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applied arbitrary assume assumptions ax(x boundary conditions boundary dQ boundary value problem CAR(p Chapter choose classical solution coefficients ax coercive concept conditions of Theorem constant continuous function convex convex set corresponds defined on Q denote derivative dF(u Dirichlet problem domain Q dx dy Example exists formal differential operator formula function defined functional F given function growth conditions Hilbert space Holder inequality holds identity implies integral Lemma Let Q Let the function lower semicontinuous mapping means minimal surface problem minimum multi-indexes nonlinear normed linear space notation obtain ordinary differential equations parameters partial differential equations plane domain point u0 possible problem for equation properties prove reader real numbers reflexive space relation Remark right-hand side second order SECTION sequence Sk u(x so-called Sobolev space solution u0 subset sufficiently symbol variables variational inequality vector weak solution x e Q zero