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.
Highlights
In this paper we study the comparison, uniqueness, and existence of viscosity solutions to the Dirichlet problem for some second order, fully nonlinear equations
The classical comparison principle between viscosity sub- and supersolutions was obtained by Jensen [24, 25] and by Ishii-Lions [23] when either (i) F is degenerate elliptic and strictly increasing in u, or (ii) F is uniformly elliptic and nondecreasing in u
The goal of this paper is to weaken the uniform ellipticity to various forms of partial nondegeneracy
Summary
In this paper we study the comparison, uniqueness, and existence of viscosity solutions to the Dirichlet problem for some second order, fully nonlinear equations. Where σ(·) is an n × m matrix-valued Lipschitz function with tr(σT σ) ≥ η and G is uniformly elliptic on Ω × IRm × Sm. In Sections 2 and 3 the three comparison principles are stated and proved in the general case of F depending on u in a nondecreasing way, and the nondegeneracy assumptions are made only at the points where F fails to be strictly increasing in u. From standard viscosity solutions theory [23, 13], we know that, under the structural conditions 6, the comparison principle holds between a supersolution v and a strict subsolution u , i.e., an u.s.c. function in Ω satisfying F (x, u , Du , D2u ) ≤.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
