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A simple proof of the generalized Leibniz rule on bounded Euclidean domains

A simple proof of the generalized Leibniz rule on bounded Euclidean domains AbstractThis paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation[P. Constantin and M. Ignatova,Critical SQG in bounded domains,Ann. PDE 2 2016, 2, Article ID 8]in bounded domains, and by two of the authors[Q.-H. Nguyen and J. L. Vázquez,Porous medium equation with nonlocal pressure in a bounded domain,Comm. Partial Differential Equations 43 2018, 10, 1502–1539]in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

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Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2020-0228
Publisher site
See Article on Publisher Site

Abstract

AbstractThis paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation[P. Constantin and M. Ignatova,Critical SQG in bounded domains,Ann. PDE 2 2016, 2, Article ID 8]in bounded domains, and by two of the authors[Q.-H. Nguyen and J. L. Vázquez,Porous medium equation with nonlocal pressure in a bounded domain,Comm. Partial Differential Equations 43 2018, 10, 1502–1539]in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

Journal

Forum Mathematicumde Gruyter

Published: Nov 1, 2021

Keywords: Fractional Laplacian operators on domains; commutator estimates; Leibniz rule; 42B37; 35J25

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