Optimal $C^{1,\alpha}$ estimates for a class of elliptic quasilinear equations

Abstract

In this article we establish sharp $C^{1,\alpha}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation $$-\,div\, a(x, \nabla u) = f,$$ which includes the standard $p$-laplacean equation with varying coefficients as a special case. The sharp exponent $\alpha$ is asymptotically optimal and is determined by the H\"older regularity of the coefficients, the exponent $p$ and the $q$-integrability of the source term $f$.