Integral inequalities of Hardy and Poincaré type

Boas, Harold P.; Straube, Emil J.

doi:10.1090/S0002-9939-1988-0938664-0

Integral inequalities of Hardy and Poincaré type
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by Harold P. Boas and Emil J. Straube

Proc. Amer. Math. Soc. 103 (1988), 172-176

DOI: https://doi.org/10.1090/S0002-9939-1988-0938664-0

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Abstract:

The Poincaré inequality $||u|{|_p} \leq C||\nabla u|{|_p}$ 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 $||u|{|_p} \leq C||{\delta ^\beta }\nabla u|{|_p}$, where $\delta$ is the distance to the boundary, and the positive exponent $\beta$ depends on the smoothness of the boundary.

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