Semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition

TL;DR

This paper proves the semiclassical limit of the Gross-Pitaevskii equation with Dirichlet boundary conditions in 3D, addressing boundary layer issues and partially resolving an open question in the field.

Contribution

It provides a rigorous justification of the semiclassical limit under small initial data assumptions, handling boundary layers in the WKB expansion.

Findings

Successfully justified the semiclassical limit with Dirichlet boundary conditions.

Addressed boundary layer challenges in the WKB expansion.

Partially solved an open problem from previous literature.

Abstract

In this paper, we justify the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition on the 3-D upper space under the assumption that the leading order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in \cite{CR2009, PNB2005} concerning the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition.