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February  2019, 39(2): 1101-1115. doi: 10.3934/dcds.2019046

## On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

 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  April 2018 Revised  August 2018 Published  November 2018

Fund Project: 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.

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.

Citation: Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
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