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 $$-diva(x,\nabla u)=f,$$ 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 Hölder regularity of the coefficients, the exponent $p$ and the $q$-integrability of the source term $f$.

arXiv comment18 pages