Transmission Problems Arising in Czochralski Process of Crystal Growth

T. FUKAO, N. KENMOCHI and I. PAWLOW

Technical Reports of Mathematical Sciences Chiba University, Vol. 17 (2001), No. 4

T. Fukao , N. Kenmochi and I. Pawlow

Abstract. We consider a transmission problem of two parabolic PDEs as a mathematical model for the so-called Czochralski crystal growth process. Originally the Czochralski process is a three-phase system of solid-liquid-gas and involves the three interfaces of solid-liquid, liquid-gas and solid-gas which are to be determined as a part of the solution from the mathematical point of view. Nevertheless, in this paper, assuming that the solidgas and liquid-gas interfaces are known, we propose a transmission problem for the couple of a linear heat equation in the gas region and a nonlinear degenerate parabolic equation in the material region as a simplified mathematical model of the Czochralski process. Moreover we prove the existence and uniqueness of a weak solution in the variational sense.

1. Introduction

The Czochralski pulling method is wildly used for the production of a column of single crystal from the melt substance. The idea for pulling method due to Czochralski is quite simple. A crucible equipped with a heating system contains the melt substance and a pul-rod with seed crystal, which moves vertically and rotates flexibly, is positioned above the crucible (see Fig.1). The rod is dipped into the melt, and then lifted slowly with an appropriate speed v_{p} so that a meniscus surface is formed below the seed crystal and the melt attached to the crystal solidifies continuously. By controlling some thermal situations in the process one obtains the growth of a single crystal column with a desired radius as well as a desired growth pattern of the solid-liquid interface and temperature pattern in the crystal in order to improve the crystal quality.

In such a model of crystal growth the shape of crystal is determined by three kinetic equations of three interfaces between solid-liquid, liquid-gas and solid-gas. But, in this paper we suppose that the crystal radius is controlled to be constant and the trijunction curve on which three interfaces meet is prescribed, too. This might be designed by a good choice of the pulling velocity. As a consequence we assume that the movement of the material domain is prescribed.

This work is partially supported by a grant in aid of JSPS.

AMS Subject Classification 35R35

[figure: Rod; Seed; Gas; Crystal; Trijunction; Meniscus; Crucible (Heater); Melt; ν; ν^{+}; ν^{−}; n^{−}; n^{+}; Ω_{ℓ}(t); S(t); Ω_{g}(t); Ω_{s}(t); Γ_{gℓ}(t); Γ_{gs}(t); S(t); Γ_{ℓ}(t); Γ_{g}(t); Fig.1; Fig.2]

We use the following notation (see Fig.2): For 0 < T < ∞ and t ∈ [0, T], Ω_{ℓ}(t) : liquid (melt) region, Ω_{s}(t) : solid (crystal) region, Ω_{g}(t) : gas region, S(t) : solid-liquid interface, Ω := {Ω_{ℓ}(t) ∪ Ω_{s}(t) ∪ S(t)} ∪ Ω_{g}(t) ∪ {Γ_{gℓ}(t) ∪ Γ_{gs}(t)}, Γ(= ∂Ω) := Γ_{g}(t) ∪ Γ_{ℓ}(t), Ω_{m}(t) := Ω_{ℓ}(t) ∪ Ω_{s}(t) ∪ S(t): material region, Γ_{m}(t)(= ∂Ω_{m}(t)) := Γ_{ℓ}(t) ∪ {Γ_{gℓ} ∪ Γ_{gs}(t)}, Γ_{gm}(t) := Γ_{gℓ}(t) ∪ Γ_{gs}(t), ν = ν(t, x): 3-dimensional unit vector normal to Γ at x ∈ Γ, ν^{+} = ν^{+}(t, x): 3-dimensional unit vector normal to Γ_{gm}(t), pointing to Ω_{g}(t), ν^{−} = ν^{−}(t, x): 3-dimensional unit vector normal to Γ_{gm}(t), pointing to Ω_{m}(t), n^{+} = n^{+}(t, x): 3-dimensional unit vector normal to S(t), pointing to Ω_{s}(t), n^{−} = n^{−}(t, x): 3-dimensional unit vector normal to S(t), pointing to Ω_{ℓ}(t), Q := (0, T) × Ω, Σ := (0, T) × Γ,

Q_{i} := ⋃_{t∈(0,T)}{t} × Ω_{i}(t) i = m, ℓ, s, g,

Σ_{i} := ⋃_{t∈(0,T)}{t} × Γ_{i}(t) i = m, ℓ, g, gm, gℓ, gs,

S := ⋃_{t∈(0,T)}{t} × S(t).

