Asymptotics of Dirichlet Problems to Fractional p-Laplacian Functionals-Approach in De Giorgi Sense

In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\to +\mathrm{\infty }$ in De Giorgi sense. In particular, we analyzed the $\mathrm{\Gamma }$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\to +\mathrm{\infty }$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $\mathrm{\Gamma }$-converges to a minimizer of H"{o}lder $\mathrm{\infty }$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-functionals when $k\to s$ from above; then we study the approximation process as $k\to s$ from below of a free fractional $p$-functional, during which we will find some special phenomenon different from the case from above. Both of the way to dispose these two asymptotic directions are in the De Giorgi sense.