Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T21:47:12.904Z Has data issue: false hasContentIssue false
  • English
  • Français

REGULARITY OF BOUNDARY POINTS IN THE DIRICHLET PROBLEM FOR THE HEAT EQUATION

Part of: Higher-dimensional theory Parabolic equations and systems

Published online by Cambridge University Press:  27 August 2014

NEIL A. WATSON
NEIL A. WATSON*
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the null limit hypothesis, in the definition of a barrier, can be relaxed for normal boundary points that satisfy a mild additional condition. We also give a simple necessary and sufficient condition for the regularity of semi-singular boundary points.

Keywords

heat equationregularityDirichlet problembarrierPWB solution

MSC classification

Secondary: 35K05: Heat equation 31B05: Harmonic, subharmonic, superharmonic functions 31B20: Boundary value and inverse problems 31B25: Boundary behavior
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 476 - 485
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Abdulla, U. G., ‘First boundary value problem for the diffusion equation. 1. Iterated logarithm test for the boundary regularity and solvability’, SIAM J. Math. Anal. 34 (2003), 1422–1434.CrossRefGoogle Scholar
Abdulla, U. G., ‘Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation’, Bound. Value Probl. 2005 (2005), 181–199.CrossRefGoogle Scholar
Bauer, H., Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics, 22 (Springer, Berlin, 1966).Google Scholar
Doob, J. L., Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der mathematischen Wissenschaften, 262 (Springer, New York, 1984).CrossRefGoogle Scholar
Effros, E. G. and Kazdan, J. L., ‘On the Dirichlet problem for the heat equation’, Indiana Univ. Math. J. 20 (1970-1), 683–693.Google Scholar
Evans, L. C. and Gariepy, R. F., ‘Wiener’s criterion for the heat equation’, Arch. Ration. Mech. Anal. 78 (1982), 293–314.Google Scholar
Lanconelli, E., ‘Sul problema di Dirichlet per l’equazione del calore’, Ann. Mat. Pura Appl. 97 (1973), 83–114.Google Scholar
Landis, E. M., ‘Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation’, Dokl. Akad. Nauk SSSR 185 (1969), 517–520; Soviet Math. Dokl. 10 (1969), 380–384.Google Scholar
Petrowsky, I., ‘Zur ersten Randwertaufgabe der Wärmeleitungsgleichung’, Compositio Math. 1 (1935), 383–419.Google Scholar
Pini, B., ‘Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico’, Rend. Semin. Mat. Univ. Padova 23 (1954), 422–434.Google Scholar
Sternberg, W., ‘Über die Gleichung der Wärmeleitung’, Math. Ann. 101 (1929), 394–398.CrossRefGoogle Scholar
Watson, N. A., ‘Green functions, potentials, and the Dirichlet problem for the heat equation’, Proc. Lond. Math. Soc. (3) 33 (1976), 251–298; Proc. Lond. Math. Soc. (3) 37 (1978), 32–34 (corrigendum).Google Scholar
Watson, N. A., Introduction to Heat Potential Theory, Mathematical Surveys and Monographs, 182 (American Mathematical Society, Providence, RI, 2012).Google Scholar