Robin Laplacian in the Large coupling limit: Convergence and spectral asymptotic

Belgacem, Fatem; BelHadjAli, Hichem; BenAmor, Ali; Thabet, Amina

doi:10.2422/2036-2145.201601_008

Cited by 2 publications

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“…It is immediately clear, for example from Figure 1, that there is little hope of extending the simple bound (26) of Theorem 4.2 to the case > 0 as the first m eigenvalues of the DtN map blow up to 1 as approaches the Dirichlet eigenvalues D of multiplicity m from below. Indeed, using the results of [12] or [4] for the asymptotics of low eigenvalues of the Robin problem with parameter ! C1, and the DtN-Robin duality, one can easily see that as !…”

Section: The Case >mentioning

confidence: 99%

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

et al. 2022

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How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.

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Section: The Case >mentioning

confidence: 99%

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

et al. 2022

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“…Let us cite, among others, their connection to parts of stochastic processes established in [Fuk80], their relationship to the construction of Dirichlet-to-Neumann operators [AtE15,Dan14] and of fractional powers of the Laplacian [MO69,CS07]. Traces of forms also appear in the study of problems related to large coupling convergence and spectral asymptotics [BBB14,BBBT18].…”

Section: Introductionmentioning

confidence: 99%

On the construction and convergence of traces of forms

BelHadjAli¹,

BenAmor²,

Seifert³

et al. 2017

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We elaborate a new method for constructing traces of quadratic forms in the framework of Hilbert and Dirichlet spaces. Our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part. We give various situations where the trace can be described more explicitly and compute it for some illustrating examples. We then show that Mosco convergence of Dirichlet forms implies Mosco convergence of a subsequence of their approximating traces.

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Robin Laplacian in the Large coupling limit: Convergence and spectral asymptotic

Cited by 2 publications

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The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

On the construction and convergence of traces of forms

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