Abstract
The aim of this paper is to discuss the main result in Gao and Lu’s paper (Z Angew Math Phys 67:62, 2016). 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 no local extrema under the mentioned constraints for \(p\in [1,\infty )\) and has (up to an additive constant) only a local maximizer for \(p=\infty \), in strong contrast with the conclusions 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 our conclusions are valid for the similar problem treated in Lu and Gao’s preprint (On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995v1).
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Zălinescu, C. On D.Y. Gao and X. Lu paper “On the extrema of a nonconvex functional with double-well potential in 1D”. Z. Angew. Math. Phys. 68, 72 (2017). https://doi.org/10.1007/s00033-017-0813-9
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DOI: https://doi.org/10.1007/s00033-017-0813-9