Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



An approximation theorem of Runge type for the heat equation

Author: B. Frank Jones
Journal: Proc. Amer. Math. Soc. 52 (1975), 289-292
MSC: Primary 35K05
MathSciNet review: 0387815
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

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

  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35K05

Retrieve articles in all journals with MSC: 35K05

Additional Information

Keywords: Nonuniqueness for initial value problem, simply connected, exponential-polynomial solutions
Article copyright: © Copyright 1975 American Mathematical Society