Indefinite elliptic boundary value problems on irregular domains

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.