Existence of minimizers for spectral problems (Q391383)

For every open subset \(G\) of \(\mathbb{R}^{n}\), denote by \(\lambda_{i}(G),\; i\in\mathbb{N}\), the eigenvalues of the Dirichlet Laplacian on \(G\). A bounded set \(A\) is defined to be quasi open if there exists an open set \(\Omega\) in which \(A\) is relatively compact and for every \(\varepsilon>0\) there exists an open subset \(\Omega_{\varepsilon}\) relatively compact in \(\Omega\) such that \(A\subseteq\Omega_{\varepsilon}\) and \(\text{cap}_{\Omega}(\Omega_{\varepsilon}\setminus A)<\varepsilon\). An unbounded set \(A\) is quasi open if the intersection of \(A\) with any ball of \(\mathbb{R}^{n}\) is a quasi open bounded set. The authors prove the following theorems: Thm. 1 Let \(k\in\mathbb{N}\) and let \(F:\mathbb{R}^{k}\rightarrow\mathbb{R}\) be an l.s.c. functional, increasing in each variables. Then there exists a bounded minimum of \(\left\{ F(\lambda_{1}(A),\ldots,\lambda_{k}(A)): A \text{ (quasi) open in } \mathbb{R}^{n}, |A|=1\right\} \). A minimizer \(A\) is contained in a cube of side \(R\), where \(R\) depends on \(k\) and on \(n\) but not on the particular functional \(F\).NEWLINENEWLINE Theorem 2. There exists \(M(k,n)\) such that for every quasi open set \(A\) one has \(\lambda_{k}(A)\leq M(k,n)\,\lambda_{1}(A)\).