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.