Counting function asymptotics and the weak Weyl-Berry conjecture for connected domains with fractal boundaries. (English) Zbl 0903.35047
Summary: We study the spectral asymptotics for connected fractal domains and Weyl-Berry conjecture. We prove, for some special connected fractal domains, the sharp estimate for second term of counting function asymptotics, which implies that the weak form of the Weyl-Berry conjecture holds for the case. Finally, we also study a naturally connected fractal domain, and we prove, in this case, the weak Weyl-Berry conjecture holds as well.
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
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