Local Lipschitz continuity of the inverse of the fractional p-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains. (April 2016)

• Record Type: Journal Article
• Authors: Warma, Mahamadi
• Is Part Of: Nonlinear analysis. Volume 135(2016)
• Page Start: 129
• Page End: 157
• Publication Date: 2016-04
• Subjects: 35R11 -- 35J60 -- 35K55 -- 35B65 -- 47J30; Fractional p-Laplace operator -- Nonlocal quasi-linear and nonlinear elliptic type equations -- Existence and regularity of solutions -- Local Lipschitz continuity -- Parabolic equations -- Continuous dependence of solutions -- Nonlinear submarkovian semigroups -- Ultracontractivity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.01.022
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let p ∈ ( 1, ∞ ), s ∈ ( 0, 1 ) and Ω ⊂ R N an arbitrary bounded open set. In the first part we consider the inverse Φ s, p : = [ ( − Δ ) p, Ω s ] − 1 of the fractional p -Laplace operator ( − Δ ) p, Ω s with the Dirichlet boundary condition. We show that in the singular case p ∈ ( 1, 2 ), the operator Φ s, 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 > s p, if 2 N N + 2 s < p < 2, 0 ≤ q ≤ N p N − s p − 2 and α, β are small constants, then the nonlinear problem ( − Δ ) p s u = α ∣ u ∣ q u + β ∣ u ∣ N p N − s p − 2 u + h in Ω, u = 0 on R N ∖ Ω, has at least one weak solution. In the second part of the paper, we prove that the operator − ( − Δ ) p, Ω s generates a (nonlinear) submarkovian semigroup ( S s, p ( t ) ) t ≥ 0 on L 2 ( Ω ) . If p ∈ [ 2, ∞ ) and s p < N, we obtain that S s, p ( t ) satisfies the following ( L q − L ∞ ) -Hölder type estimate: there exists a constant C > 0 such that for every t > 0 and u, v ∈ L q ( Ω ) ( q ≥ 2 ), we have ‖ S s, p ( t ) u − S s, p ( t ) v ‖ L ∞ ( Ω ) ≤ C ∣ Ω ∣ β ( s ) t − δ ( s ) ‖ u − v ‖ L q ( Ω ) γ ( s ) where β ( s ), δ ( s ) and γ ( s ) are explicit constants depending only on N, s, p, q .