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

Abstract

In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\to +\infty$ in De Giorgi sense. In particular, we analyzed the $\Gamma$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\to +\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $\Gamma$-converges to a minimizer of Hölder $\infty$-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.

References in corpus (1)

• Hölder estimates for viscosity solutions of equations of fractional $p$-Laplace type

Cited by in corpus (1)

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