SCIENTIFIC ACTIVITIES

October 22, 2014

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR

Back2Fields COLLOQUIUM SERIES
June 22, 2012 at 2:00p.m.
Fields Institute, Stewart L
ibrary (map)
Andrei Bir
yuk
Kuban State University, Krasnodar, Russia

Comparison theorems for $p$-elliptic equations with laden levelsets as coefficients and some other topics

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PDF file of Abstract

Part 1. Joint work with Boris E Levitskij.
We generalize Talenti's symmetrization method to new comparison theorems for a wider class of equations. Let $\Omega \subset \mathbb R^m$ be an open set with finite volume.
Let $0< q \leqslant p-1$. Let $\Phi(s)=s^{\alpha/(p-1)}$, where $\alpha< \frac{1}{m}+\frac{p-1-q}{q}$. For each measurable function $u:\Omega \to \mathbb R$ and each point $x \in \Omega$ we set $$g_0(x,u)=\Phi^{p-1}({\rm meas}\{y \in \Omega: |u(y)|>|u(x)|\})\,.$$
Let $u$ be a positive weak solution of the following equation$$
\quad \quad -\sum\limits_{k=1}^{m}\frac{\partial}{\partial x_k}\bigl(g_0(x,u)|\nabla u|^{p-2}\frac{\partial u}{\partial x_k}\bigr)=f(x)+k|\nabla u|^q\,, \quad \quad \quad(1)$$equipped with zero Dirichlet boundary conditions. Here $f \in L^1(\Omega)$ and $k \geqslant 0$. Let $\Omega^{\star}$ be thespherical symmetrization of the set $\Omega$. In other words$\Omega^{\star}$ is a ball in $\mathbb R^m$ of the same volume as the measure of$\Omega$. Let $u^{\star}:\Omega^{\star}\to \mathbb R$ and $f^{\star}:\Omega^{\star}\to \mathbb R$ denote the sphericalsymmetrization of the functions $u$ and $f$ respectively, and let
$V:\Omega^{\star}\to \mathbb R$ be the maximal weak solution for $$\quad \quad -\sum\limits_{k=1}^{m}\frac{\partial}{\partial x_k}\bigl(g_0(x,V)|\nabla V|^{p-2}\frac{\partial V}{\partial x_k}\bigr)=f^{\star}(x)+k|\nabla V|^q \quad \quad \quad (2)
$$in $\Omega^{\star}$, which is the symmetrized version of equation(1). Equation (2) is also equipped with zero Dirichlet boundaryconditions.We prove that $V$ exist and unique and that $u^{\star} \leqslant V$. Also we prove that $|\nabla u|_{L^p(\Omega)} \leqslant |\nabla
V|_{L^p(\Omega^{\star})}$.Generalizations of this result are also considered.
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Part 2. Other topics.




The Back2Fields Colloquium Series celebrates the accomplishments of former postdoctoral fellows of Fields Institute thematic programs. Over the past two decades, these programs attracted the rising stars of their field and often launches very distinguished research careers. As part of the 20th anniversary celebrations, this series of colloquium talks will allow the general mathematical public to become familiar with some of their work.




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