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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 2 u)=0 in a domain Ω n , extending standard results. Among others, the following case is considered: at any xΩ, 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,σ T (x)p,σ T (x)Mσ(x)),

where G is uniformly elliptic and σ 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.

35J70Degenerate elliptic equations
35J25Second order elliptic equations, boundary value problems
35J65Nonlinear boundary value problems for linear elliptic equations
49L25Viscosity solutions (infinite-dimensional problems)
35H20Subelliptic PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)