TY - JOUR

T1 - Solutions of elliptic equations with a level surface parallel to the boundary

T2 - Stability of the radial configuration

AU - Ciraolo, Giulio

AU - Magnanini, Rolando

AU - Sakaguchi, Shigeru

N1 - Publisher Copyright:
© 2016, Hebrew University Magnes Press.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. 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 (Formula presented.) and (Formula presented.) , 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 elaborates arguments developed by Aftalion, Busca, and Reichel.

AB - A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. 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 (Formula presented.) and (Formula presented.) , 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 elaborates arguments developed by Aftalion, Busca, and Reichel.

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U2 - 10.1007/s11854-016-0011-2

DO - 10.1007/s11854-016-0011-2

M3 - Article

AN - SCOPUS:84962422615

VL - 128

SP - 337

EP - 353

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -