The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction
Paola Mannucci
We prove a comparison principle for viscosity solutions of a fully nonlinear equation satisfying a condition of non-degeneracy in a fixed direction. We apply these results to prove that a continuous solution of the corresponding Dirichlet problem exists. To obtain the existence of barrier functions and well-posedness, we find suitable explicit assumptions on the domain and on the ellipticity constants of the operator.
keywords: degenerate elliptic equation Viscosity solution Pucci's operators comparison principle Dirichlet problem subelliptic equation Carnot groups.
On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations
Martino Bardi Paola Mannucci
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.
keywords: comparison principle Pucci operators. Viscosity solution subelliptic equation Heisenberg group degenerate elliptic equation

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