Reconstructing double-well potentials from transition layers in long-range phase coexistence models

TL;DR

This paper addresses the inverse problem of reconstructing the double-well potential in phase coexistence models from prescribed transition layers with long-range interactions, revealing how decay rates influence potential regularity.

Contribution

It introduces a method to reconstruct the potential's properties from transition layer decay, highlighting the delicate dependence in long-range interaction models.

Findings

Decay rate of transition layer determines potential regularity.

Existence of specific patterns and degeneracies in potential reconstruction.

Long-range interactions require careful analysis of layer-derivative dependence.

Abstract

In models of phase coexistence, the precise form of the double-well potential is of central importance, yet it cannot be derived from first principles. In this paper, we investigate an inverse problem: starting from a prescribed transition layer with power-type decay at infinity, we reconstruct the structural properties of the associated double-well potential. We focus on the case of long-range interactions, where the dependence of the potential on the layer and its derivatives is particularly delicate. Our analysis establishes a correspondence between the decay rate of the transition layer and the regularity of the potential, revealing the existence of specific patterns and the possible emergence of degeneracies.