具有恒流特性表面的两相热导体
Two-phase heat conductors with a surface of the constant flow property
摘要：
我们考虑一个两相热导体在$\mathbb R^N$与$N \geq 2$组成的核和壳具有不同的常数电导率。我们研究了径向对称对椭圆型和抛物型超定问题所起的作用。首先，利用隐函数定理，给出了一类Serrin型两相椭圆超定边值问题径向对称性的一个反例。之后，我们考虑以下设置为两相抛物超定问题。我们假设，最初，导体的温度为0，并且在任何时候，它的边界都保持在温度1。如果超曲面上的每一点的热流只与时间有关，则该超曲面具有常流性质。结果表明，如果在边界附近的壳体中存在恒流性质的表面，或者导热体边界的连通分量是恒流性质的表面，则导热体的结构必须是球形的。此外，通过假设导体外部的介质具有可能不同的电导率，我们考虑柯西问题，其中导体具有初始内部温度$0$和外部温度$1$。然后，我们表明，一个非常相似的对称性结果成立。
英文摘要：
We consider a two-phase heat conductor in $\mathbb R^N$ with $N \geq 2$ consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type. First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin-type. Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. A hypersurface in the domain has the constant flow property if at every of its points the heat flux across surface only depends on time. It is shown that the structure of the conductor must be spherical, if either there is a surface of the constant flow property in the shell near the boundary or a connected component of the boundary of the heat conductor is a surface of the constant flow property. Also, by assuming that the medium outside the conductor has a possibly different conductivity, we consider a Cauchy problem in which the conductor has initial inside temperature $0$ and outside temperature $1$. We then show that a quite similar symmetry result holds true.
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