# Symmetry of minimizers with a level surface parallel to the boundary

• Volume: 017, Issue: 11, page 2789-2804
• ISSN: 1435-9855

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## Abstract

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We consider the functional ${ℐ}_{\Omega }\left(v\right)={\int }_{\Omega }\left[f\left(|Dv|\right)-v\right]dx,$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if ${ℐ}_{\Omega }$ admits a minimizer in ${W}_{0}^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.

## How to cite

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Ciraolo, Giulio, Magnanini, Rolando, and Sakaguchi, Shigeru. "Symmetry of minimizers with a level surface parallel to the boundary." Journal of the European Mathematical Society 017.11 (2015): 2789-2804. <http://eudml.org/doc/277545>.

@article{Ciraolo2015,
abstract = {We consider the functional $\mathcal \{I\}\_\{\Omega \} (v) = \int \_\{\Omega \} [f(|Dv|) - v] dx,$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if $\mathcal \{I\}_\{\Omega \}$ admits a minimizer in $W_0^\{1,1\}(\Omega )$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.},
author = {Ciraolo, Giulio, Magnanini, Rolando, Sakaguchi, Shigeru},
journal = {Journal of the European Mathematical Society},
keywords = {overdetermined problems; minimizers of integral functionals; integral functionals; minimizers; symmetry; overdetermined problems},
language = {eng},
number = {11},
pages = {2789-2804},
publisher = {European Mathematical Society Publishing House},
title = {Symmetry of minimizers with a level surface parallel to the boundary},
url = {http://eudml.org/doc/277545},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Ciraolo, Giulio
AU - Magnanini, Rolando
AU - Sakaguchi, Shigeru
TI - Symmetry of minimizers with a level surface parallel to the boundary
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 11
SP - 2789
EP - 2804
AB - We consider the functional $\mathcal {I}_{\Omega } (v) = \int _{\Omega } [f(|Dv|) - v] dx,$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if $\mathcal {I}_{\Omega }$ admits a minimizer in $W_0^{1,1}(\Omega )$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.
LA - eng
KW - overdetermined problems; minimizers of integral functionals; integral functionals; minimizers; symmetry; overdetermined problems
UR - http://eudml.org/doc/277545
ER -

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