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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

A sharp pointwise bound for functions with $ L\sp 2$-Laplacians on arbitrary domains and its applications


Author: Wenzheng Xie
Journal: Bull. Amer. Math. Soc. 26 (1992), 294-298
MSC (2000): Primary 26D15; Secondary 35Q53
DOI: https://doi.org/10.1090/S0273-0979-1992-00279-3
MathSciNet review: 1126088
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Abstract: For all functions on an arbitrary open set $ \Omega \subset {R^3}$ with zero boundary values, we prove the optimal bound

$\displaystyle \mathop {{\text{sup}}}\limits_\Omega \vert u\vert \leq {(2\pi )^{... ... }\vert\nabla u{\vert^2}dx{\smallint _\Omega }\vert\Delta u{\vert^2}dx)^{1/4}}.$

The method of proof is elementary and admits generalizations. The inequality is applied to establish an existence theorem for the Burgers equation.

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DOI: https://doi.org/10.1090/S0273-0979-1992-00279-3
Article copyright: © Copyright 1992 American Mathematical Society