Asymptotic behaviour of fundamental solutions and potential theory of parabolic operators with variable coefficients
type: Item
instance of: scholarly article
title: Asymptotic behaviour of fundamental solutions and potential theory of parabolic operators with variable coefficients
zbMATH Open document ID: 0638.35003
DOI: 10.1007/BF01446432
author: Nicola Garofalo
author: Ermanno Lanconelli
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164505
review text: Let \(L=div(A(x,t)\nabla _ x)-D_ t\) be a parabolic operator in \({\mathbb{R}}^{n+1}\) with \(C^{\infty}\) coefficient matrix \(A(x,t)=(a_{ij}(x,t))\). In this paper we prove the following asymptotic expansion for the fundamental solution \(\Gamma\) of L: As \(t\to 0\) \(+\) and for x close to y \[ (1)\quad \Gamma (x,t;y,0)\sim (4\pi t)^{-n/2} \exp (-d^ 2(x,y,t)/4t)\sum ^{\infty}_{j=0}t\quad ju\quad _ j(x,y,t). \] In (1) d(x,y,t) denotes the Riemannian distance between x and y in the metric generated on \({\mathbb{R}}^ n \)at time t by \(A^{-1}(\cdot,t)\). Expansions similar to (1) hold for the derivatives of \(\Gamma\). The above result extends to time-dependent operators a classical result of Minashisundaram and Pleijel. Using (1) we address several questions in parabolic potential theory. In previous work of Fabes and Garofalo and of Garofalo and Lanconelli it was shown that solutions of \(Lu=0\) in \({\mathbb{R}}^{n+1}\) and, more in general, any \(C^{\infty}\) function on \({\mathbb{R}}^{n+1}\) can be suitably represented as weighted averages on the level sets of the fundamental solution \(\Gamma\) of L. In this paper, we prove some new mean value formulas which extend the previous cited results. The advantage of these new formulas consists in the fact that not only the kernels appearing in them are bounded, but they also possess an arbitrarily high degree of regularity. The proof of these results crucially relies on (1). Using these new formulas we are able to give an elementary proof of Harnack inequality for nonnegative solutions of \(Lu=0\) which is modelled on the classical proof for harmonic functions, and therefore does not use the parabolic BMO machinery of Moser. The last two sections of the paper are devoted to study the connection between the averaging operators introduced and L-superparabolic functions on \({\mathbb{R}}^{n+1}\). The main result states that every L-superparabolic function can be monotonically approximated by a sequence of L- superpar…
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 4040374
zbMATH Keywords: fundamental solution
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: parabolic potential theory
zbMATH Keywords: mean value formulas
zbMATH Keywords: regularity
zbMATH Keywords: Harnack inequality
zbMATH Keywords: nonnegative solutions
zbMATH Keywords: harmonic functions
zbMATH Keywords: L-superparabolic functions
zbMATH Keywords: regularity of boundary points
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: variable coefficients
MaRDI profile type: Publication
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OpenAlex ID: W2075970446
description: scientific article; zbMATH DE number 4040374