Next, we denote by v_{Σ_{m}} := v_{Σ_{m}}(t, x) the normal speed of Γ_{m}(t) at (t, x) ∈ Σ_{m}. With this v_{Σ_{m}} the 4-dimensional unit vector outward normal to Σ_{m} at each (t, x) ∈ Σ_{m} is given by

ν^{+} := (ν_{t}^{+}, ν_{x}^{+}) = \frac{1}{(|v_{Σ_{m}}|^{2} + 1)^{\frac{1}{2}}} (−v_{Σ_{m}}, ν^{+}).

Similarly, with the normal speed v_{S} := v_{S}(t, x) of S(t) at (t, x) ∈ S, the 4-dimensional unit vector normal to S, pointing to the solid region, is given by

n^{+} := (n_{t}^{+}, n_{x}^{+}) = \frac{1}{(|v_{S}|^{2} + 1)^{\frac{1}{2}}} (−v_{S}, n^{+}).

It is easily understood that by the crystal growth the shape of material domain Ω_{m}(t) changes with time and it yields a 3-dimensional convective vector field v := v(t, x) in Q. The determination of v is also one of the important questions in the mathematical modeling of the Czochralski process of crystal growth. It seems that v obeys mainly the pulling velocity v_{p} in the crystal and is a solution of the incompressible Navier-Stokes (or simply Stokes) equation in the melt (see [3, 5]). Nevertheless, in this paper, we assume that the convective field v is prescribed, too, at least satisfying that

divv = 0 in Q,  (1.1)
v · ν^{+} = v_{Σ_{m}} on Σ_{gm}(t),  (1.2)
v · ν = 0 on Σ_{i}(t), i = ℓ, g.  (1.3)

Now, from the usual energy balance laws the following system is derived, where θ_{i} := θ_{i}(t, x) is the temperature on Q_{i}, i = ℓ, s, g.

c_{i}(θ_{i,t} + v · ∇θ_{i}) − k_{i}Δθ_{i} = f  in Q_{i}, i = ℓ, s, g,  (1.4)

θ_{ℓ} = θ_{s} = 0, k_{ℓ} \frac{∂θ_{ℓ}}{∂n^{+}} + k_{s} \frac{∂θ_{s}}{∂n^{−}} = L(v · n^{+} − v_{S}) on S,  (1.5)
k_{i}θ_{i} = k_{g}θ_{g}, k_{i} \frac{∂θ_{i}}{∂ν^{+}} + k_{g} \frac{∂θ_{g}}{∂ν^{−}} = 0 on Σ_{gi}, i = ℓ, s,  (1.6)

subject to the boundary conditions

k_{i} \frac{∂θ_{i}}{∂ν} + n_{0}k_{i}θ_{i} = p on Σ_{i}, i = ℓ, g,  (1.7)

and initial conditions

θ_{i}(0, ·) = θ_{i0} on Ω_{i}(0), i = ℓ, s, g,  S(0) = S_{0}.  (1.8)

Here note that (1.5) is called the Stefan conditions, where the phase change temperature is supposed to be 0 for simplicity, and (1.6) is called the transmission conditions between the material and gas; c_{i}, k_{i}. i = ℓ, s, g, and L are positive constants which are the specific heats, the heat conductivities and latent heat, respectively; f is a given heat source on Q, p is the boundary data prescribed on Σ; θ_{i0}, i = ℓ, s, g, are the initial temperature on Ω_{i}(0) and S_{0} is the initial location of the solid-liquid interface, satisfying that

θ_{0} > 0 on Ω_{ℓ}(0), θ_{0} < 0 on Ω_{s}(0), θ_{0} = 0 on S_{0}.

As is well known, by using the enthalpy we reformulate the above system as a weak variational form. In this paper we prove its well-posedness by using the abstract theory of time-dependent subdifferential operators (cf. [1, 4, 7, 8, 9, 16]).

