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

Djivede Kelome¹ ; Andrzej Świ/ech²

¹ Department 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
² School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, U.S.A.

Studia Mathematica, Tome 176 (2006) no. 3, pp. 249-277

Résumé

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.

Détail

DOI : 10.4064/sm176-3-4

Keywords: prove perrons method method half relaxed limits barles perthame works called b continuous viscosity solutions large class fully nonlinear unbounded partial differential equations hilbert spaces perrons method extends existence b continuous viscosity solutions many equations bellman type method half relaxed limits allows limiting operations viscosity solutions without priori estimates possible applications method half relaxed limits large deviations singular perturbation problems convergence finite dimensional approximations discussed