Electron. J. Diff. Equ., Vol. 2013 (2013), No. 108, pp. 1-10.

Symmetry and regularity of an optimization problem related to a nonlinear BVP

Claudia Anedda, Fabrizio Cuccu

Abstract:
We consider the functional
$$ f\mapsto\int_\Omega \big(\frac{q+1}{2} |Du_f|^2-u_f|u_f|^q f\big) dx, $$
where $u_f$ is the unique nontrivial weak solution of the boundary-value problem
$$ -\Delta u=f|u|^q\quad \hbox{in }\Omega,\quad u\big|_{\partial\Omega}=0, $$
where $\Omega\subset\mathbb{R}^n$ is a bounded smooth domain. We prove a result of Steiner symmetry preservation and, if $n=2$, we show the regularity of the level sets of minimizers.

Submitted January 7, 2013. Published April 29, 2013.
Math Subject Classifications: 35J20, 35J60, 40K20.
Key Words: Laplacian; optimization problem; rearrangements; Steiner symmetry; regularity.

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Claudia Anedda
Dipartimento di Matematica e Informatica
Universitá di Cagliari
Via Ospedale 72, 09124 Cagliari, Italy
email: canedda@unica.it
Fabrizio Cuccu
Dipartimento di Matematica e Informatica
Universitá di Cagliari
Via Ospedale 72, 09124 Cagliari, Italy
email: fcuccu@unica.it

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