Stationary isothermic surfaces in Euclidean 3-space

Abstract

Let be a domain in R 3 with ∂ = ∂(R 3 \ ), where ∂ is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation ∂ t u = u over R 3 , where the initial data is the characteristic function of the set c = R 3 \ . We show that, if there exists a stationary isothermic surface of u with ∩ ∂ = ∅, then both ∂ and must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that ∩ ∂ = ∅ and ∂ is unbounded. To prove this result, we establish a similar theorem for uniformly dense domains in R 3 , a notion that was introduced by methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.

The Matzoh Ball Soup Problem: A complete characterization

“…We also point out that this result can be obtained under quite general regularity assumptions on ∂Ω , as shown in [16]. The case in which ∂Ω is not bounded is not settled: we mention [12,14] for some partial results in this direction and [8] for a characterization in 3-space.…”

“…In a subsequent series of papers, the same authors extended their result in several directions: spherical symmetry also holds for certain evolution nonlinear equations [11,13,15,16]; a hyperplane can be characterized as an invariant equipotential surface in the case of an unbounded solid that satisfies suitable sufficient conditions [12,14]; for a certain Cauchy problem, a helicoid is a possible invariant equipotential surface [9]; spheres, infinite cylinders and planes are characterized as (single) invariant equipotential surfaces in R 3 [8]; similar symmetry results can also be proven in the sphere and the hyperbolic space [13].…”

Symmetry Problems on Stationary Isothermic Surfaces in Euclidean Spaces

“…The stationary isothermic surfaces of solutions of the heat equation have been much studied, and it has been shown that the existence of a stationary isothermic surface forces the problems to have some sort of symmetry (see [MPeS,MPrS,MS2,MS3,MS5,MS6,MS7,S]). A balance law for stationary zeros of temperature introduced by [MS1] plays a key role in the proofs.…”