The Dirichlet problem for the Bellman equation at resonance

TL;DR

This paper extends the Donsker-Varadhan formula to the principal half-eigenvalue of certain nonlinear elliptic operators, linking measure minimizers to the solvability of the Dirichlet problem at resonance.

Contribution

It generalizes the minimax formula to fully nonlinear, positively homogeneous operators like Hamilton-Jacobi-Bellman and Pucci operators, and relates measure minimizers to problem solvability.

Findings

Extended the minimax formula to nonlinear operators

Characterized solvability of Dirichlet problem at resonance

Identified measures related to eigenvalues

Abstract

We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at resonance.