A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies

TL;DR

This paper introduces a Riemannian autocorrelation function to analyze non-local isoperimetric energies on spheres, connecting geometric measure theory with variational analysis and providing tools for limit analysis as a parameter tends to zero.

Contribution

It develops a novel Riemannian autocorrelation function linked to set geometry, characterizes finite perimeter sets via Lipschitz continuity, and applies these to analyze the asymptotic behavior of non-local energies.

Findings

The autocorrelation function relates to set perimeter via Lipschitz continuity.

The energies can be reformulated in terms of the autocorrelation function.

The limit of energies as the parameter tends to zero is characterized through $ ext{Gamma}$-convergence.

Abstract

We study a family of non-local isoperimetric energies $E_{\gamma,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2 \Delta$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_\Omega$ associated to a measurable set $\Omega\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\ge2$. This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation $BV(M)$ in terms of geodesic difference quotients, we show that $\Omega$ has finite perimeter if and only if $c_\Omega$ is Lipschitz, and we relate the Lipschitz constant to the perimeter of $\Omega$. We show that on the round sphere $E_{\gamma,\varepsilon}$ admits a reformulation in terms of $c_\Omega$, which allows us to compute the limit as $\varepsilon \to 0$ in a variational sense, that is, in the framework of $\Gamma$-convergence.