The Dirichlet problem for the Bellman equation at resonance

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.

Appendix added. 28 pages

References in corpus (1)

• Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

Cited by in corpus (6)

• The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus
• A variational characterization of the risk-sensitive average reward for controlled diffusions on ${\mathbb{R}}^d$
• A variational formula for risk-sensitive control of diffusions in ${\mathbb{R}}^d$
• Principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term
• Generalized principal eigenvalues of convex nonlinear elliptic operators in ${\mathbb{R}}^N$
• Global stability of steady states in the classical Stefan problem