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Matzoh ball soup: Heat conductors with a stationary isothermic surface. (English) Zbl 1011.35066

The main result of the paper is the following theorem: Let \(\Omega \) be a bounded domain in \(\mathbb{R}^N\), \(N\geq 2\), satisfying the exterior sphere condition and suppose that \(D\) is a domain, with boundary \(\partial D\), satisfying the interior cone condition, and such that \(\overline D\subset \Omega \). Let \(u\) be a solution of the problem \(u_t=\Delta u\) in \(\Omega \times (0,+\infty)\), \(u=1\) on \(\partial \Omega \times (0,+\infty)\), \(u=0\) on \(\Omega \times \{ 0\} \). Assume that \(u(x,t)=a(t)\) for \((x,t) \in \partial D \times (0,+\infty)\) for some function \(a:(0,+\infty) \to (0,+\infty)\). Then \(\Omega \) must be a ball.

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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