Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

摘要

We study nonlinear elliptic equations in divergence form div A( x, Du) = div G. When A has linear growth in Du, and assuming that x . A( x,.) enjoys Ban a, q smoothness, local well- posedness is found in Ba p, q for certain values of p. [ 2, n a) and q. [ 1,8]. In the particular case A( x,.) = A( x)., G = 0 and A. Ban a, q, 1 = q = 8, we obtain Du. Ba p, q for each p < n a. Our main tool in the proof is a more general result, that holds also if A has growth s - 1 in Du, 2 = s = n, and asserts local well- posedness in Lq for each q > s, provided that x . A( x,.) satisfies a locally uniform VMO condition.