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
DOI: https://doi.org/10.1090/S0002-9939-1975-0387815-9
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?)

  • [1] G. Hellwig, Partial differential equations: An introduction, Blaisdell, Waltham, Mass., 1964. MR 30 #3286. MR 0173071 (30:3286)
  • [2] L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221.
  • [3] -, An introduction to complex analysis in several variables, Van Nostrand, Princeton, N. J., 1966. MR 34 #2933. MR 0203075 (34:2933)
  • [4] B. 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. MR 19, 280. MR 0086990 (19:280a)
  • [5] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 35 #1420. MR 0210528 (35:1420)

Similar Articles

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

Retrieve articles in all journals with MSC: 35K05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0387815-9
Keywords: Nonuniqueness for initial value problem, simply connected, exponential-polynomial solutions
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society