Abstract
For any decreasing sequence of bounded finely open sets Di ⊂ RN it is shown that, for every n, the nth eigenvalue λn ( Di) of the Dirichlet laplacian A ( Di ) on Di converges to λn ( D ) (the nth eigenvalue of A ( D ) ), where D denotes the fine interior of ∩ Di. Likewise, A ( Di )-1 → A ( D )-1 in operator norm. Similar results are obtained for increasing or just order convergent sequences ( Di ). Furthermore, A ( D )-1 is identified with the integral operator on L2 ( D ) whose kernel is Green's function for D.
Similar content being viewed by others
References
Adams, D. R. and Hedberg, L. I.: Function Spaces and Potential Theory, Springer, Berlin, 1996.
Courant, R.: ‘¨Uber die Eigenwerte bei Differentialgleichungen der mathematischen Physik’, Math.Z.7(1920), 1–57.
Davies, E. B.: Spectral Theory and Differential Operators, Cambridge Univ. Press, 1995.
Debiard, A. and Gaveau, B.: ‘Potentiel fin et alg`ebres de fonctions analytiques I’, J.Functional Analysis 16(1974), 289–304.
Deny, J.: ‘Les potentiels d'´energie finie’, Acta Math. 82(1950), 107–183.
Deny, J. and Lions, J. L.: ‘Les espaces du type de Beppo Levi’, Ann.Inst.Fourier Grenoble 5(1953- 54), 305–370.
Doob, J. L.: Classical Potential Theory and Its Probalistic Counterpart, Springer, Berlin, 1984.
Feyel, D. and de La Pradelle, A.: ‘Le role des espaces de Sobolev en topologie fine’, in S´eminaire de Th´eorie du Potential, Paris, No. 2, 43–61, Lecture Notes in MathematicsNo. 563, Springer, Berlin, 1976.
Fuglede, B.: Finely Harmonic Functions, Lecture Notes in Mathematics, No. 289, Springer, Berlin, 1972.
Fuglede, B.: ‘Sur la fonction de Green pour un domaine fin’, Ann.Inst.Fourier Grenoble 25(3- 4) (1975), 201–206.
Fuglede, B.: ‘Fonctions BLD et fonctions finement surharmoniques’ in S´eminaire de Th´eorie du Potential, Paris, No. 6, 126–157, Lecture Notes in MathematicsNo. 906, Springer, Berlin, 1982.
Fuglede, B.: ‘Integral representation of fine potentials’, Math.Ann. 262(1983), 191–214.
Fuglede, B.: Continuous Domain Dependence of the Eigenvalues of the Dirichlet Laplacian and Related Operators in Hilbert Space, J. Functional Analysis (to appear).
Hansen, W.: ‘Valeurs propres pour l'op´erateur de Schroedinger’, in S´eminaire de Th´eorie du Potential, Paris, No. 9, 117–134, Lecture Notes in MathematicsNo. 1393, Springer, Berlin, 1989.
Hedberg, L. I.: ‘Approximation by harmonic functions, and stability of the Dirichlet problem’, Expo.Math. 11(1993), 193–259.
Rellich, F.: ‘Ein Satz ¨uber mittlere Konvergenz’, Nachr. Akad.Wiss. G¨ottingen, Math.Phys. Kl., 1930, 30–35.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fuglede, B. The Dirichlet Laplacian on Finely Open Sets. Potential Analysis 10, 91–101 (1999). https://doi.org/10.1023/A:1008630909423
Issue Date:
DOI: https://doi.org/10.1023/A:1008630909423