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

2019/07/18 by Li, Raphael Feng
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.08028

Abstract

In this paper we firstly study the limit of minimizers of the fractional W^s,p-norms as p→+∞ in De Giorgi sense. In particular, we analyzed the Γ-convergence of non-homogeneous Dirichlet boundary problem for fractional p-Laplacian in this approximation process, and proved that as p→+∞ the minimizers of fractional p-Laplacian with Dirichlet boundary Γ-converges to a minimizer of Hölder ∞-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→ s from above; then we study the approximation process as k→ 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.

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