Existence through convexity for the truncated Laplacians

Abstract

We study the Dirichlet problem on a bounded convex domain of ${\mathbb {R}}^N$, with zero boundary data, for truncated Laplacians ${\mathcal {P}}_k^\pm$, 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. The result and related ideas are applied to the solvability of the Dirichlet problem for the operator ${\mathcal {P}}_k^+$ with lower order term when the domain is strictly convex and the existence of principal eigenfunctions for the operator ${\mathcal {P}}_1^+$. An existence theorem is presented with regard to the principal eigenvalue for the Dirichlet problem with zero-th order term for the operator ${\mathcal {P}}_1^+$. A nonexistence result is established for the operator ${\mathcal {P}}_k^+$ with first order term when the domain has a boundary portion which is nearly flat. Furthermore, when the domain is a ball, we study the Dirichlet problem, with a constant inhomogeneous term and a possibly sign-changing first order term, and the associated eigenvalue problem.