## Abstract

We consider the Dirichlet problem for the semilinear equation δu+f(u)=0 on a bounded domain Ω⊂ℝ ^{N}. We assume that Ω is convex in a direction e and symmetric about the hyperplane H={x∈ℝ ^{N}:x{dot operator}e=0}. It is known that if N≥2 and Ω is of class C ^{2}, then any nonzero nonnegative solution is necessarily strictly positive and, consequently, it is reflectionally symmetric about H and decreasing in the direction e on the set {x∈Ω:x{dot operator}e>0}. In this paper, we prove the same result for a large class of nonsmooth planar domains. In particular, the result is valid if any of the following additional conditions on Ω holds:(i)Ω is convex (not necessarily symmetric) in the direction perpendicular to e,(ii)Ω is strictly convex in the direction e,(iii)Ω is piecewise-C ^{1,1}.

Original language | English (US) |
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Pages (from-to) | 4458-4474 |

Number of pages | 17 |

Journal | Journal of Functional Analysis |

Volume | 262 |

Issue number | 10 |

DOIs | |

State | Published - May 15 2012 |

### Bibliographical note

Funding Information:1 Supported in part by NSF grant DMS-0900947.

## Keywords

- Planar domain
- Positivity
- Semilinear elliptic equation
- Symmetry of solutions