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Some characterizations of parallel hyperplanes in multi-layered heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole space consisting of three layers with different constant conductivities, where initially the upper and middle layers have temperature 0 and the lower layer has temperature 1. Under some appropriate conditions, it is shown that, if either the interface between the lower layer and the middle layer is a stationary isothermic surface or there is a stationary isothermic surface in the middle layer near the lower layer, then the two interfaces must be parallel hyperplanes. Similar propositions hold true, either if a stationary isothermic surface is replaced by a surface with the constant flow property or if the Cauchy problem is replaced by an appropriate initial-boundary value problem.

Define two domains D, Ω in R N by (1.1) respectively. Denote by σ = σ(x) (x ∈ R N ) the conductivity distribution of the whole medium given by where σ c , σ s , σ m are positive constants with σ c = σ s . This kind of three-phase electrical conductor has been dealt with in [9] in the study of neutrally coated inclusions. Let g ∈ C 0 (R N −1 ) satisfy f (y) < g(y) < h(y) for every y ∈ R N −1 .
Let us first state two theorems concerning stationary isothermic surfaces. then ∂Ω and ∂D must be parallel hyperplanes. then ∂Ω and ∂D must be parallel hyperplanes.
In Theorems 1.1 and 1.2, the conditions (1.9) and (1.10) mean that each of ∂Ω and ∂G is a stationary isothermic surface. Thus each of Theorems 1.1 and 1.2 characterizes parallel hyperplanes as the interfaces in such a way that there exists a stationary isothermic surface in the multi-layered heat conductors. The assumptions on the function h− f are technical, and in particular the existence of its maximum value or its minimum value enables us to utilize Hopf's boundary point lemma.
Next two theorems replace a stationary isothermic surface by a surface with the constant flow property which was dealt with in [4]. then ∂Ω and ∂D must be parallel hyperplanes, where ν denotes the outward unit normal vector to ∂G.
In Theorem 1.3 the condition (1.11), together with the boundary condition (1.5), is overdetermined and it implies that the heat flow is parallel to the normal vector to ∂Ω and the amount of the flow is constant on ∂Ω for each time. Such a condition was given by [1,6] for parabolic problems, which generalizes the overdetermined condition of Serrin [17] for elliptic problems. Recently such a boundary ∂Ω was called a surface with the constant flow property in the context of the heat flow in smooth Riemannian manifolds by [13]. The condition (1.12), which was introduced by [4], is an overdetermination different from Serrin-type, and we still called it the constant flow property in [4]. Similar characterizations of concentric balls in multi-phase heat conductors were obtained in the previous papers [15,16,4], and in the present paper we deal with hyperplanes, which are not compact and need additional cares. The proofs of all the theorems consist of two steps.
In the first step we show that ∂Ω must be a hyperplane, and the second step is devoted to proving that ∂D is a hyperplane parallel to ∂Ω. We have two strategies in the first step; one applies to Theorems 1.2 and 1.4 and the other does to Theorems 1.1 and 1.3. On the other hand, the second step follows from one strategy common to all the theorems, which depends on a result concerning an elliptic overdetermined problem (see Theorem 5.1 in section 5).
