Perron's method and the method of relaxed limits for “unbounded” PDE in Hilbert spaces
Volume 176 / 2006
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.
