Sobolev spaces for singular perturbation of Laplace operato

TL;DR

This paper extends the theory of Sobolev spaces associated with singular perturbations of the Laplace operator in 2D to the $L^r$ setting, providing new representations and applications to nonlinear Schrödinger equations.

Contribution

It introduces an extended $L^r$ theory for perturbed Sobolev spaces related to singular Laplace perturbations, including function representations and well-posedness results.

Findings

Extended $L^r$ theory for perturbed Sobolev spaces.

Representation of functions in $H^{1,r}_eta$ for $r > 2$.

Application to local well-posedness of nonlinear Schrödinger equations.

Abstract

We study the perturbed Sobolev space $H^{1,r}_\alpha$, $r \in (1,\infty),$ associated with singular perturbation $\Delta_\alpha$ of Laplace operator in Euclidean space of dimension $2.$ The main results give the possibility to extend the $L^2$ theory of perturbed Sobolev space to the $L^r$ case. When $r \in (2,\infty)$ we have appropriate representation of the functions in $H^{1,r}_\alpha$ in regular and singular part. An application to local well - posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.