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On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations

On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 5, Number 4, December 2006 pp. 709–731 ON THE DIRICHLET PROBLEM FOR NON-TOTALLY DEGENERATE FULLY NONLINEAR ELLIPTIC EQUATIONS Martino Bardi and Paola Mannucci Dipartimento di Matematica Pura e Applicata Universita` degli Studi di Padova Via Belzoni, 7, 35131, Padova, Italy (Communicated by Gui-Qiang Chen) Abstract. We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic points of the boundary. The examples motivating the theory are operators of the form of sum of squares of vector fields plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group. 1. Introduction. In this paper we study the comparison, uniqueness, and exis- tence of viscosity solutions to the Dirichlet problem for some second order, fully nonlinear equations F (x, u, Du, D u) = 0, in Ω, (1) n n where Ω is an open bounded domain of IR and F : Ω×IR×IR ×S → IR satisfies the standard continuity assumptions of [13] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications on Pure and Applied Analysis Unpaywall

On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations

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Unpaywall
ISSN
1534-0392
DOI
10.3934/cpaa.2006.5.709
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Abstract

COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 5, Number 4, December 2006 pp. 709–731 ON THE DIRICHLET PROBLEM FOR NON-TOTALLY DEGENERATE FULLY NONLINEAR ELLIPTIC EQUATIONS Martino Bardi and Paola Mannucci Dipartimento di Matematica Pura e Applicata Universita` degli Studi di Padova Via Belzoni, 7, 35131, Padova, Italy (Communicated by Gui-Qiang Chen) Abstract. We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic points of the boundary. The examples motivating the theory are operators of the form of sum of squares of vector fields plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group. 1. Introduction. In this paper we study the comparison, uniqueness, and exis- tence of viscosity solutions to the Dirichlet problem for some second order, fully nonlinear equations F (x, u, Du, D u) = 0, in Ω, (1) n n where Ω is an open bounded domain of IR and F : Ω×IR×IR ×S → IR satisfies the standard continuity assumptions of [13]

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Communications on Pure and Applied Analysis – Unpaywall

Published: Jan 1, 2006

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