On the negative spectrum of the Robin Laplacian in corner domains

Bruneau, Vincent; Popoff, Nicolas

doi:10.2140/apde.2016.9.1259

Cited by 26 publications

(45 citation statements)

References 31 publications

(43 reference statements)

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“…Schrödinger operators with singular potential, modeled by δinteractions supported on smooth cones, are investigated in [4,25] whereas the case of the Laplacian with Robin boundary conditions in smooth conical domains is dealt with in [8,30]. Finally, for problems related with polyhedral geometries, let us mention [7] where the magnetic Laplacian in three-dimensional corner domains is studied as well as [9] where the bottom of the essential spectrum of the Robin Laplacian is characterized for polyhedral cones.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Dirichlet Spectrum of the Fichera Layer

Dauge

Lafranche

Ourmières-Bonafos

2018

Integr. Equ. Oper. Theory

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We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain can be viewed as an octant from which another "parallel" octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer.We show that the essential spectrum of the Laplacian on such a domain is a half-line and we characterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm-Liouville operator. By finite element computations taking singularities into account, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.2010 Mathematics Subject Classification. 35J05, 35P15, 35Q40, 81Q10, 65M60.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Dirichlet Spectrum of the Fichera Layer

Dauge

Lafranche

Ourmières-Bonafos

2018

Integr. Equ. Oper. Theory

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“…Denote q := p p−1 the Hölder conjugate of p. In this note we prove the following result: Theorem 1. As α tends to +∞ there holds (1) Λ(α) = (1 − p)α q + o(α q ).…”

Section: Introductionmentioning

confidence: 99%

“…It seems that the asymptotic behavior for large α was first addressed by Lacey, Ockedon and Sabina [8] for the linear situation (p = 2), and C 1 domains represent a borderline case. On one hand, the asymptotic behavior (1) is not valid for non-smooth domains [1,6,9]. On the other hand, for C 1,1 domains one has Λ(α) = (1 − p)α q − Hα + o(α) with H being the maximum mean curvature of the boundary [7], which is much more detailed, but the proof depends heavily on the existence of a tubular neighborhood of the boundary and on the regularity of boundary curvatures.…”

Section: Introductionmentioning

confidence: 99%

An eigenvalue estimate for a Robin $p$-Laplacian in $C^1$ domains

Pankrashkin

2020

Proc. Amer. Math. Soc.

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Let Ω ⊂ R n be a bounded C 1 domain and p > 1. For α > 0, define the quantitywith ds being the hypersurface measure, which is the lowest eigenvalue of the p-laplacian in Ω with a non-linear α-dependent Robin boundary condition. We show the asymptotics Λ(α) = (1 − p)α p/(p−1) + o(α p/(p−1) ) as α tends to +∞. The result was only known for the linear case p = 2 or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for C 1,λ domains.2010 Mathematics Subject Classification. Primary: 35J92, 35P15, 49R05, 49J40, 35J05.

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“…On the other hand, only partial results are available for domains with a non-smooth boundary, cf. [8,23,32].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of Robin Laplacians in infinite sectors

Khalile

Pankrashkin

2017

Mathematische Nachrichten

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Abstract. For α ∈ (0, π), let Uα denote the infinite planar sector of opening 2α, Uα = (x 1 , x 2 ) ∈ R 2 : arg(x 1 + ix 2 ) < α , and T γ α be the Laplacian in L 2 (Uα), T γ α u = −∆u, with the Robin boundary condition ∂ν u = γu, where ∂ν stands for the outer normal derivative and γ > 0. The essential spectrum of T γ α does not depend on the angle α and equals [−γ 2 , +∞), and the discrete spectrum is non-empty iff α < π 2 . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for α ≥ π 6 . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by κ/α with a suitable κ > 0, and the nth eigenvalue En(T γ α ) of T γ α behaves asand admits a full asymptotic expansion in powers of α 2 . The eigenfunctions are exponentially localized near the origin. The results are also applied to δ-interactions on star graphs.

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On the negative spectrum of the Robin Laplacian in corner domains

Cited by 26 publications

References 31 publications

Dirichlet Spectrum of the Fichera Layer

Dirichlet Spectrum of the Fichera Layer

An eigenvalue estimate for a Robin $p$-Laplacian in $C^1$ domains

Eigenvalues of Robin Laplacians in infinite sectors

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