Matzoh Ball Soup: Heat Conductors with a Stationary Isothermic Surface

Magnanini, Rolando; Sakaguchi, Shigeru

doi:10.2307/3597287

Cited by 42 publications

(87 citation statements)

References 14 publications

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“…The first one is due to G. Alessandrini, who proved in [As] that the ball is the only bounded conductor with the property that all its isothermic surfaces are stationary. In [MS1], it is shown that one arrives at the same conclusion even if the conductor contains only one fairly regular isothermic surface.…”

Section: Introductionsupporting

confidence: 49%

“…Therefore, it follows that Γ must be smooth, and we can complete the proof by following that of [MS1,Lemma 3.1].…”

Section: Theorem 25 Let ω Be An Open Set In R N Satisfying the Intementioning

confidence: 99%

“…We want to classify all the domains Ω such that there is an (N − 1)-dimensional surface Γ ⊂ R N which is a stationary isothermic surface, that is, such that at each time t > 0, the temperature u = u(x, t) remains constant on Γ (a constant depending on t). This problem is slightly easier than the one considered in [MS1], and, besides producing surprising results, it may be useful to build up more insight on the Matzoh Ball Soup problem.…”

Section: Introductionmentioning

confidence: 99%

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Stationary isothermic surfaces and uniformly dense domains

Magnanini¹,

Prajapat²,

Sakaguchi³

2006

Trans. Amer. Math. Soc.

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Abstract. 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 Ndimensional 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.

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Section: Introductionsupporting

confidence: 49%

“…Therefore, it follows that Γ must be smooth, and we can complete the proof by following that of [MS1,Lemma 3.1].…”

Section: Theorem 25 Let ω Be An Open Set In R N Satisfying the Intementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stationary isothermic surfaces and uniformly dense domains

Magnanini¹,

Prajapat²,

Sakaguchi³

2006

Trans. Amer. Math. Soc.

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“…The proof is complete. Still, the argument we used to obtain (2.14) does not work in the case of the initialboundary value problem for the heat equation with boundary value 1 and initial value 0-the matzoh ball soup setting considered initially in [19]. Hence, statement 3 of [21, Lemma 3.1, p. 2026] should be corrected in such a way that γ is an immersed hypersurface in R N .…”

Section: Regularity Of Uniformly Dense Setsmentioning

confidence: 99%

Stationary isothermic surfaces in Euclidean 3-space

2015

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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.

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“…The property was first identified in [MS2], motivated by the study of invariant isothermic surfaces of a nonlinear non-degenerate fast diffusion equation (designed upon the heat equation), and was used to extend to nonlinear equations the symmetry results obtained in [MS1] for the heat equation. The proof hinges on the method of moving planes developed by J. Serrin in [Se] upon A.D. Aleksandrov's reflection principle ( [Al]).…”

Section: Introductionmentioning

confidence: 99%

Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

Ciraolo

Magnanini

Sakaguchi

2016

JAMA

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Abstract. Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls Br e and Br i , with the difference re − ri (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed by Aftalion, Busca and Reichel in [ABR].

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Matzoh Ball Soup: Heat Conductors with a Stationary Isothermic Surface

Cited by 42 publications

References 14 publications

Stationary isothermic surfaces and uniformly dense domains

Stationary isothermic surfaces and uniformly dense domains

Stationary isothermic surfaces in Euclidean 3-space

Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

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