The following sections are organized as follows. In section 2, we recall one lemma and three propositions from [4,15], where we need to modify the two propositions in order to deal with the case where ∂Ω is unbounded. Indeed, we show that our case is reduced to the case where ∂Ω is bounded and of class C 2 with the aid of the maximum principle and the Gaussian bounds for the fundamental solution of u t = div(σ∇u) due to Aronson [2, Theorem 1, p. 891](see also [5, p. 328]). Section 3 is devoted to the proofs of Theorems

Preliminaries
Let us introduce the distance function δ = δ(x) of x ∈ R N to ∂Ω by where µ, M are positive constants. Then the following propositions hold true: (1) The solution u satisfies (2) For every τ > 0, there exist two positive constants B and b such that and, moreover, if u is the solution of (1.3), then (1 − u(x, t)) = 0 for every t ∈ (0, +∞). In is of class C 2 and ∂Ω divides B ε (z 0 ) into two connected components. Let σ = σ(x) (x ∈ R N ) be a general conductivity satisfying Here, κ 1 (z 0 ), . . . , κ N −1 (z 0 ) denote the principal curvatures of ∂Ω at z 0 with respect to the inward normal direction to ∂Ω and C(N, σ) is a positive constant given by Proof. It suffices to show that our case is reduced to the case where ∂Ω is bounded and We observe that the difference By comparing v with the solutions of the Cauchy problem for the heat equation with conductivity σ s and initial data ±2X N for a short time, we see that there exist two positive constants B and b such that By (2) of Lemma 2.1, we may also have Then, it follows from (2.8) and (2.9) that u also satisfies (2.3), since we already know that u * satisfies (2.3). Indeed, observing that and letting t → ∞ yield the conclusion.
Since a proposition [4,Proposition E], where the boundary of the domain is compact, also plays a key role in [4], we need to modify the proposition in order to deal with the case where ∂Ω is unbounded.

Proposition 2.3
Let Ω be a possibly unbounded domain in R N , and let z 0 ∈ ∂Ω. Assume where µ, M, σ s , and σ m are positive constants. Let u be the bounded solution of problem (1.3) for this general conductivity σ. Then, as t → +0, u converges to the number Proof. It suffices to show that our case is reduced to the case where ∂Ω is bounded and of class C 2 . As in the proof of Proposition 2.2 for problem (1.3), let u * = u * (x, t) be the bounded solution of problem (1.3) where Ω and σ are replaced with Ω * and σ * , respectively. Then u * satisfies the conclusion because of [4, Proposition E]. Therefore, since v = u − u * satisfies (2.14), u also satisfies the conclusion.
We quote another ingredient called a balance law adjusted to our use from [ Consider two points p, q ∈ W and two unit vectors ξ, η ∈ R N . Set Then the following three propositions hold true: (2) ξ · ∇u(p, t) = η · ∇u(q, t) for every t > 0 if and only if (3) ∇u(p, t) = 0 for every t > 0 if and only if Then v 1 satisfies the heat equation with conductivity σ s and v 1 (0, t) = 0 for every t > 0.
(2) is proved in [4,Lemma 4.2] with the aid of [10, Theorem 2.1]. For (2), by choosing an orthogonal matrix A satisfying Aξ = η, we consider the function Then the function ξ · ∇v 2 (x, t) satisfies the heat equation with conductivity σ s and for Thus, it follows from [10, and hence, by the divergence theorem and again integrating in r, we infer that which gives (2). (1/R − κ j (z)) is constant for z ∈ ∂Ω, where R is the distance between ∂Ω and ∂G, κ 1 (z), . . . , κ N −1 (z) denote the principal curvatures of ∂Ω at a point z ∈ ∂Ω with respect to the inward normal direction to ∂Ω, and max Once this is proved, we immediately infer that ∂Ω must be a hyperplane.  (1) there exists a number R > 0 such that where δ(x) is the distance function given by (2.1); (2) ∂Ω and ∂G are real analytic hypersurfaces; denotes the outward unit normal vector to ∂Ω at z ∈ ∂Ω; in particular ∂Ω and ∂G are parallel hypersurfaces at distance R; (4) the principal curvatures of ∂Ω satisfy Before proving this lemma, we prepare a purely geometric lemma for the proof of Theorem 1.4.
where δ(x) is the distance function given by (2.1). Then, for every ε > 0, there exists a where ν(p) denotes the outward unit normal vector to ∂G at p ∈ ∂G.
Proof. Let ε > 0. Set where qz denotes the line segment connecting q and z. Therefore we infer that .
