On some PDEs involving homogeneous operators. Spectral analysis, semigroups and Hardy inequalities
type: Item
instance of: scholarly article
title: On some PDEs involving homogeneous operators. Spectral analysis, semigroups and Hardy inequalities
published in: Journal of Differential Equations
publication date: 2022-02-10T00:00:00Z
review text: Continuing their earlier work on homogeneous operators and semigroups, the authors of the paper under review present a spectral analysis of homogeneous operators and give simple characterisations of generators of homogeneous semigroups and of homogeneous sectorial operators. Here an operator \(L\) with domain \(D(L)\) in a Banach space \(X\) of dilation-invariant functions or distributions on \(\mathbb{R}^N\) is said to be \emph{homogeneous} (of degree \(\sigma\in\mathbb{R}\)) if \[L\phi_R=R^\sigma(L\phi)_R,\qquad \phi\in D(L),\ R>0,\] where \(\phi_R(x)=\phi(Rx)\) for all \(x\in\mathbb{R}^N\) and \(R>0.\) Examples of homogeneous operators include positive fractional powers of the Laplacian \(-\Delta\) and, more generally, pseudodifferential operators with homogeneous symbol on \(\mathbb{R}^N\). Among other things the authors analyse homogeneous perturbations of homogeneous operators and semigroups, and they apply these results to study certain parabolic PDEs involving homogeneous operators, including fractional diffusion problems. The authors moreover investigate the connection between their new results and some Gagliardo-Nirenberg and Hardy type inequalities.
reviewed by: David Seifert
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47B12
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 7471769
zbMATH Keywords: homogeneity
zbMATH Keywords: semigroups of linear operators
zbMATH Keywords: spectrum
zbMATH Keywords: resolvent
zbMATH Keywords: numerical range
zbMATH Keywords: fractional diffusion
author: Jan W. Cholewa
author: Anibal Rodriguez-Bernal
MaRDI profile type: Publication
full work available at URL: j.jde.2022.01.029
OpenAlex ID: W4210704933
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DOI: 10.1016/J.JDE.2022.01.029
description: scientific article; zbMATH DE number 7471769