2. Weak formulation

The enthalpy u is defined as follows:

u := \{ \begin{matrix}c_{s}θ & if x ∈ Ω_{m}(t), θ < 0, \\ [0, L] & if x ∈ Ω_{m}(t), θ = 0, \\ c_{ℓ}θ + L & if x ∈ Ω_{m}(t), θ > 0, \\ c_{g}θ & if x ∈ \overline{Ω_{g}(t)}. \end{matrix}

Moreover we define a function β(t, x, r) on Q × R by

β(t, x, r) := \{ \begin{matrix}β_{1}(r) & := & \{ \begin{matrix}\frac{k_{s}}{c_{s}} r & if & r < 0, \\ 0 & if & 0 ≤ r ≤ L, \\ \frac{k_{ℓ}}{c_{ℓ}} (r − L) & if & r > L, \end{matrix} \} & if x ∈ Ω_{m}(t), \\ β_{2}(r) & := & \frac{k_{g}}{c_{g}} r & if x ∈ \overline{Ω_{g}(t)}. \end{matrix}

Then β(t, x, r) is a non-decreasing Lipschitz continuous function in r on R for a.e. (t, x) ∈ Q.

By using the enthalpy u our problem (1.4)-(1.8) is reformulated as an initial-boundary value problem for a degenerate parabolic equation in a cylindrical domain Q of the following form

(E) \{ \begin{matrix}u_{t} − Δβ(u) + v · ∇u = f in Q, (2.1) \\ \frac{∂β(u)}{∂ν} + n_{0}β(u) = p on Σ, \\ u(0) = u_{0} on Ω, \end{matrix}

where β(u) := β(t, x, u) and

u_{0} := c_{s}θ_{0}χ_{Ω_{s}(0)} + (c_{ℓ}θ_{0} + L)χ_{Ω_{ℓ}(0)} + c_{g}θ_{0}χ_{Ω_{g}(0)}.

with characteristic functions χ_{Ω_{i}(0)} of Ω_{i}(0), i = ℓ, s, g. In fact, multiply equations (1.4) by any test function η ∈ C^{2}(\overline{Q}) with η(T) := η(T, ·) = 0 on Ω, and then integrate the resultants over Q_{i}, i = ℓ, s, g, respectively, to get the following identities

∫_{Q_{i}} c_{i}θ_{t}ηdxdt − ∫_{Q_{i}} k_{i}Δθηdxdt + ∫_{Q_{i}} c_{i}(v · ∇θ)ηdxdt = ∫_{Q_{i}} fηdxdt, i = ℓ, s, g, (2.2)

where the subindex i of θ_{i} is omitted, since no confusion arises. With the help of (1.2), (1.3) and the Green-Stokes’ formula as well as the relations dΣ_{i} = (|v_{Σ_{m}}|^{2}+1)^{\frac{1}{2}} dΓ_{i}(t)dt, i = gm, gℓ, gs, dΣ_{i} = dΓ_{i}(t)dt, i = ℓ, g, and dS = (|v_{S}|^{2} + 1)^{\frac{1}{2}} dS(t)dt, we have

∫_{Q_{ℓ}} c_{ℓ}θ_{t}ηdxdt + ∫_{Q_{s}} c_{s}θ_{t}ηdxdt + ∫_{Q_{g}} c_{g}θ_{t}ηdxdt

= ∫_{Σ_{gℓ}} c_{ℓ}θην_{t}^{+}dΣ_{gℓ} + ∫_{Ω_{ℓ}(0)} c_{ℓ}θ_{0}η(0)(−1)dx − ∫_{Q_{ℓ}} c_{ℓ}θη_{t}dxdt

+ ∫_{Σ_{gs}} c_{s}θην_{t}^{+}dΣ_{gs} + ∫_{Ω_{s}(0)} c_{s}θ_{0}η(0)(−1)dx − ∫_{Q_{s}} c_{s}θη_{t}dxdt

+ ∫_{Σ_{gℓ}} c_{g}θην_{t}^{−}dΣ_{gℓ} + ∫_{Σ_{gs}} c_{g}θην_{t}^{−}dΣ_{gs} + ∫_{Ω_{g}(0)} c_{g}θ_{0}η(0)(−1)dx − ∫_{Q_{g}} c_{g}θη_{t}dxdt

= − ∫_{Q} uη_{t}dxdt − ∫_{Ω} u_{0}η(0)dx

+ ∫_{Q_{ℓ}} Lη_{t}dxdt + ∫_{Ω_{ℓ}(0)} Lη(0)dx + ∫_{Σ_{gℓ}} c_{ℓ}θη(−v_{Σ_{m}})dΓ_{gℓ}(t)dt

