Polygonal Faber–Krahn inequality: Local minimality via validated computing

Source abstract

The main result of the paper shows that the regular [Formula: see text]-gon is a local minimizer for the first Dirichlet–Laplace eigenvalue among [Formula: see text]-gons having fixed area for [Formula: see text]. The eigenvalue is seen as a function of the coordinates of the vertices in [Formula: see text]. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right-hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for [Formula: see text]. All computations, including the finite element computations, are realized using interval arithmetic.