The Dirichlet problem for the Bellman equation at resonance

Details The Dirichlet problem for the Bellman equation at resonance The Dirichlet problem for the Bellman equation at resonance

2008-12-07

arxiv.org/abs/0812.1327v2

J. Differential Equations 247 (2009) 931--955

Mathematics

Analysis of PDEs

Appendix added. 28 pages

Scientific paper

10.1016/j.jde.2009.03.007

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.

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