Extremal eigenvalue problems for composite membranes. I
The class of structures that is considered here are membranes on \(\Omega \in {\mathbb{R}}^ N\) with densities in \(ad_{\gamma}=\{\alpha \chi +\beta (1-\chi):\chi\) is the characteristic function of a subset of \(\Omega\) with measure \(\gamma\), \(\alpha\) and \(\beta\) are given\(\}\). The investigated composite membranes are such that their material density is \(\alpha\) in volume fraction \(\gamma /_{| \Omega |}\) and \(\beta\) in volume fraction \(1-\gamma /_{| \Omega |}\). An optimum problem is formulated for the clamped membranes, the cost of membrane with density \(\rho \in ad_{\gamma}\) is measured in terms of \(\lambda_ k(\rho)\), the membrane kth eigenvalue. This paper establishes a topology for \(ad_{\gamma}\) and studies the continuity of \(\rho \to \lambda_ k(\rho)\) on the associated closure of \(ad_{\gamma}\). Existence of extremizers for \(\lambda_ k\) is derived. Rayleigh's principle, Minmax principle, Auchmuty's principle are used in combination with convex analysis to prove some properties of the largest or smallest kth natural frequency. The obtained results are the generalizations of Krein's one- dimensional work to the N dimensional case.
scientific article; zbMATH DE number 4167168
scientific article