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Indefinite elliptic boundary value problems on irregular domains


Authors: Jacqueline Fleckinger and Michel L. Lapidus
Journal: Proc. Amer. Math. Soc. 123 (1995), 513-526
MSC: Primary 35P15; Secondary 28A75, 35J25, 47F05
DOI: https://doi.org/10.1090/S0002-9939-1995-1231296-2
MathSciNet review: 1231296
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish estimates for the remainder term of the asymptotics of the Dirichlet or Neumann eigenvalue problem

$\displaystyle - \Delta u(x) = \lambda r(x)u(x),\quad x \in \Omega \subset {\mathbb{R}^n},$

defined on the bounded open set $ \Omega \subset {\mathbb{R}^n}$; here, the "weight" r is a real-valued function on $ \Omega $ which is allowed to change sign in $ \Omega $ and the boundary $ \partial \Omega $ is irregular. We even obtain error estimates when the boundary is "fractal".

These results--which extend earlier work of the authors [particularly, J. Fleckinger & M. L. Lapidus, Arch. Rational Mech. Anal. 98 (1987), 329-356; M. L. Lapidus, Trans. Amer. Math. Soc. 325 (1991), 465-529]--are already of interest in the special case of positive weights.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1231296-2
Keywords: Dirichlet and Neumann Laplacians, indefinite elliptic problems, variational boundary value problems on irregular domains, indefinite weight functions, asymptotics of eigenvalues, remainder estimate, interplay between the irregularity of the weight and of the boundary, Whitney-type coverings, "fractal" boundaries, vibrations of drums with fractal boundary and with variable mass density
Article copyright: © Copyright 1995 American Mathematical Society

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