On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

Vladimir Georgiev; Koichi Taniguchi

doi:10.3934/dcds.2019046

Article Contents

2019, Volume 39, Issue 2: 1101-1115. Doi: 10.3934/dcds.2019046

This issue Previous Article Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey Next Article Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations

On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

Vladimir Georgiev^1,2,3, and
Koichi Taniguchi^4,

1.
Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5 Pisa, 56127, Italy
2.
IMI–BAS, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria
3.
Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
4.
Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Received: March 31, 2018

Revised: July 31, 2018

Published: January 2019

The first author was supported in part by Project 2017 "Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari" of INDAM, GNAMPA - Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University and the Project PRA 2018 49 of University of Pisa

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Abstract

The goal of the work is to verify the fractional Leibniz rule for Dirichlet Laplacian in the exterior domain of a compact set. The key point is the proof of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

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Mathematics Subject Classification: Primary: 35K05, 42B37, 35B20; Secondary: 35K20.

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