Density results and trace operator in weighted Sobolev Spaces defined on the half line equipped with power weights

TL;DR

This paper investigates the properties and boundary behaviors of weighted Sobolev spaces on the half-line, providing analytic characterizations and insights relevant for boundary value problems and interpolation theory.

Contribution

It offers a comprehensive analysis of the trace operators and asymptotic behavior in weighted Sobolev spaces, extending understanding of their structure and applications.

Findings

Characterization of $W_0^{1,p}( plus,t^eta)$ for all $eta$

Analysis of trace operators at zero and infinity

Applications to boundary value problems and interpolation theory

Abstract

We study properties of $W_0^{1,p}(\mathbb{R}_+,t^\beta)$ - the completion of $C_0^\infty(\mathbb{R}_+)$ in the power-weighted Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^\beta)$, where $\beta\in\mathbb{R}$. Among other results, we obtain the analytic characterization of $W_0^{1,p}(\mathbb{R}_+,t^\beta)$ for all $\beta\in \mathbb{R}$. Our analysis is based on the precise study the two trace operators: $Tr^{0}(u):= \lim_{t\to 0} u(t)$ and $Tr^{\infty}(u):= \lim_{t\to \infty} u(t)$, which leads to the analysis of the asymptotic behavior of functions from $W_0^{1,p}(\mathbb{R}_+,t^\beta)$ near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems in ODE's, or PDE's with the radial symmetries. We can also apply our results to some questions in the complex interpolation theory, raised by M. Cwikel and A. Einav in 2019, which we discuss within the particular case of Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^\beta)$.