## 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)].

### MSC:

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

Zbl 1401.35223
Full Text:

### References:

 [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
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