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Perron's method and the method of relaxed limits for “unbounded” PDE in Hilbert spaces

Volume 176 / 2006

Djivede Kelome, Andrzej Świ/ech Studia Mathematica 176 (2006), 249-277 MSC: 49L25, 35R15, 35J60. DOI: 10.4064/sm176-3-4

Abstract

We prove that Perron's method and the method of half-relaxed limits of Barles–Perthame works for the so called $B$-continuous viscosity solutions of a large class of fully nonlinear unbounded partial differential equations in Hilbert spaces. Perron's method extends the existence of $B$-continuous viscosity solutions to many new equations that are not of Bellman type. The method of half-relaxed limits allows limiting operations with viscosity solutions without any a priori estimates. Possible applications of the method of half-relaxed limits to large deviations, singular perturbation problems, and convergence of finite-dimensional approximations are discussed.

Authors

  • Djivede KelomeDepartment of Mathematics and Statistics
    University of Massachusetts
    Amherst, MA 01003, U.S.A.
    and
    Department of Mathematics and Statistics
    McGill University
    805 Sherbrooke St West
    Montreal, QC, Canada H3A-2K6
    e-mail
  • Andrzej Świ/echSchool of Mathematics
    Georgia Institute of Technology
    Atlanta, GA 30332, U.S.A.
    e-mail

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