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

We investigate the convergence as $p\searrow 1$ 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.

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