Potential estimates for a class of fully nonlinear elliptic equations

Abstract

We study the pointwise properties of $k$-subharmonic functions, that is, the viscosity subsolutions to the fully nonlinear elliptic equations $F_k[u]=0$, where $F_k[u]$ is the elementary symmetric function of order $k,1\leq k\leq n$, of the eigenvalues of $[D\sp 2u]$, $F_1[u]=\Delta u,F_n[u]=\det D^2u$. Thus $1$-subharmonic functions are subharmonic in the classical sense; $n$-subharmonic functions are convex. We use a special capacity to investigate the typical questions of potential theory: local behaviour, removability of singularities, and polar, negligible, and thin sets, and we obtain estimates for the capacity in terms of the Hausdorff measure. We also prove the Wiener test for the regularity of a boundary point for the Dirichlet problem for the fully nonlinear equation $F_k[u]=0$. The crucial tool in the proofs of these results is the Radon measure $F_k[u]$ introduced recently by N. Trudinger and X.-J. Wang for any $k$-subharmonic $u$. We use ideas from the potential theories both for the complex Monge-Ampère and for the $p$-Laplace equations.