### 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 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.

Original language | English (US) |
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Pages (from-to) | 931-955 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 247 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1 2009 |

### Keywords

- Dirichlet problem
- Fully nonlinear elliptic equation
- Hamilton-Jacobi-Bellman equation
- Principal eigenvalue

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Armstrong, S. N. (2009). The Dirichlet problem for the Bellman equation at resonance.

*Journal of Differential Equations*,*247*(3), 931-955. https://doi.org/10.1016/j.jde.2009.03.007