Shigeru Sakaguchi

Matzoh ball soup: heat conductors with a stationary isothermic surface

We consider a bounded heat conductor that satisfies the exterior sphere condition. Suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. We show that if the conductor contains a proper sub-domain, satisfying the interior cone condition and having constant boundary temperature at each given time, then the conductor must be a ball.

Nonlinear diffusion with a bounded stationary level surface

Abstract We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C 2 . Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C 2 -subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole R N of some substance whose density is initially a characteristic function of the complement of a domain with bounded C 2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere S N and the hyperbolic space H N .

Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain Ω in the N-dimensional Euclidean space R N is said to be uniformly dense in a surface Γ ⊂ R N of codimension 1 if, for every small r > 0, the volume of the intersection of Ω with a ball of radius r and center x does not depend on x for x ∈ Γ. We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary ∂Ω, and we show that the principal curvatures of ∂Ω satisfy certain identities. The case in which the surface Γ coincides with ∂Ω is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if N = 2, it must be either a circle or a straight line and (ii) if N = 3, it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

Symmetry of minimizers with a level surface parallel to the boundary

We consider the functional

Stationary Level Surfaces and Liouville-Type Theorems Characterizing Hyperplanes

I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,

Two-phase heat conductors with a stationary isothermic surface

where

Symmetry Problems on Stationary Isothermic Surfaces in Euclidean Spaces

\Omega

When does the heat equation have a solution with a sequence of similar level sets

is a bounded domain and