The Variational Approach to Linear Boundary and Eigenvalue Problems

Part of the Texts and Monographs in Physics book series (TMP)


One of the first areas in which the basic variational concepts and methods are applied is the solution of large, important classes of linear boundary and eigenvalue problems. The results discussed in this chapter follow in part from the results on nonlinear boundary and eigenvalue problems which we shall discuss in Chap. 8; nonetheless we present proofs in this chapter which emphasise the simplicity and elementary character of the variational approach. We start with a variational proof of the spectral theorem for compact self-adjoint operators, followed by a general version of the projection theorem (for convex closed sets).


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  1. 6.1
    Achieser, H.T., Glasmann, T. M.: Theory of linear operators in Hilbert spaces. Pitman, Boston 1981Google Scholar
  2. 6.2
    Hirzebruch, F., Scharlau, W.: Einführung in die Funktionalanalysis. BI Taschenbücher 296, BI, Mannheim 1971Google Scholar
  3. 6.3
    Adams, R. A.: Sobolev spaces. Academic Press, New York 1975zbMATHGoogle Scholar
  4. 6.4
    Courant, R.: Dirichlet’s principle, conformal mapping and minimal surfaces. Springer, Berlin Heidelberg 1977CrossRefGoogle Scholar
  5. 6.5
    Courant, R., Hilbert, D.: Methods of mathematical physics. Wiley-Interscience, New York 1966Google Scholar
  6. 6.6
    Aubin, J.P.: Applied functional analysis. Wiley, New York 1979zbMATHGoogle Scholar
  7. 6.7
    Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Springer, Berlin Heidelberg 1972Google Scholar
  8. 6.8
    Temam, R.: Navier-Stokes equations. North-Holland, Amsterdam 1977zbMATHGoogle Scholar

Further Reading

  1. Necas, J.: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967zbMATHGoogle Scholar
  2. Rektory, K.: Variational methods in mathematics, science and engineering. Reidel, Dordrecht 1980Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

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