The Cauchy-Dirichlet problem for a class of linear parabolic differential equations with unbounded coefficients in an unbounded domain

Abstract

We consider the Cauchy-Dirichlet problem in [0, ∞) × D for a class of linear parabolic partial differential equations. We assume that D ∪ ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains. Copyright © 2011 Gerardo Rubio.