove saddle surface satisfy Schauder Schwarz inequality smoothing process Sobolev square integrable functions step functions strong derivatives subdomain G subregion G summable functions termwise test functions topology tuple uniformly variable vector field weak sense Weyl's lemma