A Haar-rado Type Theorem For Minimizers In Sobolev Spaces

Let u is an element of phi + W(0)(1,1) (Omega) be a minimum for I(v) = integral(Omega)g(x, v( x)) + f(del v(x))dx where f is convex, v bar right arrow g(x, v) is convex for a. e. x. We prove that u shares the same modulus of continuity of f whenever Omega is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds for all gamma is an element of partial derivative Omega |u(x) - phi(gamma)| <= omega(|x - gamma|) a. e. x is an element of Omega. This result generalizes the classical Haar-Rado theorem for Lipschitz functions.