Finite propagation speed and kernels of strictly elliptic operators
type: Item
instance of: scholarly article
title: Finite propagation speed and kernels of strictly elliptic operators
zbMATH Open document ID: 0578.47039
DOI: 10.1155/S0161171285000072
author: D. Gurarie
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/45989
review text: Resolvent and semigroup kernels are constructed and estimated in \(L^ p\)-spaces for a class of global elliptic operators on \({\mathbb{R}}^ n\) of the form \(A=\rho A_ 0+B\), where \(A_ 0=\sum_{| \alpha | =m}a_{\alpha}(x)D^{\alpha}\) is uniformly elliptic of order m, \(\rho\) satisfies a ''finite propagation speed'' condition, \(\rho (x)=O(| x|^ m)\), and perturbation \(B=\sum_{| \alpha | \leq m}b_{\alpha}D^{\alpha}\) may have \(L^ p\)-singular coefficients. This class is modeled after Schrödinger-type operators: \(A=-\nabla \cdot \rho \nabla +B\cdot \nabla +V\). Applications are given to closedness and essential selfadjointness of A, its \(L^ p\)-domains, ''resolvent summability'' (i.e. \(L^ p\)-convergence: \(\zeta (\zeta -A)^{-1}f\to f\) as \(\zeta\) \(\to \infty)\), and existence of a holomorphic semigroup \(\{e^{-tA}\}Re t>0\). Most results are valid in \(L^ p\)-spaces, \(1<p<\infty.\) The basic techniques of the paper fail in the limiting case \(p=1\). The ensuring problems and examples are briefly discussed at the end.
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 3925648
zbMATH Keywords: Resolvent and semigroup kernels
zbMATH Keywords: global elliptic operators
zbMATH Keywords: uniformly elliptic
zbMATH Keywords: finite propagation speed
zbMATH Keywords: perturbation
zbMATH Keywords: \(L^ p\)-singular coefficients
zbMATH Keywords: Schrödinger-type operators
zbMATH Keywords: closedness
zbMATH Keywords: essential selfadjointness
zbMATH Keywords: resolvent summability
zbMATH Keywords: holomorphic semigroup
Wikidata QID: Q61675448
MaRDI profile type: Publication
OpenAlex ID: W2171815022
description: scientific article; zbMATH DE number 3925648