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In this paper, we obtain an explicit formula for the discrepancy between thelimit of the second inner variations of $p$-Laplace Allen-Cahn energies and thesecond inner variation of their $\Gamma$-limit which is the area functional.Our analysis explains the mysterious discrepancy term found in our previouspaper \cite{Le} in the case $p=2$. The discrepancy term turns out to be relatedto the convergence of certain 4-tensors which are absent in the usualAllen-Cahn functional. These (hidden) 4-tensors suggest that, in thecomplex-valued Ginzburg-Landau setting, we should expect a differentdiscrepancy term which we are able to identify. Along the way, we partiallyanswer a question of Kohn and Sternberg \cite{KS} by giving a relation betweenthe limit of second variations of the Allen-Cahn functional and the secondinner variation of the area functional at local minimizers. Moreover, ouranalysis reveals an interesting identity connecting second inner variation andPoincar\'e inequality for area-minimizing surfaces with volume constraint inthe work of Sternberg and Zumbrun \cite{SZ2}.

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