For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of suitably defined viscosity solutions of Dirichlet problems and we further show that it is a Lipschitz continuous function.
We study the Dirichlet problem on a bounded convex domain of RN, with zero boundary data, for truncated Laplacians Pk±, which are degenerate elliptic operators, for k< N, defined by the upper and respectively lower partial sum of k eigenvalues of the Hessian matrix. We establish a necessary and sufficient condition (Theorem 1) in terms of the “flatness” of domains for existence of a solution for general inhomogeneous term. This result, in particular, shows that the strict convexity of the domain is sufficient for the solvability of the Dirichlet problem.
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