We study the spectral gap phenomena for weighted $L^p$-Hardy inequalities on $C^{1,\gamma}$-domain with a compact boundary, where $\gamma\in (0,1]$. We show that the weighted Hardy constant is attained by some appropriate minimizer if and only if the spectral gap (the difference between the weighted Hardy constant and the weighted Hardy constant at infinity ) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. In this talk, we will try to understand how the ideas in criticality theory help us to extend the spectral gap phenomena from $C^2$-domains to $C^{1,\gamma}$-domains. This talk is based on the joint work with Yehuda Pinchover, Baptiste Devyver.

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Last updated: 11 Dec 2023