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Existence and regularity of minimizers of nonconvex integrals with \(p-q\) growth. (English) Zbl 1124.49031

This paper deals with integral functional of the form \[ F(u)= \int_{\Omega} \left[f(Du(x))+g(x,u(x)) \right] \,dx \] with \(f\) is a convex function with \(p-q\) growth satisfying a qualified convexity condition at infinity and \(g\) is a Lipschitz continuous function with respect to \(u\). The authors prove that the local minimizers are locally Lipschitz and then they apply the regularity in order to obtain an existence theorem when \(f\) is a not convex function and \(g=a(x)h(u)\).

MSC:

49N60 Regularity of solutions in optimal control
49J10 Existence theories for free problems in two or more independent variables
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