Isolated Singularities of Solutions of Fully Nonlinear Elliptic Equations

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 ${P}^+_{\lambda,\Lambda}(D^2u)$ $=\sup(\sum A_{ij} D_{ij}u) = 0$ , ( ${P}^-_{\lambda,\Lambda}(D^2u) =\inf(\sum A_{ij} D_{ij}u) = 0$ ), 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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