TY - JOUR

T1 - Symmetry of minimizers with a level surface parallel to the boundary

AU - Ciraolo, Giulio

AU - Magnanini, Rolando

AU - Sakaguchi, Shigeru

PY - 2015

Y1 - 2015

N2 - We consider the functional $I_\Omega(v)=\int_\Omega [f(|Dv|)-v] dx$; where $\Omega$ is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if $I_\Omega$ 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. 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.

AB - We consider the functional $I_\Omega(v)=\int_\Omega [f(|Dv|)-v] dx$; where $\Omega$ is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if $I_\Omega$ 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. 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.

KW - Applied Mathematics

KW - Mathematics (all)

KW - Minimizers of integral functionals

KW - Overdetermined problems

KW - Applied Mathematics

KW - Mathematics (all)

KW - Minimizers of integral functionals

KW - Overdetermined problems

UR - http://hdl.handle.net/10447/150504

UR - http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=17&iss=11&rank=3

M3 - Article

VL - 17

SP - 2789

EP - 2804

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

ER -