On spectral asymptotics for domains with fractal boundaries
The authors consider the spectral properties of the Laplacian for domains with fractal boundaries. They determine the second term in the spectral asymptotics for the counting function \(N(\lambda)\) for three classes of fractal boundaries. They show that the Minkowski dimension gives the appropriate answer for cabbage type of boundaries due to the one-dimensional structure of the cabbage type fractal boundaries. The answer is different for bubble type domains and for planar bubble fractals. One of the main points of the present work is to single out fractals for which spectral asymptotics can be found without assuming self-similarity or geometrical conditions which permit separation of variables. The second point is to determine if the order of the second term of spectral asymptotics is always related to the Minkowski dimension of the boundary. There are many other excellent papers on the subject of spectral asymptotics for manifolds with fractal boundaries; in particular it is known that the modified Weyl-Berry conjecture is false. See, for example, 1) \textit{M. van den Bergh}, Proc. Symp. Pure Math. 57, 11-22 (1995; Zbl 0851.35050), 2) \textit{J. Fleckinger, M. Levitin}, and \textit{D. Vassiliev}, Proc. Lond. Math. Soc., III. Ser. 71, No. 2, 372-396 (1995; Zbl 0836.35058), 3) \textit{M. van den Bergh}, Probab. Theory Relat. Fields 100, No. 4, 439-456 (1994; Zbl 0812.60068), 4) \textit{M. Levitin} and \textit{D. Vassiliev}, Proc. Lond. Math. Soc., III. Ser. 72, No. 1, 188-214 (1996; Zbl 0837.35105).
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