+ ∫_{Σ_{gs}} c_{s}θη(−v_{Σ_{m}})dΓ_{gs}(t)dt + ∫_{Σ_{gℓ}} c_{g}θη(v_{Σ_{m}})dΓ_{gℓ}(t)dt + ∫_{Σ_{gs}} c_{g}θη(v_{Σ_{m}})dΓ_{gs}(t)dt,

and by (1.6) we have

− ∫_{Q_{ℓ}} k_{ℓ}Δθηdxdt − ∫_{Q_{s}} k_{s}Δθηdxdt − ∫_{Q_{g}} k_{g}Δθηdxdt

= − ∫_{0}^{T} \{∫_{Γ_{ℓ}(t)} k_{ℓ} \frac{∂θ}{∂ν} ηdΓ_{ℓ}(t) + ∫_{Γ_{gℓ}(t)} k_{ℓ} \frac{∂θ}{∂ν^{+}} ηdΓ_{gℓ}(t) + ∫_{S(t)} k_{ℓ} \frac{∂θ}{∂n^{+}} ηdS(t)\} dt

+ ∫_{Q_{ℓ}} k_{ℓ}(∇θ · ∇η)dxdt

− ∫_{0}^{T} \{∫_{Γ_{gs}(t)} k_{s} \frac{∂θ}{∂ν^{+}} ηdΓ_{gs}(t) + ∫_{S(t)} k_{s} \frac{∂θ}{∂n^{−}} ηdS(t)\} dt + ∫_{Q_{s}} k_{s}(∇θ · ∇η)dxdt

− ∫_{0}^{T} \{∫_{Γ_{g}(t)} k_{g} \frac{∂θ}{∂ν} ηdΓ_{g}(t) + ∫_{Γ_{gℓ}(t)} k_{g} \frac{∂θ}{∂ν^{−}} ηdΓ_{gℓ}(t) + ∫_{Γ_{gs}(t)} k_{g} \frac{∂θ}{∂ν^{−}} ηdΓ_{gs}(t)\} dt

+ ∫_{Q_{g}} k_{g}(∇θ · ∇η)dxdt

= ∫_{Q} ∇β(u) · ∇ηdxdt − ∫_{Σ} \frac{∂β(u)}{∂ν}ηdΓdt − ∫_{0}^{T} ∫_{S(t)} (k_{ℓ} \frac{∂θ}{∂n^{+}} + k_{s} \frac{∂θ}{∂n^{−}} ) ηdS(t)dt.

Moreover it follows from (1.1) that

∫_{Q_{ℓ}} c_{ℓ}(v · ∇θ)ηdxdt + ∫_{Q_{s}} c_{s}(v · ∇θ)ηdxdt + ∫_{Q_{g}} c_{g}(v · ∇θ)ηdxdt

= ∫_{0}^{T} {∫_{Γ_{ℓ}(t)} c_{ℓ}θ(v · ν)ηdΓ_{ℓ}(t) + ∫_{Γ_{gℓ}(t)} c_{ℓ}θ(v · ν^{+})ηdΓ_{gℓ}(t)} dt − ∫_{Q_{ℓ}} c_{ℓ}θ(v · ∇η)dxdt

+ ∫_{0}^{T} {∫_{Γ_{gs}(t)} c_{s}θ(v · ν^{+})ηdΓ_{gs}(t)} dt − ∫_{Q_{s}} θ(v · ∇η)dxdt

+ ∫_{0}^{T} {∫_{Γ_{g}(t)} c_{g}θ(v · ν)ηdΓ_{g}(t) + ∫_{Γ_{gℓ}(t)} c_{g}θ(v · ν^{−})ηdΓ_{gℓ}(t)

+ ∫_{Γ_{gs}(t)} c_{g}θ(v · ν^{−})ηdΓ_{gs}(t)} dt − ∫_{Q_{g}} c_{g}θ(v · ∇η)dxdt

= − ∫_{Q} u(v · ∇η)dxdt + ∫_{Q_{ℓ}} L(v · ∇η)dxdt

+ ∫_{Σ_{gℓ}} c_{ℓ}θ(v · ν^{+})ηdΓ_{gℓ}(t)dt + ∫_{Σ_{gs}} c_{s}θ(v · ν^{+})ηdΓ_{gs}(t)dt
+ ∫_{Σ_{gℓ}} c_{g}θ(v · ν^{−})ηdΓ_{gℓ}(t)dt + ∫_{Σ_{gs}} c_{g}θ(v · ν^{−})ηdΓ_{gs}(t)dt.

