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

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