# zbMATH — the first resource for mathematics

A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains. (English) Zbl 0822.31005
It is shown that resolvents of Dirichlet boundary problems depend on the domain in a continuous manner. More precisely, two notions of convergence of the associated operators are investigated: Convergence in the strong resolvent sense is proven using monotone convergence of forms and a suitable representation of Dirichlet boundary conditions. To establish norm resolvent convergence we assume that the domains vary only in a region of finite capacity. Basic to the continuity with respect to norm resolvent sense is a convergence theorem for measure perturbations of Dirichlet forms, which is the main new result of the present article.

##### MSC:
 31C25 Dirichlet forms 60J45 Probabilistic potential theory 47A40 Scattering theory of linear operators 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text:
##### References:
 [1] Adams, R.: Sobolev Spaces. New York: Academic Press 1975 · Zbl 0314.46030 [2] Albeverio, S., Ma, Z.: Perturbation of Dirichlet forms-Lower semiboundedness, Closability, and Form Cores J. Funct. Anal.,99 (1991), 332–356 · Zbl 0743.60071 [3] Baumgärtel, Demuth, M.: Decoupling by a projection. Rep. Math. Phys.15 (1979), 173–186 · Zbl 0426.47007 · doi:10.1016/0034-4877(79)90017-X [4] Baxter, J., DalMaso, G., Mosco, U.: Stopping times and $$\Gamma$$-convergence Trans. Amer. Math. Soc.303 (1987), 1–38 · Zbl 0627.60071 [5] Davies, E. B.: Heat kernels and spectral theory. Cambridge: Cambridge University Press 1989 · Zbl 0699.35006 [6] Defant, A., Floret, K.: Tensor Norms and Operator Ideals. Amsterdam: North Holland 1993 · Zbl 0774.46018 [7] Demuth, M.: On topics in spectral and stochastic analysis for Schrödinger operators. In: Proceedings ”Recent Developments in Quantum Mechanics”, A. Boutet de Monvel et al. (eds.). Kluwer, 1991 · Zbl 0733.60088 [8] Demuth, M. On large coupling operator norm convergence of resolvent differences. J. Math. Phys.32 (1991), 1522–1530 · Zbl 0732.47014 · doi:10.1063/1.529260 [9] Demuth, M., van Casteren, J.: On spectral theory of selfadjoint Feller generators. Rev. Math. Phys.1 (4) (1989), 325–414 · Zbl 0715.60093 · doi:10.1142/S0129055X89000158 [10] Feyel, D.: Ensembles singuliers associées aux espaces de Banach réticules. Ann. Inst. Fourier31, 1 (1981), 192–223 · Zbl 0442.46019 [11] Feyel, D., de la Pradelle, A.: Espaces de Sobolev sur les ouvert fins. C. R. Acad. Sci.280, série A (1975), 1125–1127 · Zbl 0303.35028 [12] Fuglede, B.: The quasi-topology associated with a countably subadditive set-function. Ann. Inst. Fourier21 (1971), 123–179 · Zbl 0197.19401 [13] Fukushima, M.: Dirichlet Forms and Markov Processes Amsterdam: North Holland 1980 · Zbl 0422.31007 [14] Fukushima, M., Le Jan, Y.: On quasi-supports of smooth measures and closability of pre-Dirichlet forms. Osaka J. Math.28 (1991), 837–845 · Zbl 0766.31008 [15] Kato, T.: Perturbation Theory for Linear Operators. 2nd ed. Berlin: Springer-Verlag 1976 · Zbl 0342.47009 [16] Rauch, J., Taylor, M.: Potential and Scattering Theory on Wildly perturbed domains. J. Funct. Anal.18 (1975), 27–59 · Zbl 0293.35056 · doi:10.1016/0022-1236(75)90028-2 [17] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. New York: Academic Press 1978 · Zbl 0401.47001 [18] Simon, B.: A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal.28 (1978), 377–385 · Zbl 0413.47029 · doi:10.1016/0022-1236(78)90094-0 [19] Stollmann, P.: Smooth perturbations of regular Dirichlet forms. Proc. Amer. Math. Soc.117 (1992), 747–752 · Zbl 0768.31009 · doi:10.1090/S0002-9939-1992-1107277-3 [20] Stollmann, P.: Closed ideals in Dirichlet spaces. Potential Analysis2 (1993), 263–268 · Zbl 0784.31009 · doi:10.1007/BF01048510 [21] Stollmann, P.: Scattering at obstacles of finite capacity. J. Funct. Anal.121 (1994), 416–425 · Zbl 0803.47015 · doi:10.1006/jfan.1994.1054 [22] Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Analysis, to appear · Zbl 0861.31004 [23] Sturm, T.: Measures charging no polar sets and additive functionals of Brownian motion. Forum Math4 (1992), 257–297 · Zbl 0745.60080 · doi:10.1515/form.1992.4.257 [24] Weidmann, J.: Stetige Abhängigkeit der Eigenwerte und Eigenfunktionen elliptischer Differen-tialoperatoren vom Gebiet. Math. Scand.54 (1984), 51–69 · Zbl 0526.35061 [25] Weidmann, J.: unpublished. 1992
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.