Unique continuation for fully nonlinear elliptic equations

TL;DR

This paper proves that viscosity solutions to certain fully nonlinear elliptic equations that vanish on an open set must be identically zero, using boundary Harnack inequality and regularity results without convexity assumptions.

Contribution

It establishes a unique continuation principle for fully nonlinear elliptic equations without requiring convexity or concavity of the nonlinearity.

Findings

Viscosity solutions vanish on an open set imply they are identically zero.

Regularity at boundary points is achieved without a priori $C^2$ estimates.

The result extends classical unique continuation to fully nonlinear equations.

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.