## 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:

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