Tohoku Mathematical Journal
2023
March
SECOND SERIES VOL. 75, NO. 1

Tohoku Math. J.
75 (2023), 67-88

Title THE LEIBNIZ RULE FOR THE DIRICHLET AND THE NEUMANN LAPLACIAN

Author Tsukasa Iwabuchi

(Received December 25, 2018, revised November 5, 2021)
Abstract. We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.

Mathematics Subject Classification. Primary 46E35; Secondary 42B35, 42B37.
Key words and phrases. Bilinear estimates, Fractional Laplacian, Dirichlet Laplacian, Neumann Laplacian, Sobolev spaces, Besov spaces.

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