Abstract
Let L be a uniformly elliptic second order differential operator with nice coefficients, defined on a smooth, bounded domain in ℝd, d ≥ 2, with either the Dirichlet or an oblique-derivative boundary condition. In this work we study the asymptotics for the principal eigenvalue of L under hard and soft obstacle perturbations. The hard obstacle perturbation of L is obtained by making a finite number of holes with the Dirichlet boundary condition on their boundaries. The main result gives the asymptotic shift of the principal eigenvalue as the holes shrink to points. The rates are expressed in terms of the Newtonian capacity of the holes and the principal eigenfunctions for the unperturbed operator and its formal adjoint. The soft obstacle corresponds to a finite number of compactly supported finite potential wells. Here we only consider the oblique-derivative Laplacian. The main difference from the hard obstacle problem is that phase transitions occur, due to the various scaling possibilities. Our results generalize known results on similar perturbations for selfadjoint operators. Our approach is probabilistic.
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References
M. Flucher, Approximation of Dirichlet eigenvalues on domains with small holes, Journal of Mathematical Analysis and Applications 193 (1995), 169–199.
R. Z. Hasminskii, Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, Translated from the Russian by D. Louvish.
M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain Journal of Mathematics 4 (1974), 511–537, Notes by Jack Macki and Reuben Hersh, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972).
V. A. Marchenko and E. Ya. Khruslov, Homogenization of partial differential equations, Progress in Mathematical Physics, vol. 46, Birkhäuser Boston Inc., Boston, MA, 2006, Translated from the 2005 Russian original by M. Goncharenko and D. Shepelsky.
V. Maz’ya, S. Nazarov and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I, Operator Theory: Advances and Applications, vol. 111, Birkhäuser Verlag, Basel, 2000, Translated from the German by Georg Heinig and Christian Posthoff.
S. Ozawa, Spectra of random media with many randomly distributed obstacles, Osaka Journal of Mathematics 30 (1993), 1–27.
R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995.
R. G. Pinsky, Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture, Journal of Functional Analysis 200 (2003), 177–197.
J. Rauch, The mathematical theory of crushed ice, in Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 370–379.
J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, Journal of Functional Analysis 18 (1975), 27–59.
M. R. Spiegel, Mathematical Handbook of Formulas and Tables, International edition, McGraw-Hill Book Co., Singapore, 1990.
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Ben-Ari, I. The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles. Isr. J. Math. 169, 181–220 (2009). https://doi.org/10.1007/s11856-009-0009-x
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DOI: https://doi.org/10.1007/s11856-009-0009-x