Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Sobolev-type lower bounds on $ \parallel \nabla \psi \parallel ^{2}$ for arbitrary regions in two-dimensional Euclidean space


Author: Gerald Rosen
Journal: Quart. Appl. Math. 34 (1976), 200-202
MSC: Primary 26A86
DOI: https://doi.org/10.1090/qam/473125
MathSciNet review: 473125
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Abstract: This note reports the derivation of lower bounds of the Sobolev type on $ {\left\Vert {\nabla \psi } \right\Vert^2} \equiv \smallint {}_R{(\partial \psi /\partial {x_1})^2} + {(\partial \psi /\partial {x_2})^2})d{x_1}d{x_2}$ for generic real scalar $ \psi = \psi ({x_1},{x_2})$ of function class $ {C^0}$ piecewise $ {C^2}$ which vanish over the boundary of the (bounded or unbounded) region $ R$ in Euclidean 2-space.


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DOI: https://doi.org/10.1090/qam/473125
Article copyright: © Copyright 1976 American Mathematical Society


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