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Poincare's variational problem in potential theory
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2007 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 185, no 1, 143-184 p.Article in journal (Refereed) Published
Abstract [en]

One of the earliest attempts to rigorously prove the solvability of Dirichlet's boundary value problem was based on seeking the solution in the form of a "potential of double layer", and this leads to an integral equation whose kernel is ( in general) both singular and non-symmetric. C. Neumann succeeded with this approach for smoothly bounded convex domains, and H. Poincare, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was ( according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Unfortunately, a rigorous treatment of this problem was not possible with the tools available at that time ( as Poincare was well aware). So far as we know, the only one to propose a rigorous treatment of Poincare's problem was T. Carleman ( in the two-dimensional case) in his doctoral dissertation. Thanks to later developments ( especially concerning Sobolev spaces, and spectral theory of operators on Hilbert space) we can now give a complete, general and rigorous account of Poincare's variational problem, and that is the main object of the present paper. As a by-product, we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar Bergman - Schiffer singular integral equation. We interpret Poincare's variational principle as a non-selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials. We also prove a series of novel spectral analysis facts ( some of them conjectured by Poincare) related to the Poincare - Neumann integral operator. One of the earliest attempts to rigorously prove the solvability of Dirichlet's boundary value problem was based on seeking the solution in the form of a "potential of double layer", and this leads to an integral equation whose kernel is (in general) both singular and non-symmetric. C. Neumann succeeded with this approach for smoothly bounded convex domains, and H. Poincare, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was (according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Unfortunately, a rigorous treatment of this problem was not possible with the tools available at that time (as Poincare was well aware). So far as we know, the only one to propose a rigorous treatment of Poincare's problem was T. Carleman (in the two-dimensional case) in his doctoral dissertation. Thanks to later developments (especially concerning Sobolev spaces, and spectral theory of operators on Hilbert space) we can now give a complete, general and rigorous account of Poincare's variational problem, and that is the main object of the present paper. As a by-product, we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensins the basic properties of the analogue of the planar Bergman-Schiffer singular integral equation. We interpret Poincare's variational principle as a non-selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials. We also prove a series of novel spectral analysis facts (some of them conjectured by Poincare) related to the Poincare-Neumann integral operator.

Place, publisher, year, edition, pages
2007. Vol. 185, no 1, 143-184 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-37048DOI: 10.1007/s00205-006-0045-1ISI: 000246221300004Scopus ID: 2-s2.0-34248171616OAI: oai:DiVA.org:kth-37048DiVA: diva2:431907
Available from: 2011-07-27 Created: 2011-07-27 Last updated: 2017-12-08Bibliographically approved

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