Existence through convexity for the truncated Laplacians
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.