On the minimization of Dirichlet eigenvalues. (English) Zbl 1317.49052
Two main results are proven in this paper. Given an open convex set \(\Omega\subset {\mathbb R}^m\) of finite Lebesgue measure with Dirichlet eigenvalues \(\lambda_j(\Omega)\), \(j\in {\mathbb N}\), the following problem is considered:
\[
I_k(c)=\inf\{\lambda_k(\Omega):\Omega\subset {\mathbb R}^m \text{ open and convex}, \mathcal{T}(\Omega)=c\}. \tag{*}
\]
Here \(\mathcal{T}\) is a function defined on the space of open convex sets in \({\mathbb R}^m\), satisfying some of the following hypothesis
The second result in this paper is Theorem 2, providing relations between \(\pi_k\), \(\mu_k\) and \(\omega_m\), for all integers \(m\geq 2\) and \(k\geq 1\). Here \(\omega_m\) is the Lebesgue measure of the unit ball in \(\mathbb{R}^m\), and \(\pi_k\) and \(\mu_k\) are defined by \[ \pi_k=\inf\{|\Omega|:\Omega\in\mathfrak{B}_k\},\quad\mu_k=\inf\{|\Omega|:\Omega\in \mathfrak{M}_k\}, \] where \(\mathfrak{B}_k\) is the set of minimizers of \(\inf\{\lambda_k(\Omega):\Omega\subset\mathbb{R}^m \text{ open}, \mathcal{P}(\Omega)=1,|\Omega|<\infty\}\), and \(\mathfrak{M}_k\) is the set of minimizers of \(\inf\{\lambda_k(\Omega):\Omega\subset\mathbb{R}^m \text{ quasi-open},|\Omega|=1\}\).
- (a)
- \(\mathcal{T}\) is invariant under isometries, monotone (\(\Omega_1\subset\Omega_2\Rightarrow\mathcal{T}(\Omega_1)\leq \mathcal{T}(\Omega_2)\)), and non-negative (\(\mathcal{T}(\Omega)=0 \Leftrightarrow \Omega=\emptyset\)).
- (b)
- \(\mathcal{T}(\alpha\Omega)=\alpha^\tau\mathcal{T}(\Omega)\) for some \(\tau>0\) and every \(\alpha>0\).
- (c1)
- \(T^*=\inf\{\mathcal{T}(\Omega): \Omega\subset\mathbb{R}^m \text{ open and convex}, |\Omega|=1\}>0\).
- (c2)
- There exists an open convex set \(D\subset\mathbb{R}^m\) with \(|D|=1\), unique up to isometries, such that \(T^*=\mathcal{T}(D)\).
- (d)
- There exist constants \(K<\infty\) and \(t>1/\tau\) such that, if \(\Omega\subset\mathbb{R}^m\) is an open bounded convex set, then \(\text{diam}(\Omega)\leq K\mathcal{T}(\Omega)^t|\Omega|^{(1-t\tau)/m}\).
The second result in this paper is Theorem 2, providing relations between \(\pi_k\), \(\mu_k\) and \(\omega_m\), for all integers \(m\geq 2\) and \(k\geq 1\). Here \(\omega_m\) is the Lebesgue measure of the unit ball in \(\mathbb{R}^m\), and \(\pi_k\) and \(\mu_k\) are defined by \[ \pi_k=\inf\{|\Omega|:\Omega\in\mathfrak{B}_k\},\quad\mu_k=\inf\{|\Omega|:\Omega\in \mathfrak{M}_k\}, \] where \(\mathfrak{B}_k\) is the set of minimizers of \(\inf\{\lambda_k(\Omega):\Omega\subset\mathbb{R}^m \text{ open}, \mathcal{P}(\Omega)=1,|\Omega|<\infty\}\), and \(\mathfrak{M}_k\) is the set of minimizers of \(\inf\{\lambda_k(\Omega):\Omega\subset\mathbb{R}^m \text{ quasi-open},|\Omega|=1\}\).
Reviewer: Manuel Ritoré (Granada)
MSC:
49Q10 | Optimization of shapes other than minimal surfaces |
49R05 | Variational methods for eigenvalues of operators |
35J25 | Boundary value problems for second-order elliptic equations |
35P15 | Estimates of eigenvalues in context of PDEs |
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