Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T16:43:24.942Z Has data issue: false hasContentIssue false
  • English
  • Français

The spectrum of the self-adjoint partial differential equation in any domain

Published online by Cambridge University Press:  24 October 2008

S. W. P. Steen
S. W. P. Steen
Affiliation:
Christ's College
Get access Rights & Permissions [Opens in a new window]

Extract

The object of this paper is to obtain the general solution to the self-adjoint partial differential equation in n dimensions

where pij, q and ρ and bounded, continuous functions of (x1,…, xn) in a domain D and on its boundary, and where ∑pijXiXj≥0 for all (x1,…, xn) of D and all X1,…, Xn. The domain D is an n-dimensional domain and may be either the whole or part of a Riemann surface space of n dimensions. Its boundary is to consist of any number, zero, finite or enumerable, of continuous continua of n − 1 dimensions. These terms will be explained in paragraph II. The solution u = u(x1,…, xn; t) will be valid for (x1,…, xn) in D and t ≥ 0, and will satisfy boundary conditions of the type or similar, these conditions becoming identical at any part of the boundary of D that lies at infinity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 1 , January 1933 , pp. 23 - 44
Copyright
Copyright © Cambridge Philosophical Society 1933

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

* Hellinger, E., Journ. f. Math. 136 (1909), 210–271Google Scholar; Hahn, H., Monats. Math. Phys. 23 (1912), 161–224CrossRefGoogle Scholar; Hyslop, J., Proc. London Math. Soc. (2), 24 (1926), 264–304.CrossRefGoogle Scholar

* Carathéodory, , Math. Ann. 73 (1913), 323–370.CrossRefGoogle Scholar

* The conditions of continuity as used in a former paper are not bounded (Steen, , Proc. Camb. Phil. Soc. 28 (1932), 31–32)CrossRefGoogle Scholar, they should therefore be altered to the above form.

* See Steen, , Proc. Camb. Phil. Soc. 28 (1932), 27–29.CrossRefGoogle Scholar

* I am indebted to Mr Cornock of the General Electric Company for bringing to my notice systems of equations.