Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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.

References [Enhancements On Off] (What's this?)

  • [1] G. Rosen, Minimum value for c in the Sobolev inequality, SIAM J. Appl. Math. 21, 30-32 (1971) MR 0289739
  • [2] R. Courant and D. Hilbert, Methods of mathematical physics, vol. I, Interscience, New York, 1953, pp. 297-306 MR 0065391
  • [3] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, Academic, New York, 1965, p. 672

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

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