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Integral Equations and Eigenvalue Problems

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Partial Differential Equations

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Abstract

An equation of the type

$$u(x) = \lambda \int\limits_\Omega {K(x;y)u(y)dy + f(x)}$$
(1.1)

is a Fredholml integral equation with kernel K(·; ·).Here Ω is a bounded region of R N, N ≥ 1, with boundary ∂Ω of class C 1.

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References

  1. Erik Ivar Fredholm, 1866–1927.

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  2. The assumptions (1.3)—(1.4), and their analog for (1.1)’, arise naturally from the integral equations (6.2), (7.4) and (9.2) of Chapter III. However, they are somewhat stronger than needed. The theory of existence and uniqueness of solutions, developed in the next sections, requires only that K(.; •) satisfies (1.2). Indeed the same theory could be developed for kernels satisfying only one of (1.2). See Riesz-Nagy, [32] pp. 143–192.

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  3. Precompact here is meant in the topology of the uniform convergence in Ω.

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  4. Kernels of this kind are also referred to as degenerate or kernels of finite rank. The reason for this terminology will be apparent from the argument of Section 14 (see also Remark 14.1), We have preferred the less standard but more suggestive terminology of separable.

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  5. See 4.1 of the Complements.

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  6. We say that a complex number)r E C is an eigenvalue of the matrix [aij] if it is a solution of the algebraic equation A more common definition is that a complex number ii E Cis an eigenvalue of [a] if it is a solution of the algebraic equation det (µI— [aij]) = O. The motivation for our definition will be apparent as we proceed and affords a more streamlined presentation of the theory of existence of solutions to integral equations.

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  7. See Cartan, [2], page 41.

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  8. I. Fredholm, Sur une nouvelle méthode pour la résolution du problème de Dirichlet, Kong. Vetenskaps-Akademiens Fröh. Stockholm, (1900), pp. 39–46; I. Fredholm, Sur une classe d’équations fonctionnelles, Acta Math. Vol. 27 (1903), pp. 365–390. See also the monographs of Tricorni, [38] and Mikhlin, [29].

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  9. See Section 6 of the Complements.

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  10. The method of approximating a general kernel with a separable one in some suitable topology is due to E. Schmidt, Anflösung der allgemeinen linearen integralgleichungen, Math. Annalen, Band 64, (1907), pp. 161–174. See also J. Radon, Über lineare Funktionaltransformationen und Funktionalgleichungen, Sitzsber. Akad. Wiss. Wien, #128 (1919), pp. 1083–1121.

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  11. For N = 2 see Problem 7.1 of the Complements.

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  12. See 8.5 of the Complements of Chapter II.

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  13. See (2.4) and (2.5) of Chapter II.

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  14. See Section 3.3 of the Complements in the Preliminaries.

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  15. This is meant in the sense of (i) and (ii) of Section 10 of Chapter III. Actually A compact implies that also A* is compact. See, for example, F. Riesz—B. Nagy, [32] or K. Yoshida, [41].

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  16. See Section 4 of the Complements in the Preliminaries.

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  17. The inner product of vectors a = (al, aZ,…, a n ) and b =(b1 bZ,…, bn) in Cn, is defined.

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  18. Any kernel K(·; ·). that can be decomposed, as in (9.11)-(9.12), for all s E (0, 1), generates, via (8.5), a compact operator in L2(t2). See Riesz-Nagy, [32], page 177. Compactness methods in integral equations are due to Frigyes (Frédéric) Riesz, 1880–1956; F. Riesz, Über lineare Functionalgleichungen, Acta Math. #41, (1918), pp. 71–98.

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  19. Even though K(-;.) is symmetric, the kernel K 0 (.; •) need not be symmetric, and in general.

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  20. See Remark 10.1.

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  21. See Problem 10.2 of the Complements.

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  22. The idea of using the extremal problem (11.2) to find the first eigenvalue is due to Hilbert. The method applies to general, linear, symmetric, compact operators in L 2 (S2); D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen,(Leipzig, 1912). See also F. Reillich, Spektraltheorie in nicht-separablen Räumen, Math. Annalen, Band 110, (1934), pp. 342–356.

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  24. See also 5.9 of the Complements of Chapter II.

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  26. Angles are counted counterclockwise starting from the positive direction of the horizontal axis.

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  27. Vito Volterra, 1860–1940.

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  29. See Cartan, [2], the example on page 107.

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  31. It is natural to ask whether integral equations of the type of (1.9c)’, set in the unbounded domain (0, oo), have a solution if the kernel K (·; ·). does not vanish for y > x. It turns out that some decay has to imposed on K (·; ·). For kernels of the type K(x; y) = K (x — y) and a theory is developed: N. Wiener and E. Hopf, Über eine Klasse singulärer Integralgleichungen, Sitzungsber. Preuss. Akad. der Wiss.,1931, pp. 695. See also G. Talenti, Sulle equazioni integrali di Wiener—Hopf, Boll. Un. Mat. Ital., (4) #7, Suppl. fasc. 1 (1973), pp. 18–118.

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  33. Karl Theodor Wilhelm Weierstrass, 1815–1897. The theorem was proved by Weierstrass for functions of one variable: K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Mathematische Werke, Band 3, Abhandlugen III, pp. 1–37 (Sitzungsberichte, Kön. Preussischen Akad. der Wissenschaften, July 9–30, 1885), and extended to functions of several variables by Marshall H. Stone, 1903–1989; M.H. Stone, Generalized Weierstrass approximation theorem, Math. Magazine, Vol. 21, 1947/1948, pp. 167–184, and pp. 237–254.

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  34. Friedrich Wilhelm Bessel, 1784–1817.

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  35. Marc-Antoine Parseval des Chêmes, 1755–1833.

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  36. See Section 12 of Chapter II.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 60208-2730, Evanston, IL, USA

    Emmanuele DiBenedetto

  2. Dipartimento di Ingegneria, IIa Università di Roma, Tor Vergata, 00133, Rome, Italy

    Emmanuele DiBenedetto

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  1. Emmanuele DiBenedetto
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DiBenedetto, E. (1995). Integral Equations and Eigenvalue Problems. In: Partial Differential Equations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-2840-5_5

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  • DOI: https://doi.org/10.1007/978-1-4899-2840-5_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4899-2842-9

  • Online ISBN: 978-1-4899-2840-5

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