The location of hot spots and other extremal points

TL;DR

This paper provides lower bounds on the distance from extremal points of solutions to elliptic and parabolic equations within a domain, relating these bounds to geometric properties of the domain, with improvements over previous estimates.

Contribution

It introduces new bounds for extremal points of solutions to elliptic and parabolic equations, extending previous results to non-convex domains and more general equations.

Findings

Bounds depend on domain geometry such as inradius and mean curvature.

Improved estimates for the first eigenmode in non-convex domains.

Results are consistent with stationary cases and extend to quasilinear and semilinear equations.

Abstract

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we first consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with a mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. Also in the case of the first mode, the relevant bound only depends on the space dimension. Moreover, it largely improves on an earlier estimate obtained by the first author and co-authors, for convex domains. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.