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 ${\left \| {\nabla \psi } \right \|^2} \equiv \smallint {}_R{(\partial \psi /\partial {x_1})^2} + {(\partial \psi /\partial {x_2})^2})d{x_1}d{x_2}$ 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.
- Gerald Rosen, Minimum value for $c$ in the Sobolev inequality $\phi ^{3}\|\leq c\nabla \phi \|^{3}$, SIAM J. Appl. Math. 21 (1971), 30–32. MR 289739, DOI https://doi.org/10.1137/0121004
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, Academic, New York, 1965, p. 672
G. Rosen, Minimum value for c in the Sobolev inequality, SIAM J. Appl. Math. 21, 30–32 (1971)
R. Courant and D. Hilbert, Methods of mathematical physics, vol. I, Interscience, New York, 1953, pp. 297–306
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, Academic, New York, 1965, p. 672
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Article copyright:
© Copyright 1976
American Mathematical Society