Analysis and Operator Theory Seminar - F. Brock

Title: Symmetry to degenerate elliptic problems via continuous rearrangement

 

Speaker: F. Brock - University of Rostock 

 

Abstract: We consider non-negative distributional solutions $u\in C^1 (\overline{B_R } )$ to the equation \\ $-\mbox{div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u)$ in a ball $B_R$, with $u=0$ on $\partial B_R $, where $f$ is continuous and non-increasing in the first variable and $g\in C^1 (0,+\infty )\cap C[0, +\infty )$, with $g(0)=0$ and $g'(t)>0$ for $t>0$. We first show that the solutions satisfy a 'local' type of symmetry. The proof is based on the method of continuous Steiner symmetrization. \\ In a joint work with P. Taka\v{c} (Rostock) we use this result and the Strong Maximum Principle for the elliptic operator, to prove that the solutions are radially symmetric, provided that $f$ satisfies appropriate growth conditions near its zeros. \\ Finally we discuss stability issues in the autonomous case $f=f(u)$. Note that the solutions are critical points for an associated variation problem. We then show that global and local minimizers of the variational problem are radial under rather mild conditions.