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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 correspondingEuler-Lagrange equations, and the equivalence between minimizers and weaksolutions. As ancillary results, we study some regularity issues regardingminimizers of the $W^{s,1}$-energy.

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