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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

Citations:

Zbl 1401.35223
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References:

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