Optimisation of the lowest Robin eigenvalue in the exterior of a compact set. II: Non-convex domains and higher dimensions. (English) Zbl 1456.35147

Isoperimetric problems for the lowest eigenvalue of the Robin Laplacian on the exterior \(\Omega^{\mathrm{ext}}\) of a bounded, smooth open set \(\Omega \subset \mathbb{R}^d\), \(d \geq 2\), are studied. Denote by \(\lambda_1^\alpha (\Omega^{\mathrm{ext}})\) the minimum of the spectrum of the negative Laplacian in \(L^2 (\Omega^{\mathrm{ext}})\) subject to the boundary condition \[ \frac{\partial u}{\partial n} = \alpha u \quad \text{on}~\partial \Omega, \] where the Robin parameter \(\alpha < 0\) is a constant and \(\frac{\partial}{\partial n}\) denotes the derivative with respect to the outer unit normal vector to \(\Omega\) (i.e. the normal pointing inside \(\Omega^{\mathrm{ext}}\)); in dimension \(d = 2\), \(\lambda_1^\alpha (\Omega^{\mathrm{ext}})\) is always a discrete, negative eigenvalue, while for \(d \geq 3\) this is true for all \(\alpha\) below a certain threshold.
In the first main result of this article, it is shown for \(d = 2\), fixed \(\alpha < 0\) and fixed \(c > 0\) that \[ \max_{\frac{|\partial \Omega|}{N_\Omega} = c} \lambda_1^\alpha (\Omega^{\mathrm{ext}}) = \lambda_1^\alpha (B^{\mathrm{ext}}), \] where the maximum is taken over all smooth, bounded open sets \(\Omega\) consisting of a finite number of connected components (the latter number denoted by \(N_\Omega\)) such that \(\frac{|\partial \Omega|}{N_\Omega} = c\) and \(B\) is the disk with perimeter \(c\). This improves upon an earlier result by the same authors where only convex \(\Omega\) where allowed.
The second main result concerns the higher-dimensional case \(d \geq 3\); here convexity of \(\Omega\) is required. With the notation \[ \mathcal{M} (\partial \Omega) := \frac{1}{|\partial \Omega|} \int_{\partial \Omega} \left( \frac{\kappa_1 + \dots + \kappa_{d - 1}}{d - 1} \right)^{d - 1}, \] where \(\kappa_1, \dots, \kappa_{d - 1}\) denote the principle curvatures of \(\partial \Omega\), the authors prove that, for each \(\alpha < 0\) and \(c > 0\), \[ \max_{\mathcal{M} (\partial \Omega) = c} \lambda_1^\alpha (\Omega^{\mathrm{ext}}) = \lambda_1^\alpha (B^{\mathrm{ext}}), \] where the maximum is taken over all convex, smooth, bounded open sets \(\Omega\) such that \(\mathcal{M} (\partial \Omega) = c\), and \(B\) is the ball with \(\mathcal{M} (\partial B) = c\).
For Part I, see [the authors, J. Convex Anal. 25, No. 1, 319–337 (2018; Zbl 1401.35223)].


35P15 Estimates of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds


Zbl 1401.35223
Full Text: DOI arXiv


[1] Abramowitz, MS; Stegun, IA, Handbook of Mathematical Functions (1964), New York: Dover, New York
[2] Alexandrov, AD, A characteristic property of spheres, Ann. Mat. Pura Appl., 58, 4, 303-315 (1962) · Zbl 0107.15603
[3] Antunes, PRS; Freitas, P.; Krejčiřík, D., Bounds and extremal domains for Robin eigenvalues with negative boundary parameter, Adv. Calc Var., 10, 357-380 (2017) · Zbl 1375.35284
[4] Bareket, M., On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal., 8, 280-287 (1977) · Zbl 0359.35060
[5] Behrndt, J.; Langer, M.; Lotoreichik, V.; Rohleder, J., Quasi boundary triples and semi-bounded self-adjoint extensions, Proc. Roy. Soc. Edinburgh Sect. A, 147, 895-916 (2017) · Zbl 1386.35290
[6] Bossel, M-H, Membranes élastiquement liées: Extension du théoréme de Rayleigh-Faber-Krahn et de l’inégalité de Cheeger, C. R. Acad. Sci. Paris Sér. I Math., 302, 47-50 (1986) · Zbl 0606.73018
[7] Burago, YD; Zalgaller, VA, Geometric Inequalities (1988), Berlin: Springer, Berlin
[8] Daners, D., A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann., 335, 767-785 (2006) · Zbl 1220.35103
[9] Daners, D., Principal eigenvalues for generalised indefinite Robin problems, Potential Anal., 38, 1047-1069 (2013) · Zbl 1264.35152
[10] Freitas, P.; Krejčiřík, D., The first Robin eigenvalue with negative boundary parameter, Adv. Math., 280, 322-339 (2015) · Zbl 1317.35151
[11] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. 2nd. 1st. paperback ed. Cambridge University Press, Cambridge (1988)
[12] Henrot, A., Extremum Problems for Eigenvalues of Elliptic Operators (2006), Basel: Birkhäuser, Basel · Zbl 1109.35081
[13] Henrot, A.: Shape Optimization and Spectral Theory. De Gruyter, Warsaw (2017) · Zbl 1369.49004
[14] Kato, T., Perturbation Theory for Linear Operators (1966), Berlin: Springer, Berlin · Zbl 0148.12601
[15] Klingenberg, W., A Course in Differential Geometry (1978), New York: Springer, New York
[16] Krejčiřík, D.; Lotoreichik, V., Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, J. Convex Anal., 25, 319-337 (2018) · Zbl 1401.35223
[17] Krejčiřík, D.; Raymond, N.; Tušek, M., The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces, J. Geom. Anal., 25, 2546-2564 (2015) · Zbl 1337.35042
[18] Lu, G.; Ou, B., A Poincaré inequality on \(\mathbb{R}^n\) ℝn and its application to potential fluid flows in space, Commun. Appl. Nonlinear Anal., 12, 1-24 (2005) · Zbl 1060.26017
[19] McLean, W., Strongly Elliptic Systems and Boundary Integral Equations (2000), Cambridge: Cambridge University Press, Cambridge
[20] Pankrashkin, K.; Popoff, N., An effective Hamiltonian for the eigenvalue asymptotics of a Robin Laplacian with a large parameter, J. Math. Pures Appl., 106, 615-650 (2016) · Zbl 1345.35068
[21] Payne, LE; Weinberger, HF, Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl., 2, 210-216 (1961) · Zbl 0098.39201
[22] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, IV. Analysis of Operators (1978), New York: Academic Press, New York · Zbl 0401.47001
[23] Savo, A., Lower bounds for the nodal length of eigenfunctions of the Laplacian, Ann. Glob. Anal. Geom., 16, 133-151 (2001) · Zbl 1010.58025
[24] Schneider, R., Convex bodies: The Brunn-Minkowski Theory (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0798.52001
[25] Segura, J., Bounds for ratios of modified Bessel functions and associated Turán-type inequalities, J. Math. Anal. Appl., 374, 516-528 (2011) · Zbl 1207.33009
[26] Sz.-Nagy, B., Über Parallelmengen nichtkonvexer ebener Bereiche, Acta Sci. Math., 20, 36-47 (1959) · Zbl 0101.14701
[27] Willmore, TJ, Riemannian Geometry (1993), Oxford: Clarendon Press, Oxford
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.