REGULARITY FOR NONLINEAR ELLIPTIC EQUATIONS AND SYSTEMS

We study the regularity of weak solutions to the elliptic system in divergence form divA(x, Du) = 0 in an open set Omega of R-n, n >= 2. The vector field A(x.xi), A : Omega x R-mxn -> R-mxn, has a variational nature in the sense that A(x, xi) = D xi f (x, xi), where f : Omega x R-mxn > R is a convex Caratheodory integrand; i.e., f = f (x, xi) is measurable with respect to x is an element of R-n and it is a convex function with respect to xi is an element of R-mxn. If m = 1 then the system reduces to a partial differential equation. In the context m > 1 of general vector-valued maps and systems, a classical assumption finalized to the everywhere regularity of the weak solutions is a modulus-dependence in the energy integrand; i.e., we require that f (x, xi) = g(x, vertical bar xi vertical bar), where g : Omega x [0, infinity) -> [0, infinity) is measurable with respect to x is an element of R-n and it is a convex and increasing function with respect to the gradient variable t is an element of [0, infinity).