Hence, by the inverse mapping theorem and (3.5), there exists an infinite solid cylinder U , whose axis is the line containing qz, such that If ∇δ(p) · ν(p) = 0, then the conclusion follows from (3.5). Thus, let us consider the case Then, as long as x(s) exists, x(s) ∈ ∂G and moreover, since ∇δ(x) = p−z |p−z| for every x ∈ pz, we have from the uniqueness of the solution of the Cauchy problem (3.6) These contradict the fact that δ(x(s)) ≥ R and δ(x(s)) = −s + δ(p). Thus there exists a Proof of Lemma 3.1. First, it follows from the assumption (1.8) that  (2) of Lemma 2.1 that Suppose that there exists a point q ∈ ∂G with δ(p) < δ(q). Then, (1.12) gives (2) of Proposition 2.4. In particular, we choose r = δ(p), ξ = ν(p) and η = ν(q) to infer that u(x, t)(x − q) dx for every t > 0. (3.8) On the other hand, it follows from (2) of Lemma 2.1 that the right-hand side of (3.8) tends to 0 as t → +0, which contradicts (3.7). Therefore, we conclude that δ(q) ≤ δ(p) for every q ∈ ∂G. Moreover, (3.2) yields that δ(q) = R for every q ∈ ∂G and R > 0. Thus assertion (1) holds also under the assumption (1.12).
Once we have (1) under the assumption (1.12) of Theorem 1.4, we infer that for every x ∈ ∂G there exists a unique z = z(x) ∈ ∂Ω satisfying Thus, we infer that the formula (3.7) holds if we set p = x(z) ∈ ∂G with z ∈ γ and Hence, combining (2) of Proposition 2.4 with this formula (3.10) yields that there exists a number c > 0 satisfying Then, since ∂Ω is of class C 2 , combining (3.9) with (3.11) yields that γ is closed in ∂Ω.
On the other hand, the inverse mapping theorem implies that γ is also open in ∂Ω and the mapping γ ∋ z → x(z) ∈ ∂G is a local diffeomorphism. Therefore γ = ∂Ω, since ∂Ω is connected. Thus the others (3) w| − = w| + and σ s ∂w ∂ν Then Hence it follows from the maximum principle that v * ≡ 0, that is, in R N \ G × (0, +∞), the solution u depends only on δ(x) and t since Θ ∈ R N is an arbitrary vector parallel to the hyperplane ∂Ω. Therefore w depends only on δ(x) in R N \ G and hence (5.3) holds true. (3.16) gives the fact that 0 < α < 1 in (5.3), and (3.14), (3.15) and (3.17) yield that β > 0. Indeed, by solving (3.14), we get for some positive number 0 < c 0 < 1. This together with (3.15) yields that β > 0.
Therefore Theorem 5.1 implies the conclusion of Theorems 1.

Consider the function
Then Either v * = 0 on ∂G × (0, +∞) or ∂v * ∂ν = 0 on ∂G × (0, +∞), Hence it follows from the maximum principle that v * ≡ 0, that is, in (Ω \ G) × (0, +∞), the solution u depends only on δ(x) and t since Θ ∈ R N is an arbitrary vector parallel to the hyperplane ∂Ω. Therefore w depends only on δ(x) in Ω \ G and hence ( Since ∂Ω is uniformly of class C 6 , there exists two positive numbers r and K such that, for every point p ∈ ∂Ω, there exist an orthogonal coordinate system z and a function ϕ ∈ C 6 (R N −1 ) such that the z N coordinate axis lies in the inward normal direction to ∂Ω at p, the origin is located at p, C 6 norm of ϕ in R N −1 is less than K, ϕ(0) = 0, ∇ϕ(0) = 0 and the set B r (p) ∩ Ω is written as in the z coordinate system {z ∈ B r (0) : z N > ϕ(z 1 , . . . , z N −1 )}.
As in the proofs of [4, Theorems 1.4 and 1.5 in section 5], we introduce the function

An elliptic overdetermined problem
In this section, we assume that ∂Ω is a hyperplane, that is, f is an affine function in (1.1).
