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

Abstract

We consider the functional $$ fmapstoint_Omega ig(frac{q+1}{2} |Du_f|^2-u_f|u_f|^q fig) dx, $$ where $u_f$ is the unique nontrivial weak solution of the boundary-value problem $$ -Delta u=f|u|^qquad ext{in }Omega,quad uig|_{partialOmega}=0, $$ where $Omegasubsetmathbb{R}^n$ 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.

Keywords

• Laplacian
• optimization problem
• rearrangements
• Steiner symmetry
• regularity