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

Giulio Ciraolo; Rolando Magnanini; Shigeru Sakaguchi

Journal of the European Mathematical Society (2015)

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

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topCiraolo, 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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