Using (1.2) and (1.3) we get

− ∫_{Q} uη_{t}dxdt + ∫_{Q} ∇β(u) · ∇ηdxdt − ∫_{Σ} \frac{∂β(u)}{∂ν}ηdΓdt − ∫_{Q} u(v · ∇η)dxdt

+ ∫_{Q_{ℓ}} Lη_{t}dxdt + ∫_{Ω_{ℓ}(0)} Lη(0)dx − ∫_{S} (k_{ℓ} \frac{∂θ}{∂n^{+}} + k_{s} \frac{∂θ}{∂n^{−}} ) ηdS(t)dt + ∫_{Q_{ℓ}} L(v · ∇η)dxdt

= ∫_{Q} fηdxdt + ∫_{Ω} u_{0}η(0)dx.

Moreover on account of the Stefan condition (1.5), we get the following equation:

∫_{Q_{ℓ}} Lη_{t}dxdt + ∫_{Ω_{ℓ}(0)} Lη(0)dx − ∫_{S} (k_{ℓ} \frac{∂θ}{∂n^{+}} + k_{s} \frac{∂θ}{∂n^{−}} ) ηdS(t)dt + ∫_{Q_{ℓ}} L(v · ∇η)dxdt

= ∫_{Σ_{gℓ}} Lην_{t}^{+}dΣ_{gℓ} + ∫_{S} Lηn_{t}^{+}dS + ∫_{Ω_{ℓ}(0)} Lη(0)dx

− ∫_{S} L(v · n^{+} − v_{S})dS(t)dt + ∫_{Σ_{gℓ}} Lη(v · ν^{+})dΓ_{gℓ}(t)dt + ∫_{S} Lη(v · n^{+})dS(t)dt

= 0.

Therefore, using (1.7), we arrive at the following equality

− ∫_{Q} uη_{t}dxdt + ∫_{Q} ∇β(u) · ∇ηdxdt + n_{0} ∫_{Σ} β(u)ηdΓdt − ∫_{Q} u(v · ∇η)dxdt
= ∫_{Q} fηdxdt + ∫_{Σ} pηdΓdt + ∫_{Ω} u_{0}η(0)dx for all η ∈ W,  (2.3)

where W := {η ∈ H^{1}(Q); η(T) = 0 on Ω}. This is the variational form of (E) which is regarded as a weak formulation of (1.4)-(1.8) as usual. Problem (E) is called the enthalpy formulation of (1.4)-(1.8).

Remark 2.1 In the transmission condition (1.6), equalities k_{i}θ_{i} = k_{g}θ_{g}, i = ℓ, s, imply that the temperature is possibly discontinuous on Σ_{gm}, since k_{i} ≠ k_{g} in general. Therefore, this may not be acceptable from some physical points of view. When the continuity of the temperature is required in Q, k_{i}θ_{i} = k_{g}θ_{g} should be replaced by θ_{i} = θ_{g} on Σ_{gm}. In this case, the weak variational identity is different from the above (2.3) and we need a different mathematical treatment of the problem and this will be studied in a forthcoming paper of the authors. Also, some mathematical modelings of this sort of crystal growth have been discussed by Pawlow [13] in more general settings.

For the solvability of two-phase Stefan problems without convection in cylindrical domains, the time-dependent subdifferential operator theory was skillfully applied by Damlamian [4]. The case of non-cylindrical domains with convection was treated by Fukao, Kenmochi and Pawlow [6] by applying the theory of quasi-linear equations of parabolic type.

3. Statements of main results

In this section we formulate our main result with some functional settings. First we introduce a function space V := H^{1}(Ω) equipped with norm

|u|_{V}^{2} := |∇u|_{L^{2}(Ω)}^{2} + n_{0} ∫_{Γ} |u|^{2}dΓ for all u ∈ V,

which is an equivalent norm to the standard one of H^{1}(Ω). The dual space of V is denoted by V^{∗} and the duality pairing between V^{∗} and V is denoted by ⟨·, ·⟩_{V^{∗},V}; note that if u^{∗} ∈ L^{2}(Ω), then

⟨u^{∗}, u⟩_{V^{∗},V} = (u^{∗}, u)_{L^{2}(Ω)}, for all u ∈ V,

where (·, ·)_{L^{2}(Ω)} denotes the inner product in L^{2}(Ω), The duality mapping F : V → V^{∗} is defined by

⟨F u_{1}, u_{2}⟩_{V^{∗},V} := ∫_{Ω} ∇u_{1} · ∇u_{2}dx + n_{0} ∫_{Γ} u_{1}u_{2}dΓ for all u_{2} ∈ V.

