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Analysis and Operator Theory Seminar - F. Brock

Analysis and Operator Theory Seminar
September 17, 2020
11:00AM - 11:55AM
Zoom

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Add to Calendar 2020-09-17 11:00:00 2020-09-17 11:55:00 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.   Zoom Department of Mathematics math@osu.edu America/New_York public
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.
 

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