TY - JOUR

T1 - Unique continuation for fully nonlinear elliptic equations

AU - Armstrong, Scott N.

AU - Silvestre, Luis

PY - 2011/9

Y1 - 2011/9

N2 - 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 C1,1. We do not assume that the nonlinearity is convex or concave, and thus a priori C2 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.

AB - 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 C1,1. We do not assume that the nonlinearity is convex or concave, and thus a priori C2 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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U2 - 10.4310/MRL.2011.v18.n5.a9

DO - 10.4310/MRL.2011.v18.n5.a9

M3 - Article

AN - SCOPUS:84856087931

SN - 1073-2780

VL - 18

SP - 921

EP - 926

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 5

ER -