Regularity in Campanato Spaces for Solutions of Fully Nonlinear Elliptic Systems

The authors investigate weak solutions u 2 H2 \H1 0( ,RN) of the fully nonlinear elliptic system (1) F(x,D2u(x)) = f(x), with Rn, n 2, being a bounded convex open set with C2,1-boundary @ , x 7! F(x,M) being H¨older continuous with exponent b 2 (0, 1) for every symmetric M and M 7! F(x,M) satisfying the Campanato condition Ax for almost every x 2 , which means that there exist constants , , with > 0, 0, and + < 1, and a positive measurable function : !R, > 0 almost everywhere in , 2 C0,b( ), such that for all symmetric N-tuples of (n×n)- matricesM,Q and for almost every x 2 the inequality Xn i=1 Qii −(x)[F(x,M +Q)−F(x,M)] kQk+ Xn i=1 Qii holds. The authors prove global regularity of solutions of (1) in Campanato spaces in the sense that, for data f 2 L2,( ;RN), with n < n+2 and F satisfying the above structure assumptions, D2u 2 L2,μ( ,Rn2N) for some exponent μ 2 (0, ] depending on the structural parameters b, and .