Dirichlet problem for some hypoelliptic operators
type: Item
instance of: scholarly article
title: Dirichlet problem for some hypoelliptic operators
zbMATH Open document ID: 0915.35030
DOI: 10.1007/BF02837224
author: K. Appert
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1999-05-18T00:00:00Z
review text: Let \[ L=\sum_{i=1}^n \sum_{j=1}^n a_{ij}(x) D_{ij}+ \sum_{i=1}^n b_i(x)D_i+ c(x) \] be a degenerate elliptic operator of second order with \(C^\infty\) coefficients with \(c\leq 0\) defined on an open set \(\Omega\) of \(\mathbb{R}^n\). Let us suppose that there exist vector fields \(X_1, X_2,\dots, X_r\), \(Y\) such that \(L\) can be written in the form \[ L=\sum_{k=1}^{k=r} X_k^2+ Y+c. \] Further let us assume that the Lie algebra generated by \(X_1, X_2,\dots, X_r\), \(Y\) at any point is of rank \(n\). For these operators, with the addition assumption that \(\exists \lambda<0\) such that \(c(x)\leq \lambda\), \(\forall x\in \Omega\), \textit{J. M. Bony} [Ann. Inst. Fourier 19, No. 1, 277-304 (1969; Zbl 0176.09703)] has proved the existence and uniqueness of the Dirichlet problem \(Lu=f\) in \(\omega\), \(u=\varphi\) on \(\partial\omega\), where \(f\) and \(\varphi\) are continuous and \(\omega\) is a relatively compact open subset of \(\Omega\), \(\overline{\omega} \subset\Omega\), such that \(\partial\omega\) admits a special kind of barrier. In this article we remove the above restriction on \(c\) and assume just that \(c\leq 0\), but assume that \(L\) satisfies maximum principles and prove the existence and uniqueness of the Dirichlet problem when \(f=0\).
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1210564
zbMATH Keywords: maximum principles
MaRDI profile type: Publication
cites work: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérées
cites work: Hypoelliptic second order differential equations
full work available at URL: https://doi.org/10.1007/bf02837224
OpenAlex ID: W2003676415
description: scientific article; zbMATH DE number 1210564