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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 \(F(x,u,Du, D^2u)= 0\) in a domain \(\Omega\subset\mathbb{R}^n\), 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 \(F(x,u,p,M+ rI)\) 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.

MSC:

35J70 Degenerate elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35H20 Subelliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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