Local Lipschitz continuity of the inverse of the fractional [formula omitted]-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains

Abstract

Let p∈(1,∞), s∈(0,1) and Ω⊂RN an arbitrary bounded open set. In the first part we consider the inverse Φs,p:=[(−Δ)p,Ωs]−1 of the fractional p-Laplace operator (−Δ)p,Ωs with the Dirichlet boundary condition. We show that in the singular case p∈(1,2), the operator Φs,p is locally Lipschitz continuous on L∞(Ω) and that global Lipschitz continuity cannot be achieved. We use this result to show that in the case N>sp, if 2NN+2s<p<2, 0≤q≤NpN−sp−2 and α,β are small constants, then the nonlinear problem (−Δ)psu=α∣u∣qu+β∣u∣NpN−sp−2u+hin Ω,u=0 on RN∖Ω, has at least one weak solution. In the second part of the paper, we prove that the operator −(−Δ)p,Ωs generates a (nonlinear) submarkovian semigroup (Ss,p(t))t≥0 on L2(Ω). If p∈[2,∞) and sp<N, we obtain that Ss,p(t) satisfies the following (Lq−L∞)-Hölder type estimate: there exists a constant C>0 such that for every t>0 and u,v∈Lq(Ω) (q≥2), we have ‖Ss,p(t)u−Ss,p(t)v‖L∞(Ω)≤C∣Ω∣β(s)t−δ(s)‖u−v‖Lq(Ω)γ(s) where β(s),δ(s) and γ(s) are explicit constants depending only on N,s,p,q.