Traces for homogeneous Sobolev spaces in infinite strip-like domains

Abstract

In this paper we construct a trace operator for homogeneous Sobolev spaces defined on infinite strip-like domains. We identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse. The intrinsic seminorm involves two features not encountered in the trace theory of bounded Lipschitz domains or half-spaces. First, due to the strip-like structure of the domain, the boundary splits into two infinite disconnected components. The traces onto each component are not completely independent, and the intrinsic seminorm contains a term that measures the difference between the two traces. Second, as in the usual trace theory, there is a term in the seminorm measuring the fractional Sobolev regularity of the trace functions with a difference quotient integral. However, the finite width of the strip-like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation. The screened homogeneous fractional Sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest, and we study their basic properties. We conclude the paper with applications of the trace theory to partial differential equations.