In view of the assumption, we may deal with the following two cases in the original coordinate system x: Let us consider case (I) first. (5.2) yields that c 2 = 0 and hence 0 < c 1 ≤ 1 by (5.5). Thus Then we notice that w ′ + (ρ) < 0 for all ρ ∈ R and lim ρ→+∞ w + (ρ) = 0. Let v * = v * (ρ) (ρ ≥ z * N ) be the unique solution of the Cauchy problem: Hence we have for some constants c * 3 , c * Distinguish two cases: In case (I-1) we have from (5.8) that We also have w ≡ v * and w ≤ v * on ∂D.
(5.12) Therefore, since −σ c ∆w + w = −σ c ∆v * + v * = 0 and 0 < w < 1 in D, c * 4 ≥ 0 and min{v * , 1} is a bounded supersolution in D, it follows from the comparison principle (5.13) Here we applied Proposition A.3 to the function min{v * , 1} − w in D. Thus, with the aid of Hopf's boundary point lemma at z * ∈ ∂D, this contradicts the fact that v * = w and ∂v * ∂ν where ν denotes the outward unit normal vector to ∂D and − denotes the limit from inside of D. Here we used (5.4).
In case (IIa) (5.6) yields (5.8). Then the same arguments as in case (I) work and we get a contradiction, that is, case (IIa) does not occur.
With (5.17) in hand, in case (IIb-2) by the same arguments as in case (I-2) we notice that c * 4 < 0 and hence we obtain (5.15) which gives a contradiction with the aid of Hopf's boundary point lemma at z * ∈ ∂D. In case (IIb-1), if c * 4 ≥ 0, then the same arguments as in case (I-1) also work and one can get a contradiction. Thus it suffices to take care of case (IIb-1) with c * 4 < 0. Let us consider case (IIb-1) with c * 4 < 0. For every r ≥ z * N , we introduce the solution v r = v r (ρ) (ρ ∈ R) of the Cauchy problem: Hence we have for some constants c 3 (r), In particular, we have and v * are given in (5.9). Set Distinguish two cases: In case (IIb-1-1), with (5.17) in hand, the same arguments as in (I) also work and (5.11) is replaced with v ♯ (ρ) < w + (ρ) for every ρ < z ♯ N . In case (IIb-1-2), in view of (5.17) and (5.19), we observe that there exists R > 0 satisfying z * N < z ♯ N < R, c * 4 = c 4 (z * N ) < 0, c ♯ 4 = c 4 (z ♯ 4 ) > 0 and c 4 (R) < 0.
By (5.19), c 4 (r) is continuous in r. Therefore, it follows from the intermediate value theorem that there exist two numbers r 1 and r 2 satisfying z * N < r 1 < z ♯ N < r 2 < R and c 4 (r 1 ) = c 4 (r 2 ) = 0, and hence in particular both the functions v r j (j = 1, 2) are bounded in [0, ∞). Introduce two functions w j = w j (ρ) (j = 1, 2) for ρ ≥ 0 by Then we can apply Proposition A.2 to these w j = w j (ρ) (j = 1, 2) and obtain that r 1 = r 2 , which is a contradiction. Therefore, case (IIb-1-2) does not occur.
In case (IIc) it follows that there exists a unique r 0 > 0 satisfying Distinguish three cases: Let us first consider case (IIc-1). Distinguish two cases: In case (IIc-1-1), we employ v ♯ . It follows from (1) of Proposition A.1 that Moreover, by integrating the ordinary differential equations which w + and v ♯ satisfy, we This implies that v ♯ must have a critical point and hence c ♯ 4 > 0. We also have from (5.24) that w ≡ v ♯ and w ≤ v ♯ on ∂D. since min{v ♯ , 1} is a bounded supersolution in D because of the fact that c ♯ 4 > 0. Thus we get a contradiction with the aid of Hopf's boundary point lemma at z ♯ ∈ ∂D. Therefore, case (IIc-1-1) does not occur.
