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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.
Forum Mathematicum – de Gruyter
Published: Nov 1, 2021
Keywords: Fractional Laplacian operators on domains; commutator estimates; Leibniz rule; 42B37; 35J25
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