Speaker
Description
We propose a notion of mean convexity of a function in R^n with respect to all k-dimensional subspaces, k integer between 1 and n. We show its connection with the truncated Laplacian of order k of the function, i.e., the sum of the first k eigenvalues of the Hessian matrix.
We consider the Hessian PDE that prescribes the k-truncated Laplacian, in viscosity sense. We study existence and uniqueness of solutions to the Dirichlet problem under suitable conditions on the data, in particular convexity properties of the domain. Here we exploit some results by Birindelli, Galise and Ishii.
Our main result is about the inverse problem of the single-valuedness of the truncated Laplacian in viscosity sense. Its solution allows us to characterise k-mean convexity of continuous functions in terms of inequalities for the Hessian equation.
Joint work with P. Mannucci