Communications on Pure and Applied Analysis (CPAA)

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

Pages: 709 - 731, Volume 5, Issue 4, December 2006      doi:10.3934/cpaa.2006.5.709

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Martino Bardi - Dipartimento di Matematica P. e A., Universita di Padova, via Belzoni 7, 35131 Padova, Italy (email)
Paola Mannucci - Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Belzoni, 7, 35131, Padova, Italy (email)

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.

Keywords:  Viscosity solution, comparison principle, degenerate elliptic equation, subelliptic equation, Heisenberg group, Pucci operators.
Mathematics Subject Classification:  Primary: 35J70, 35J25, 35J60; Secondary: 49L25, 35H20.

Received: December 2005;      Revised: April 2006;      Available Online: September 2006.