Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

Savo, Alessandro

doi:10.1016/j.jde.2019.09.013

Cited by 15 publications

(17 citation statements)

References 20 publications

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“…If the domain is convex, such a bound can be inferred from the literature on the Laplacian associated with standard positive Robin boundary condition (e.g. [8,48,61,66]). For example, if Ω is convex then [61,Corollary 3]…”

Section: Propositionmentioning

confidence: 99%

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

Nordmann

2021

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We call pattern any non-constant solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [20] and Matano [50] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the stability of patterns, and the C 1 norm of the nonlinearity.In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the L 2 -norm of the gradient of a solution is controlled by the L 2 -norm of f (u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates.Our approach relies on the introduction of what we call the Robincurvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions.Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [54] and to extend some results on De Giorgi's conjecture to a larger class of domains.

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Section: Propositionmentioning

confidence: 99%

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

Nordmann

2021

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…The latter, which is an immediate consequence of a new lower bound for the first Robin-eigenvalue (Corollary 3.7), was proved by D. Daners [11,Thm. 1.1] for domains in R n and by A. Savo in [40,Thm. 4] for more general manifolds with suitable curvature bounds.…”

Section: Introductionmentioning

confidence: 99%

“…In this case, one can deduce estimates for the first eigenvalues of the Dirichlet and Robin Laplacian, see [19,Cor. 3.2], [40,Cor. 3], [29], [30].…”

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confidence: 99%

New eigenvalue estimates involving Bessel functions

Chami¹,

Ginoux²,

Habib³

2021

Publicacions Matemàtiques

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Given a compact Riemannian manifold (M n , g) with boundary ∂M , we give an estimate for the quotient ∂M f dµ g M f dµ g, where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

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“…Upper and lower bounds on the first eigenvalue in terms of inradius have been developed by Kovařík [34,Theorem 4.5]. His lower bound was recently sharpened by Savo [42,Corollary 3]. For rectangular domains, the latest developments include an analysis of Courant-sharp Robin eigenvalues on the square by Gittins and Helffer [24], and of Pólya-type inequalities for disjoint unions of rectangles by Freitas and Kennedy [18].…”

Section: Introductionmentioning

confidence: 99%

The Robin Laplacian—Spectral conjectures, rectangular theorems

Laugesen

2019

Journal of Mathematical Physics

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The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes.Conjectures with fixed Robin parameter include: a strengthened Rayleigh-Bossel inequality for the first eigenvalue of a convex domain under area normalization; a Szegő-type upper bound on the second eigenvalue of a convex domain; the gap conjecture saying the line segment minimizes the spectral gap under diameter normalization; and the Robin-PPW conjecture on maximality of the spectral ratio for the ball. Questions for a varying Robin parameter include monotonicity of the spectral gap and the spectral ratio, as well as concavity of the second eigenvalue.Results for rectangular domains include that: the square minimizes the first eigenvalue among rectangles under area normalization, when the Robin parameter α ∈ R is scaled by perimeter; that the square maximizes the second eigenvalue for a sharp range of α-values; that the line segment minimizes the Robin spectral gap under diameter normalization for each α ∈ R; and the square maximizes the spectral ratio among rectangles when α > 0. Further, the spectral gap of each rectangle is shown to be an increasing function of the Robin parameter, and the second eigenvalue is concave with respect to α.Lastly, the shape of a Robin rectangle can be heard from just its first two frequencies, except in the Neumann case.

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Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

Cited by 15 publications

References 20 publications

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

New eigenvalue estimates involving Bessel functions

The Robin Laplacian—Spectral conjectures, rectangular theorems

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