In case (IIc-1-2), we employ v * instead of v ♯ . It follows from (1) of Proposition A.1 that w + (ρ) < v * (ρ) if ρ > z * N , and hence c * 4 > 0. (5.27) Here positivity of c * 4 comes from that of c 2 . Thus the same comparison arguments yield a contradiction with the aid of Hopf's boundary point lemma at z * ∈ ∂D, and hence case (IIc-1-2) does not occur. Eventually, case (IIc-1) does not occur. We easily know that the same manner as in case (IIc-1) works also in case (IIc-3).
Because of (5.16) there exists a point z 0 ∈ ∂D with z 0 N = r 0 and moreover v r 0 = w and ∇v r 0 = ∇w = 0 at the point z 0 ∈ ∂D.
Then the same comparison arguments yield a contradiction with the aid of Hopf's boundary point lemma at z 0 ∈ ∂D. Thus, case (IIc-2-2) does not occur.
In case (IIc-2-1), we employ v * . It follows from (2) of Proposition A.1 that Remark that this inequality is not sufficient for the previous comparison arguments, because of (5.16). For the sake of this reason, by integrating the ordinary differential equations which w + and v * satisfy, we have from (5.28) we introduce a function v * * = v * * (ρ) for ρ ≥ 0 given by Hence we have in particular Indeed, for z N * < ρ ≤ r 0 , by integrating the ordinary differential equations which w + and v * satisfy, we have from (5.28) Therefore, for z N * < ρ ≤ r 0 , inequality (5.29) holds true. For r 0 < ρ < z ♯ N , since v ′ * * (ρ) < 0 and w ′ + (ρ) > 0, inequality (5.29) follows easily.
where we set Then we can apply Proposition A.5 to w 1 = w, w 2 = v * * , ℓ = z * N and L = z ♯ N and conclude that w ≤ v * * in Ω, and hence w < v * * in D.
Therefore, this yields a contradiction with the aid of Hopf's boundary point lemma at z * ∈ ∂D, and case (IIc-2-1) does not occur. The proof of Theorem 5.1 is complete.
Thus the conclusion follows from the same argument as in (1).
We have a proposition concerning the unique determination of discontinuity of the conductivity for an ordinary differential equation with a nontrivial Cauchy data (see Lemma 3.1 in [16] for the case dealing with bounded domains).
Transmission conditions yield that w j (j = 1, 2) are continuous on [0, ∞) and Hence we have Thus we obtain Changing the roles of w j (j = 1, 2) yields that In the same way we also have Therefore by combing (A.2) and (A.3) with the initial condition we obtain since σ s = σ c . Then it follows from these equalities and the initial condition that which yields that w 1 ≡ w 2 . Moreover, since w 1 is not constant because of the initial condition, it follows that r 1 = r 2 .
Let us next give a maximum principle for an elliptic equation in unbounded domains in R N , whose proof can be modified in proving the next key proposition.
Proposition A.3 Let D ⊂ R N be an unbounded domain, and let σ = σ(x) (x ∈ D) be a general conductivity satisfying where µ, M are positive constants. Assume that w ∈ H 1 loc (D) ∩ L ∞ (D) ∩ C 0 (D) satisfies −div(σ∇w) + λw ≥ 0 in D and w ≥ 0 on ∂D for some constant λ > 0. Then w ≥ 0 in D, and moreover, either w > 0 in D or w ≡ 0 in D.

Proof of Proposition
where δ > 0 is a constant which will be chosen later.
Finally, we give a comparison principle for two solutions of differential inequalities with different conductivities on a half-space of R N (see Lemma 3.3 in [16] for the case dealing with bounded domains).
we have that w 1 (z) ≤ w 2 (z N ) for z ∈ Ω.