Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T01:50:00.806Z Has data issue: false hasContentIssue false
  • English
  • Français

A Friedrichs inequality and an application

Published online by Cambridge University Press:  14 November 2011

Roger T. Lewis
Roger T. Lewis
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, Birmingham, Alabama 35294, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

An inequality whose origins date to the work of G. H. Hardy is presented. This Hardy-type inequality applies to derivatives of arbitrary order of functions whose domain is a subset of ℝn. The Friedrichs inequality is a corollary. The result is then used to establish lower bounds on the essential spectra of even-order elliptic partial differential operators on unbounded domains.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 185 - 191
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Birman, M. S.. The spectrum of singular boundary problems. Amer. Math. Soc. Transl. 53 (1966), 23–80; Math. Sb. 55 (1961), 125–174.Google Scholar
2Evans, W. D., Lewis, R. T. and Zettl, A.. Non-self-adjoint operators and their essential spectra. Lecture Notes in Mathematics 1032 (Berlin: Springer, 1984). Edited by Everitt, W. N. and Lewis, R. T.. (Proceedings of the 1982 Dundee Symposium on Differential Equations.)Google Scholar
3Friedrichs, K. O.. Die Randwert- und Eigenwertprobleme aus der Theorie der elastischen Platten. (Anwendung der direkten Methoden der Variationsrechnung.) Math. Ann. 98 (1928), 205–247.CrossRefGoogle Scholar
4Friedrichs, K. O.. Spektraltheorie halbbeschränkter Operatoren mit Anwendung aud die Spektralzerlegung von Differentialoperatoren, I. Math. Ann. 109 (1934), 465–487.Google Scholar
5Friedrichs, K. O.. Spektraltheorie halbbeschränkter Operatoren mit Anwendung auf die Spektralzerlegung von Differentialoperatoren, II. Math. Ann. 109 (1934), 685–713.Google Scholar
6Friedrichs, K. O.. Spektraltheorie halbbesehränkter Operatoren mit Anwendung auf die Spektralzerlegung von Differentialoperatoren, III. Math. Ann. 110 (1935), 777–779.CrossRefGoogle Scholar
7Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge: University Press, 1934).Google Scholar
8Hinton, D. B. and Lewis, R. T.. Discrete spectra criteria for singular differential operators with middle terms. Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347.CrossRefGoogle Scholar
9Kalf, H. and Walter, J.. Strongly singular potentials and essential self-adjointness of singular elliptic operators in . J. Funct. Anal. 10 (1972), 114–130.CrossRefGoogle Scholar
10Lewis, R. T.. Singular elliptic operators of second order with purely discrete spectra. Trans. Amer. Math. Soc. 271 (1982), 653–666.CrossRefGoogle Scholar
11Lewis, R. T.. Applications of a comparison theorem for quasi-accretive operators in a Hilbert space. Lecture Notes in Mathematics 964, 422–434 (Berlin: Springer, 1982). Edited by Everitt, W. N. and Sleeman, B. D..Google Scholar
12Showalter, R. E.. Hilbert space methods for partial differential equations (New York: Pitman, 1977).Google Scholar