Gaussian estimates for degenerate diffusion
type: Item
instance of: scholarly article
title: Gaussian estimates for degenerate diffusion
zbMATH Open document ID: 1284.35206
DOI: 10.1090/S0002-9939-2012-11313-9
author: Michal Chovanec
published in: Proceedings of the American Mathematical Society
publication date: 2013-09-03T00:00:00Z
Mathematics Subject Classification ID: 35K08
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 6204266
zbMATH Keywords: kernel estimates
zbMATH Keywords: weighted Lebesgue space
MaRDI profile type: Publication
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review text: The author shows Gaussian estimates for the kernel of the semigroup generated by the operator \(m \Delta\) on the weighted Lebesgue space \(L^p(\Omega, m^{-1}(x) \, dx)\) for \(p \in [1,\infty)\) and bounded measurable functions \(m: \Omega \to (0,\infty)\) on arbitrary open sets \(\Omega \subset \mathbb{R}^N\). In the main result it is shown that if \(m^{-1} \in L^q(\Omega)\) for some \(q > \max(N/2,1)\), the operator \(m\Delta\) generates a strongly continuous semigroup on \(L^p(\Omega, m^{-1}(x) \, dx)\) for all \(p \in [1, \infty)\). Furthermore, the semigroup \((e^{m\Delta t})_{t \geq 0}\) is then given by a non-negative bounded measurable kernel \(K(t,x,y)\) which satisfies NEWLINE\[NEWLINEK(t,x,y) \leq C_{\hat{N},q} \cdot t^{-\frac{\hat{N}(q-1)}{2q-\hat{N}}} e^{-\frac{\|x-y\|^2}{4\tilde{c}t}} \qquad \text{for all } t > 0,NEWLINE\]NEWLINE where the constants \(\tilde{c}\) and \(\hat{N}\) depend on \(\|m\|_{\infty}\) and \(N\) respectively. The proof mainly relies on the form methods introduced by Kato and Lions.
reviewed by: Stephan Fackler
description: scientific article; zbMATH DE number 6204266