Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants

TL;DR

This paper introduces a numerical method to compute guaranteed two-sided bounds for eigenvalues of symmetric elliptic operators, with applications to fundamental inequalities and constants in mathematical analysis.

Contribution

The paper develops a general approach combining Galerkin, a priori-a posteriori inequalities, and complementarity to bound eigenvalues and constants in key inequalities.

Findings

Successfully computes tight bounds for eigenvalues.

Provides fully computable bounds for Friedrichs', Poincaré, and trace constants.

Demonstrates accuracy through numerical examples.

Abstract

We present a general numerical method for computing guaranteed two-sided bounds for principal eigenvalues of symmetric linear elliptic differential operators. The approach is based on the Galerkin method, on the method of a priori-a posteriori inequalities, and on a complementarity technique. The two-sided bounds are formulated in a general Hilbert space setting and as a byproduct we prove an abstract inequality of Friedrichs'-Poincar\'e type. The abstract results are then applied to Friedrichs', Poincar\'e, and trace inequalities and fully computable two-sided bounds on the optimal constants in these inequalities are obtained. Accuracy of the method is illustrated on numerical examples.