Elliptic Problems on Unbounded Domains

Abstract

This paper considers elliptic boundary value problems on unbounded domains with possibly unbounded boundary using the variational method. Since a nonvanishing harmonic function is not square integrable on $R^n $, the construction of solutions in the usual Sobolev spaces, so successful for bounded domains, must fail. Kudrjavcev [8], [9] showed how to circumvent this difficulty by introducing weighted Sobolev spaces. See Besov et al. [3] for a survey on this method. Benci, Fortunate [2], Cantor [4], Janl3en [5], Maulen [10], Owen [15], Vogelsang [16], and others, apparently independently, have seized upon this method for treating elliptic problems on unbounded domains. This paper generalizes the method in two directions: 1) We treat operators of all orders with Dirichlet, Neumann and mixed boundary conditions. 2) We impose none of the usual restrictions on the coefficients of the operator (e.g. that they should converge to a constant as $| x | \to \infty $). The main tools are variants of the Poincaré and Friedrichs inequality, respectively, and compact embeddings in weighted Sobolev spaces.