Non variational basic elliptic systems of second order

Abstract

Denoting byu a vector in R N defined on a bounded open set Ω ⊂ R n , we setH(u)={Dij u} and consider a basic differential operator of second ordera(H(u)) wherea(ξ) is a vector in R N , which is elliptic in the sense that it satisfies the condition (A). After a rapid comparison between this condition (A) and the classical definition of ellipticity, we shall prove that, if seu∈H 2 (Ω) is a solution of the elliptic systema(H(u))=0 in Ω thenH(u)∈H loc 2, q for someq>2. We then deduce from this the so called fundamental internal estimates for the matrixH(u) and for the vectorsDu andu. We shall then present a first risult on holder regularity for the solutions of the system withf holder continuous in Ω, and a partial holder continuity risult for solutionsu∈H 2 (Ω) of a differential systema (x, u, Du, H (u))=b(x, u, Du)