Indefinite elliptic boundary value problems on irregular domains

Fleckinger, Jacqueline; Lapidus, Michel L.

doi:10.1090/S0002-9939-1995-1231296-2

Indefinite elliptic boundary value problems on irregular domains
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Proc. Amer. Math. Soc. 123 (1995), 513-526 Request permission

Abstract:

We establish estimates for the remainder term of the asymptotics of the Dirichlet or Neumann eigenvalue problem \[ - \Delta u(x) = \lambda r(x)u(x),\quad x \in \Omega \subset {\mathbb {R}^n},\] 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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