On the maximum and minimum of some functionals for the eigenvalue problem of Sturm-Liouville type
One considers the eigenvalues \(\lambda_ n= \lambda_ n(q)\) of the Sturm-Liouville equation \(y''+ (\lambda- q(x))y= 0\), with the boundary conditions \(y(-\ell)= y(\ell)=0\). Let \(E(h,H,M,\ell)\) be the set of potential functions \(q\) such that \(h\leq q(x)\leq H\), \(\int^ \ell_ \ell q(x)dx= M\). One determines the potential functions in \(E(h,H,M,\ell)\) which extremize \(\lambda_ n(q)\) and minimize \(\lambda_ 2(q)- \lambda_ 1(q)\).
scientific article; zbMATH DE number 522668