On the second inner variations of Allen–Cahn type energies and applications to local minimizers

Abstract

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 [8] 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 [6] 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 [16].