Regularity for nonlinear elliptic equations and systems

We study the regularity of weak solutions to the elliptic system in divergence form divA( x , D u)=0 in an open set Ω of R n , n ≥2. The vector field A( x .ξ), A: Ω×R m×n →R m×n , has a variational nature in the sense that A( x ,ξ)= D ξ f ( x ,ξ), where f :Ω×R m×n →R is a convex Caratheodory integrand ; i.e., f = f ( x ,ξ) is measurable with respect to x ∈R n and it is a convex function with respect to ξ∈R m×n . 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 ,ξ)= g ( x ,|ξ|), where g :Ω×[0,∞)→[0,∞) is measurable with respect to x∈  R n and it is a convex and increasing function with respect to the gradient variable t∈[0,∞).