TY - JOUR

T1 - Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

AU - Ciraolo, Giulio

AU - Magnanini, Rolando

AU - Sakaguchi, Shigeru

PY - 2016

Y1 - 2016

N2 - Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls Bre and Bri, with the difference re−ri (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel.

AB - Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls Bre and Bri, with the difference re−ri (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel.

KW - Harnack’s inequality.

KW - Parallel surfaces

KW - method of moving
planes

KW - overdetermined problems

KW - stability

KW - stationary surfaces

KW - Harnack’s inequality.

KW - Parallel surfaces

KW - method of moving
planes

KW - overdetermined problems

KW - stability

KW - stationary surfaces

UR - http://hdl.handle.net/10447/150508

UR - http://link.springer.com/article/10.1007/s11854-016-0011-2

M3 - Article

VL - 128

SP - 337

EP - 353

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

ER -