  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.
