2014/08/27 by Nam Q. Le, Le, Nam Q.
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1408.6481
To be published in Journal de Mathématiques Pures et Appliquées; final version incorporating comments/suggestions from referee reports
arxiv created 2014/10/08 · arxiv updated 2014/10/09
In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of p-Laplace Allen-Cahn energies and the second inner variation of their Γ-limit which is the area functional. Our analysis explains the mysterious discrepancy term found in our previous paper \citeLe in the case p=2. The discrepancy term turns out to be related to the convergence of certain 4-tensors which are absent in the usual Allen-Cahn functional. These (hidden) 4-tensors suggest that, in the complex-valued Ginzburg-Landau setting, we should expect a different discrepancy term which we are able to identify. Along the way, we partially answer a question of Kohn and Sternberg \citeKS by giving a relation between the limit of second variations of the Allen-Cahn functional and the second inner variation of the area functional at local minimizers. Moreover, our analysis reveals an interesting identity connecting second inner variation and Poincaré inequality for area-minimizing surfaces with volume constraint in the work of Sternberg and Zumbrun \citeSZ2.