### 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 921-926 |

Number of pages | 6 |

Journal | Mathematical Research Letters |

Volume | 18 |

Issue number | 5 |

State | Published - Sep 2011 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*18*(5), 921-926.

**Unique continuation for fully nonlinear elliptic equations.** / Armstrong, Scott; Silvestre, Luis.

Research output: Contribution to journal › Article

*Mathematical Research Letters*, vol. 18, no. 5, pp. 921-926.

}

TY - JOUR

T1 - Unique continuation for fully nonlinear elliptic equations

AU - Armstrong, Scott

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 C 1,1. We do not assume that the nonlinearity is convex or concave, and thus 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.

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 C 1,1. We do not assume that the nonlinearity is convex or concave, and thus 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.

UR - http://www.scopus.com/inward/record.url?scp=84856087931&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856087931&partnerID=8YFLogxK

M3 - Article

VL - 18

SP - 921

EP - 926

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 5

ER -