On D.Y. Gao and X. Lu paper "On the extrema of a nonconvex functional with double-well potential in 1D"

TL;DR

This paper critically analyzes Gao and Lu's 2016 work on nonconvex functionals with double-well potentials, clarifying the role of function space norms and correcting their conclusions about local extrema.

Contribution

It provides a detailed correction and clarification of the original results, emphasizing the importance of the chosen function space norm in the analysis.

Findings

No local extrema for p in [1,4)

Only a local maximizer for p=∞

Original conclusions about multiple extrema are incorrect

Abstract

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995].