$(s,p)$-harmonic approximation of functions of least $W^{s,1}$-seminorm

TL;DR

This paper studies the convergence of minimizers of fractional Sobolev energies as the integrability parameter approaches 1, establishing pointwise, $ ext{Gamma}$-convergence, and analyzing related Euler-Lagrange equations and regularity.

Contribution

It provides a comprehensive analysis of the limit behavior of $W^{s,p}$-energy minimizers as $p$ approaches 1, including convergence, Euler-Lagrange equations, and regularity results.

Findings

Minimizers of $W^{s,p}$-energy converge to those of $W^{s,1}$-energy as $p o 1$

Established $ ext{Gamma}$-convergence of the energies

Analyzed the convergence of Euler-Lagrange equations and regularity of minimizers

Abstract

We investigate the convergence as $p\searrow1$ of the minimizers of the $W^{s,p}$-energy for $s\in(0,1)$ and $p\in(1,\infty)$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $\Gamma$-convergence. We also address the convergence of the corresponding Euler-Lagrange equations, and the equivalence between minimizers and weak solutions. As ancillary results, we study some regularity issues regarding minimizers of the $W^{s,1}$-energy.