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On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. (English) Zbl 1142.35041

The authors prove some variants of the comparison principle for viscosity sub- and supersolutions of fully nonlinear elliptic equations $F\left(x,u,Du,{D}^{2}u\right)=0$ in a domain ${\Omega }\subset {ℝ}^{n}$, 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 $F\left(x,u,p,M+rI\right)$ 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

$F\left(x,u,p,M\right)=G\left(x,u,{\sigma }^{T}\left(x\right)p,{\sigma }^{T}\left(x\right)M\sigma \left(x\right)\right),$

where $G$ is uniformly elliptic and $\sigma$ is an $n×m$ 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.

##### MSC:
 35J70 Degenerate elliptic equations 35J25 Second order elliptic equations, boundary value problems 35J65 Nonlinear boundary value problems for linear elliptic equations 49L25 Viscosity solutions (infinite-dimensional problems) 35H20 Subelliptic PDE 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)