Potential Estimates for a Class of Fully Nonlinear Elliptic Equations
Denis A. Labutin
Abstract:
We study the pointwise properties of k-subharmonic
functions, that is the viscosity subsolutions to the fully
nonlinear elliptic equations Fk[u]=0, where
Fk[u] is the
elementary symmetric function of order k, 1\leq k\leq
n, of
the eigenvalues of [ D2u ],
F1[u]=\Delta u,
Fn[u]=\det D2u. 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, polar, negligible and thin sets, and 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
Fk[u]=0. The crucial tool in the proofs of
these results is
the Radon measure Fk[u] introduced recently by
Trudinger and
Wang for any k-subharmonic u. We use ideas from the
potential theories both for the complex Monge-Ampère and the
p$-Laplace equations.
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