The location of hot spots and other extremal points

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 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 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, in this case 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. For the first mode, the relevant bound only depends on the space dimension, as well. Moreover, it largely improves on an earlier estimate obtained for convex domains by the first author and co-authors. 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.