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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An approximation theorem of Runge type for the heat equation
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by B. Frank Jones PDF
Proc. Amer. Math. Soc. 52 (1975), 289-292 Request permission


If $\Omega$ is an open subset of ${{\mathbf {R}}^{n + 1}}$, the approximation problem is to decide whether every solution of the heat equation on $\Omega$ can be approximated by solutions defined on all of ${{\mathbf {R}}^{n + 1}}$. The necessary and sufficient condition on $\Omega$ which insures this type of approximation is that every section of $\Omega$ taken by hyperplanes orthogonal to the $t$-axis be an open set without “holes,” i.e., whose complement has no compact component. Part of the proof involves the Tychonoff counterexample for the initial value problem.
  • Günter Hellwig, Partial differential equations: An introduction, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. Translated by E. Gerlach. MR 0173071
  • L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221.
  • Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
  • Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990
  • Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 52 (1975), 289-292
  • MSC: Primary 35K05
  • DOI:
  • MathSciNet review: 0387815