Two-step minimization approach to an $L^\infty$-constrained variational problem with a generalized potential

TL;DR

This paper introduces a two-step minimization method for solving an $L^ abla$-constrained variational problem involving generalized potentials, with applications to Dirac delta potentials and trapped modes.

Contribution

It develops a novel two-step minimization approach for $L^ abla$-constrained problems with generalized potentials, including new theoretical tools and applications.

Findings

Established decomposition, comparison, and perturbation principles.

Analyzed minimizers for delta-type potentials.

Studied trapped modes in potential wells.

Abstract

We study a variational problem on $H^1({\mathbb R})$ under an $L^\infty$-constraint related to Sobolev-type inequalities for a class of generalized potentials, including $L^p$-potentials, non-positive potentials, and signed Radon measures. We establish various essential tools for this variational problem, including the decomposition principle, the comparison principle, and the perturbation theorem, which are the basis of the two-step minimization method. As for their applications, we present precise results for minimizers of minimization problems, such as the study of potentials of Dirac's delta measure type and the analysis of trapped modes in potential wells.