The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
type: Item
instance of: scholarly article
title: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
zbMATH Open document ID: 0654.35031
DOI: 10.1007/BF02392544
author: Luis Caffarelli
author: Louis Nirenberg
author: Joel Spruck
published in: Acta Mathematica
publication date: 1985-01-01T00:00:00Z
review text: This paper is a continuation of parts I and II [Commun. Pure Appl. Math. 37, 369-402 (1984; Zbl 0598.35047) and 38, 209-252 (1985; Zbl 0598.35048)]. Here is studied the solvability of Dirichlet's problem in a bounded domain \(\Omega \subset R^ n\) with smooth boundary \(\partial \Omega:\) \[ F(D^ 2u)=\psi \quad in\quad \Omega;\quad u=\phi \quad on\quad \partial \Omega, \] where the function F is defined by a smooth symmetric function \(f(\lambda_ 1,...,\lambda_ n)\) of the eigenvalues \(\lambda =(\lambda_ 1,...,\lambda_ n)\) of the Hessian matrix \(D^ 2u=\{u_{ij}\}\). It is assumed that the equation is elliptic, i.e. \(\partial f/\partial x_ i>0\), for all i, and that f is a concave function.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 4067434
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
zbMATH Keywords: Dirichlet's problem
zbMATH Keywords: bounded domain
zbMATH Keywords: smooth boundary
zbMATH Keywords: eigenvalues
zbMATH Keywords: Hessian matrix
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4067434