Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 108, pp. 1-10.
Title: Symmetry and regularity of an optimization problem related
to a nonlinear BVP
Authors: Claudia Anedda (Univ. di Cagliari, Italy)
Fabrizio Cuccu (Univ. di Cagliari, Italy)
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
$$
-\Delta u=f|u|^q\quad \text{in }\Omega,\quad
u\big|_{\partial\Omega}=0,
$$
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.