Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-21T22:55:12.500Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hardy–Rellich inequality for polyharmonic operators

Published online by Cambridge University Press:  14 November 2011

Mark P. Owen
Mark P. Owen
Affiliation:
Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Extract

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies, from the case of the Dirichlet Laplacian of a region Ω ⊆ ℝN to that of the higher-order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 825 - 839
Copyright
Copyright © Royal Society of Edinburgh 1999

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

1Davies, E. B.. One-parameter semigroups. London Mathematical Society, Monograph no. 15 (Academic Press, 1980).Google Scholar
2Davies, E. B.. Some norm bounds and quadratic form inequalities for Schrödinger operators II. J. Oper. Theory 12 (1984), 177–196.Google Scholar
3Davies, E. B.. Trace properties of the Dirichlet Laplacian. Math. Z. 188 (1985), 245–251.CrossRefGoogle Scholar
4Davies, E. B.. Heat kernels and spectral theory. Cambridge Tracts in Mathematics, no. 92 (Cambridge University Press, 1989).CrossRefGoogle Scholar
5Davies, E. B.. Spectral theory and differential operators. Cambridge Studies in Advanced Mathematics, no. 42 (Cambridge University Press, 1996).Google Scholar
6Davies, E. B. and Hinz, A. M.. Explicit constants for Rellich inequalities in Lp(Ω). Math. Z. 227 (1998), 511–523.CrossRefGoogle Scholar
7Hardy, G. H.. Note on a theorem of Hilbert. Math. Z. 6 (1920), 314–317.CrossRefGoogle Scholar
8Lenard, A.. Generalization of the Golden-Thompson inequality Tr(eAeB) ≥ TreA+B. Math. J.Indiana Univ. 21 (1971), 457–467.CrossRefGoogle Scholar
9Maz'ya, V. G.. Sobolev spaces (Springer, 1985).Google Scholar
10Opic, B. and Kufner, A.. Hardy-type inequalities. Pitman Research Notes in Mathematics, no. 219 (Longman Scientific and Technical, 1990).Google Scholar
11Reed, M. and Simon, B.. Methods of modern mathematical physics. IV. Analysis of operators (Academic Press, 1978).Google Scholar
12Reed, M. and Simon, B.. Methods of modern mathematical physics. III. Scattering theory (Academic Press, 1979).Google Scholar
13Rellich, F.. Halbbeschränkte Differentialoperatoren höherer Ordnung. In Proc. Int. Congr. Math. 1954, PP. 243–250 (1956).Google Scholar
14Safarov, Yu. and Vassiliev, D.. The asymptotic distribution of eigenvalues of partial differential operators. Translations of Mathematical Monographs, no. 155 (AMS, 1996).CrossRefGoogle Scholar