Moreover, defining an inner product (·, ·)_{V^{∗}} in V^{∗} by

(u_{1}^{∗}, u_{2}^{∗})_{V^{∗}} := ⟨u_{1}^{∗}, F^{−1}u_{2}^{∗}⟩_{V^{∗},V} , for all u_{1}^{∗}, u_{2}^{∗} ∈ V^{∗}.

we see that V^{∗} is a Hilbert space with the following dense and compact imbeddings:

V ↪ L^{2}(Ω) ↪ V^{∗}.

Now we give the definition of a weak solution of (E).

Definition 3.1 u : [0, T] → L^{2}(Ω) is called a weak solution of (E) if u satisfies following conditions:

(D1) u ∈ W^{1,2}(0, T; V^{∗}) ∩ L^{∞}(0, T; L^{2}(Ω)).

(D2) β(u) ∈ L^{2}(0, T; V).

(D3) u satisfies the variational identity (2.3).

Our main result is stated under the following geometric assumption (⋆) on the time dependence of the prescribed material domain Ω_{m}(t):

(⋆) Assume that Ω is a bounded domain in R^{3} with smooth boundary Γ and assume that there exists a one-parameter family of C^{2}-diffeomorphims Θ := Θ(t, ·) on \overline{Ω}, 0 ≤ t ≤ T, such that Θ ∈ C^{2}(\overline{Q}) and

Θ(t, \overline{Ω_{i}(t)}) = \overline{Ω_{i}(0)}, i = m, g, 0 ≤ t ≤ T.

Theorem 3.1 Assume that condition (⋆) holds, f ∈ L^{2}(Q), p ∈ C^{1}(\overline{Σ}), u_{0} ∈ L^{2}(Ω) and β(u_{0}) ∈ H^{1}(Ω). Also assume that v ∈ C^{1}(\overline{Q})^{3} and (1.1), (1.2) and (1.3) hold. Then there exists one and only one weak solution u of (E).

The proofs of existence and uniqueness of a weak solution of (E) will be given in sections 4 and 5, respectively.

4. Existence proof

Let us consider the existence question of a weak solution of (E). Our approach is based on the time-dependent subdifferential operator theory. To do so we define a function ϕ^{t} : L^{2}(Ω) → (−∞, ∞], for each t ∈ [0, T] by this formula:

ϕ^{t}(z) = \{ \begin{matrix}∫_{Ω_{m}(t)} β̂_{1}(z(x))dx + ∫_{Ω_{g}(t)} β̂_{2}(z(x))dx & if & z ∈ L^{2}(Ω), \\ +∞ & if & z ∈ V^{∗} \ L^{2}(Ω), \end{matrix}

where β̂_{i}(r) := ∫_{0}^{r} β_{i}(s)ds, i = 1, 2. Clearly ϕ^{t} is a proper, lower semi-continuous and convex function on V^{∗}. We denote by ∂ϕ^{t} the subdifferential of ϕ^{t}; by definition (cf. [2]), z^{∗} ∈ ∂ϕ^{t}(z) in V^{∗} means that

(z^{∗}, w − z)_{V^{∗}} ≤ ϕ^{t}(w) − ϕ^{t}(z), for all w ∈ V^{∗}.

The characterization of ∂ϕ^{t} is given in the following lemma. Lemma 4.1 Let z ∈ D(∂ϕ^{t}) and β(z) := β_{1}(z)χ_{Ω_{m}(t)} + β_{2}(z)χ_{Ω_{g}(t)}. Then z^{∗} ∈ ∂ϕ^{t}(z) in V^{∗} if and only if

(F^{−1}z^{∗}, w)_{L^{2}(Ω)} = (β(z), w)_{L^{2}(Ω)} for all w ∈ L^{2}(Ω).  (4.1)

Moreover, this is equivalent to the statement that β(z) ∈ V and

⟨z^{∗}, η⟩_{V^{∗},V} = ∫_{Ω} ∇β(z) · ∇ηdx + n_{0} ∫_{Γ} β(z)ηdΓ for all η ∈ V.  (4.2)

Proof. We have z ∈ L^{2}(Ω) by D(∂ϕ^{t}) ⊂ D(ϕ^{t}) = L^{2}(Ω) and by definition

(z^{∗}, w − z)_{V^{∗}} ≤ ∫_{Ω_{m}(t)} (β̂_{1}(w(x)) − β̂_{1}(z(x))) dx + ∫_{Ω_{g}(t)} (β̂_{2}(w(x)) − β̂_{2}(z(x))) dx,

for all w ∈ L^{2}(Ω).

