Bounds for the \(N\) lowest eigenvalues of fourth-order boundary value problems
The author presents a method for calculating inclusion intervals to the first \(N\) (real) eigenvalues of certain fourth-order elliptic differential equations on bounded domains \(\Omega\) and under Dirichlet boundary conditions. The problem is treated in variational form in the Sobolev space \(H^2_0(\Omega)\), so that error bounds in terms of defects in the dual space \(H^{-2}(\Omega)\) may be introduced. In order to obtain a lower bound for the remainder of the spectrum, a boundary homotopy method is proposed. Impressing numerical results are presented for the 100 lowest eigenvalues of the clamped plate problem in the unit square, where rounding errors were controlled by interval analysis.
fourth-order elliptic differential equations
scientific article; zbMATH DE number 1057395
scientific article