# A symmetry problem in the calculus of variations

• Volume: 008, Issue: 1, page 139-154
• ISSN: 1435-9855

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## Abstract

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We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\left(|Du|\right)-u\right]dx$, $u\in {W}_{0}^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^{n}$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^{n}$ with smooth boundary, and $ℝ\ni s↦f\left(|s|\right)$ is a convex, differentiable function. We prove that if $J$ 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.

## How to cite

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Crasta, Graziano. "A symmetry problem in the calculus of variations." Journal of the European Mathematical Society 008.1 (2006): 139-154. <http://eudml.org/doc/277557>.

@article{Crasta2006,
abstract = {We consider the integral functional $J(u)=\int _\Omega [f(|Du|)−u]dx$, $u\in W^\{1,1\}_0(\Omega )$, where $\Omega \subset \mathbb \{R\}^n$, $n\ge 2$, is a nonempty bounded connected open subset of $\mathbb \{R\}^n$ with smooth boundary, and $\mathbb \{R\}\ni s\mapsto f(|s|)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W^\{1,1\}_0(\Omega )$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.},
author = {Crasta, Graziano},
journal = {Journal of the European Mathematical Society},
keywords = {minimizers of integral functionals; distance function; Euler equation; calculus of variations; distance function; symmetry of solutions},
language = {eng},
number = {1},
pages = {139-154},
publisher = {European Mathematical Society Publishing House},
title = {A symmetry problem in the calculus of variations},
url = {http://eudml.org/doc/277557},
volume = {008},
year = {2006},
}

TY - JOUR
AU - Crasta, Graziano
TI - A symmetry problem in the calculus of variations
JO - Journal of the European Mathematical Society
PY - 2006
PB - European Mathematical Society Publishing House
VL - 008
IS - 1
SP - 139
EP - 154
AB - We consider the integral functional $J(u)=\int _\Omega [f(|Du|)−u]dx$, $u\in W^{1,1}_0(\Omega )$, where $\Omega \subset \mathbb {R}^n$, $n\ge 2$, is a nonempty bounded connected open subset of $\mathbb {R}^n$ with smooth boundary, and $\mathbb {R}\ni s\mapsto f(|s|)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W^{1,1}_0(\Omega )$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.
LA - eng
KW - minimizers of integral functionals; distance function; Euler equation; calculus of variations; distance function; symmetry of solutions
UR - http://eudml.org/doc/277557
ER -

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