The Dirichlet problem for sub-elliptic second order equations
type: Item
instance of: scholarly article
title: The Dirichlet problem for sub-elliptic second order equations
zbMATH Open document ID: 0934.35031
DOI: 10.1007/BF01783470
author: Chao-Jiang Xu
author: Alberto Parmeggiani
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2000-01-04T00:00:00Z
review text: Let \(\Omega\) be a smooth bounded open subset of \(\mathbb{R}^n\), and let \(L\) be a second-order differential operator: \[ L-\sum_{1\leq i,j\leq n} \partial_{x_i}(a_{ij} \partial_{x_j}u) +cu, \] where the matrix \((a_{ij})\) is positive semidefinite on a neighbourhood of \(\overline\Omega\), \(\partial\Omega\) is noncharacteristic for \(L\), \(c(x)\geq c_0>0\), the coefficients \(a_{ij},c\) are \({\mathcal C}^\infty\) real functions, and \(L\) satisfies the Fefferman-Phong's condition. Then, by using the Dirichlet principle associated with the operator \(L\) and a comparison principle for \(L\) with an operator of the form sum-of-squares, the authors prove that there exists a unique solution \(u\in{\mathcal C}^\infty (\overline\Omega)\) of the Dirichlet problem \[ Lu=f\text{ in }\Omega,\;u|_{\partial\Omega}=g, \quad f\in{\mathcal C}^\infty (\overline\Omega),\quad g\in {\mathcal C}^\infty (\partial\Omega). \]
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 35G15
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1383226
zbMATH Keywords: Dirichlet principle
zbMATH Keywords: comparison principle
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01783470
OpenAlex ID: W2027092809
description: scientific article; zbMATH DE number 1383226