Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T17:25:50.529Z Has data issue: false hasContentIssue false
  • English
  • Français

Phragmèn-Lindelöf and Comparison Theorems for Elliptic-Parabolic Differential Equations

Published online by Cambridge University Press:  20 November 2018

J. K. Oddson
J. K. Oddson*
Affiliation:
University of Maryland
Rights & Permissions [Opens in a new window]

Extract

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.

Theorems of Phragmèn-Lindelöf type and other related results for solutions of elliptic-parabolic equations have been given by numerous authors in recent years. Many of these results are based upon the maximum principle and the use of auxiliary comparison functions which are constructed as supersolutions of the equations under various conditions on the coefficients. In this paper we present an axiomatized treatment of these topics, replacing specific hypotheses on the nature of the coefficients of the equations by a single assumption concerning the maximum principle and another concerning the existence of positive supersolutions, in terms of which the theorems are stated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 864 - 871
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Friedman, A., On two theorems of Phragmèn-Lindelöf for linear elliptic and parabolic differential equations of the second order, Pacific J. Math., 7 (1957), 1563–1575.Google Scholar
2. Gilbarg, D., The Phragmèn-Lindelöf theorem for elliptic partial differential equations, J. Rat. Mech. Anal., 1 (1952), 411–417.Google Scholar
3. Gilbarg, D. and Serrin, J., On isolated singularities of solutions of second order elliptic equations, J. Analyse Math., 4 (1956), 309–340.Google Scholar
4. Hopf, E., Remarks on the preceding paper by D. Gilbarg, J. Rat. Mech. Anal., 1 (1952), 418–424.Google Scholar
5. Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus., Sitzungsb. Preuss. Akad. Wiss. Berlin, 19 (1927), 147–152.Google Scholar
6. Meyers, N. and Serrin, J., The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech., 9 (1960), 513–538.Google Scholar
7. Miranda, C., Equazioni aile derivate parziali di tipo ellitico (Berlin, 1955).Google Scholar
8. Oddson, J. K., Maximum principles and Phragmèn-Lindelöf theorems for second order differential equations, Technical Note BN-409, Inst, for Fluid Dynamics and Appl. Math., Univ. of Maryland (1965).Google Scholar
9. Pucci, C., Operatori ellittici estremanti, Ann. Mat. Pura Appl., 72 (1966), 141–170.Google Scholar