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Integral inequalities of Hardy and Poincaré type

Authors: Harold P. Boas and Emil J. Straube
Journal: Proc. Amer. Math. Soc. 103 (1988), 172-176
MSC: Primary 46E35; Secondary 26D10, 35H05
MathSciNet review: 938664
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Abstract: The Poincaré inequality $ \vert\vert u\vert{\vert _p} \leq C\vert\vert\nabla u\vert{\vert _p}$ in a bounded domain holds, for instance, for compactly supported functions, for functions with mean value zero and for harmonic functions vanishing at a point. We show that it can be improved to $ \vert\vert u\vert{\vert _p} \leq C\vert\vert{\delta ^\beta }\nabla u\vert{\vert _p}$, where $ \delta $ is the distance to the boundary, and the positive exponent $ \beta $ depends on the smoothness of the boundary.

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Keywords: Poincaré's inequality, Hardy's inequality, hypoelliptic partial differential equations
Article copyright: © Copyright 1988 American Mathematical Society

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