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Research Report MRR99-002
Isolated Singularities of Solutions of Fully Nonlinear Elliptic Equations
Denis A. Labutin
Abstract:
We obtain Serrin type characterization of
isolated singularities for solutions of
Pucci equations\break
${P}^+_{\lambda,\Lambda}(D^2u)$
$=\sup(\sum A_{ij} D_{ij}u) = 0$,
($
{P}^-_{\lambda,\Lambda}(D^2u)
=\inf(\sum A_{ij} D_{ij}u) = 0
$), where the supremum, (infimum), is taken over all symmetric
matrices $A=[A_{ij}]$ with the eigenvalues in the segment
$[\lambda, \Lambda]$, $0<\lambda<\Lambda$. The main result states
that any solution to the equation in the punctured ball bounded
from one side is either extendable to the solution in the entire
ball or can be controlled near the centre of the ball by means
of special fundamental solutions. In comparison with the semi- and
quasilinear results the new element in the proof is based on using
the viscosity notion of generalized solution rather than the
distributional or the Sobolev weak solutions. We also discuss one
way of defining the expression $-{P}^+_{\lambda,\Lambda}(D^2u)$,
(${P}^-_{\lambda,\Lambda}(D^2u)$), as a measure for the viscosity
supersolutions (subsolutions) of the corresponding equation.
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