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Zbl 1142.35041
Bardi, Martino; Mannucci, Paola
On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. (English)
[J] Commun. Pure Appl. Anal. 5, No. 4, 709-731 (2006). ISSN 1534-0392; ISSN 1553-5258/e

The authors prove some variants of the comparison principle for viscosity sub- and supersolutions of fully nonlinear elliptic equations $F(x,u,Du, D^2u)= 0$ in a domain $\Omega\subset\bbfR^n$, extending standard results. Among others, the following case is considered: at any $x\in\Omega$, $F$ is strictly increasing with respect to $u$ or $F$ is non-totally degenerate, what roughly means that $F(x,u,p,M+ rI)$ is a strictly decreasing function of the real parameter $r$, with $I$ being the identity matrix and $M$ an arbitrary symmetric matrix. A further case refers to operators of the form $$F(x, u, p, M)= G(x,u,\sigma^T(x)p, \sigma^T(x)M\sigma(x)),$$ where $G$ is uniformly elliptic and $\sigma$ is an $n\times m$ matrix-valued function satisfying a non-degeneracy condition. A number of more specific equations is considered including Bellman-Isaacs equations, quasilinear subelliptic equations, and equations involving Pucci-type oerators. The results are applied to deduce existence theorems for the Dirichlet problem in the viscosity setting.
[F. Tomi (Heidelberg)]

MSC 2000:: *35J70 Elliptic equations of degenerate type
35J25 Second order elliptic equations, boundary value problems
35J65 (Nonlinear) BVP for (non)linear elliptic equations
49L25 Viscosity solutions
35H20 Subelliptic equations
35B05 General behavior of solutions of PDE

Keywords: comparison principles; viscosity solutions; nonlinear degenerate elliptic equations; subelliptic equation; Dirichlet problems; Pucci operator; Heisenberg group

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Zentralblatt MATH Berlin [Germany]