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

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.