Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations
type: Item
instance of: scholarly article
title: Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations
zbMATH Open document ID: 0878.46024
DOI: 10.1007/BF02110147
author: S. K. Vodop'yanov
published in: Siberian Mathematical Journal
publication date: 1997-12-11T00:00:00Z
review text: Let \(X_1,X_2,\dots,X_k\) be \(C^\infty\)-vector fields in \(\mathbb{R}^N\) satisfying Hörmander's conditions. The author studies regularity theory and potential theory for the quasilinear degenerate hypoelliptic equation of the form \[ -\text{div} {\mathcal A}(x,\nabla_{\mathcal L} u)=0, \tag{1} \] where \(\nabla_{\mathcal L}u=(X_1u,X_2u,\dots,X_ku),\;{\mathcal A}:\Omega\times{\mathbb{R}}^k \rightarrow {\mathbb{R}}^k\) is a mapping satisfying certain structural assumptions, in particular \({\mathcal A}(x,\xi)\approx w(x)|\xi|^p\), \(\Omega\) is a domain in \({\mathbb{R}}^N, w(x)\) is a \(p\)-admissible weight. In the particular case of Euclidean spaces and standard vector fields the corresponding theory can be found in the recent monograph by \textit{J. Heinonen, T. Kilpeläinen} and \textit{O. Martio} [``Nonlinear potential theory of degenerate elliptic equations'', Oxford (1993; Zbl 0780.31001)] where the notion of \(p\)-admissibility and conditions on \(\mathcal A\) are given. The author studies also boundary behavior of the solutions to (1) introducing a new notion of an improper boundary. The generalization of the Wiener criterion of regularity of a boundary point is proved.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 923498
zbMATH Keywords: Hörmander's conditions
zbMATH Keywords: regularity
zbMATH Keywords: potential theory
zbMATH Keywords: quasilinear degenerate hypoelliptic equation
zbMATH Keywords: vector fields
zbMATH Keywords: \(p\)-admissibility
zbMATH Keywords: improper boundary
zbMATH Keywords: Wiener criterion
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02110147
OpenAlex ID: W2315880237
description: scientific article; zbMATH DE number 923498