JournalsrlmVol. 21, No. 1pp. 59–78

The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions

  • Alessio Porretta

    Università di Roma, Italy
The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions cover
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Abstract

We study the limit, when λ tends to 0, of the solutions u_λ of the Dirichlet problem
-∆_u + λu
+ |∇_u_|q = f(x) in Ω
u = 0 on ∂ Ω,
when 1 < q ≤ 2 and f is bounded. In case the limit problem does not have any solution, we prove that _u_λ has a complete blow-up (_u_λ → -∞) and its behaviour is described in terms of the corresponding ergodic problem with state constraint conditions. In particular, _λu_λ converges to the ergodic constant _c_0 and _u_λ + ||_u_λ||∞ converges to the boundary blow-up solution _ν_0 of the ergodic problem associated to the stochastic optimal control with state constraint.

Cite this article

Alessio Porretta, The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 21 (2010), no. 1, pp. 59–78

DOI 10.4171/RLM/561