Extension of an inequality of Pólya-Szegö to wedge-like domains
This is an interesting note which establishes an isoperimetric inequality for the first eigenvalue of the clamped membrane problem: \[ \Delta u+\lambda u=0\quad\text{in } D,\quad u=0\quad\text{on }\partial D, \] where \(D\) is a domain contained in a wedge of angle \(\pi/\alpha\). The precise result proved is: The first eigenvalue \(\lambda_ 1\) satisfies for any \(\alpha\geq 1/2\): \[ \lambda_ 1\leq{j^ 2_ \alpha\over 2A_ \alpha}\left[1+\left({\pi\over 2B_ \alpha}- \alpha\right){{\partial J_ \alpha\over\partial\alpha}(j_ \alpha)\over j_ \alpha J_{\alpha+1}(j_ \alpha)}\right] \] where \(j_ \alpha\) is the first positive root of \(J_ \alpha\), and \(A_ \alpha\), \(B_ \alpha\) are integrals depending on the domain and its boundary. It is important to observe that this inequality is best possible in the sense that equality holds when \(D\) is a circular sector of angle \(\pi/\alpha\).
isoperimentric inequality for the first eigenvalue
scientific article; zbMATH DE number 168469