Mathematical Research Letters

Volume 18 (2011)

Number 5

Unique Continuation for Fully Nonlinear Elliptic Equations

Pages: 921 – 926

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n5.a9

Authors

Scott N. Armstrong (Department of Mathematics, The University of Chicago, 5734 S. University Avenue Chicago, IL 60637, USA)

Luis Silvestre (Department of Mathematics, The University of Chicago, 5734 S. University Avenue Chicago, IL 60637, USA)

Abstract

We show that a viscosity solution of a uniformly elliptic, fully nonlinear equation which vanishes on an open set must be identically zero, provided that the equation is $C^{1,1}$. We do not assume that the nonlinearity is convex or concave, and thus \emph{a priori} $C^2$ estimates are unavailable. Nevertheless, we use the boundary Harnack inequality and a regularity result for solutions with small oscillations to prove that the solution must be smooth at an appropriate point on the boundary of the set on which it is assumed to vanish. This then permits us to conclude with an application of a classical unique continuation result for linear equations.

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