Here, choose z + ε(w̃ − z), ε > 0, as w in the above inequality for each w̃ ∈ L^{2}(Ω), divide the both sides by ε and finally pass to the limit ε → 0. Then we get

(z^{∗}, w̃ − z)_{V^{∗}} ≤ (β_{1}(z)χ_{Ω_{m}(t)} + β_{2}(z)χ_{Ω_{g}(t)}, w̃ − z)_{L^{2}(Ω)} for all w̃ ∈ L^{2}(Ω).

From the arbitrariness of w̃ ∈ L^{2}(Ω) and the relation (z^{∗}, w̃ − z)_{V^{∗}} = (F^{−1}z^{∗}, w̃ − z)_{L^{2}(Ω)} it follows that

(F^{−1}z^{∗}, w)_{L^{2}(Ω)} = (β_{1}(z)χ_{Ω_{m}(t)} + β_{2}(z)χ_{Ω_{g}(t)}, w)_{L^{2}(Ω)} for all w ∈ L^{2}(Ω),

namely (4.1) holds. It is easy to show the converse assertion. Also, by the definition of the duality mapping F, (4.1) and (4.2) are equivalent to each other. □

Now we are ready to apply the existence result in [16] (or [7]) for nonlinear evolution equations including time-dependent subdifferential operators and small perturbations. Consider the following initial value problem (K):

(K) \{ \begin{matrix}u_{t}(t) + ∂ϕ^{t}(u(t)) + g(t, u(t)) ∋ f^{∗}(t) in V^{∗} for a.e. t ∈ [0, T], \\ u(0) = u_{0} in V^{∗}, \end{matrix}

where g(t, z) : L^{2}(Ω) → V^{∗} is an operator defined by:

⟨g(t, z), η⟩_{V^{∗},V} := − ∫_{Ω} z(x)(v(t) · ∇η)(x)dx for all η ∈ V.

for each t ∈ [0, T] and z ∈ L^{2}(Ω). In the next lemma we check the time-dependence of ϕ^{t} and the conditions required in order to apply the abstract result in [16] (or [7]) to problem (K).

Lemma 4.2 (i) There exists a positive constant α_{0} with the following property: for each s, t ∈ [0, T], s ≤ t and z ∈ D(ϕ^{s}) there exists z_{1} ∈ D(ϕ^{t}) such that

|z_{1} − z|_{V^{∗}} ≤ α_{0}|t − s|(1 + |ϕ^{s}(z)|^{\frac{1}{2}} ),  (4.3)
|ϕ^{t}(z_{1}) − ϕ^{s}(z)| ≤ α_{0}|t − s|(1 + |ϕ^{s}(z)|).  (4.4)

Furthermore the set L_{r} := ⋃_{0≤t≤T} {z ∈ V^{∗}; ϕ^{t}(z) + |z|_{V^{∗}}^{2} ≤ r} is bounded in L^{2}(Ω) and hence relatively compact in V^{∗} for each r ≥ 0. (ii) The operator g(t, ·) : D(g(t, ·)) = L^{2}(Ω) → V^{∗}, t ∈ [0, T], satisfies that

|g(t, z)|_{V^{∗}}^{2} ≤ C_{0}ϕ^{t}(z) + C_{1} for all t ∈ [0, T] and all z ∈ L^{2}(Ω),  (4.5)

where C_{0} and C_{1} are positive constants, and that if {z_{n}} ⊂ L^{2}(Ω), {t_{n}} ⊂ [0, T], {ϕ^{t_{n}}(z_{n})} is bounded, z_{n} → z in V^{∗} and t_{n} → t, as n → ∞, then g(t_{n}, z_{n}) → g(t, z) weakly in V^{∗} as n → ∞. Proof. (i) By our geometric assumption (⋆) the mapping Θ_{t,s} := Θ(s, ·)^{−1} ∘ Θ(t, ·) is a C^{2}-diffeomorphism on \overline{Ω} for any s, t ∈ [0, T] such that Θ_{t,s}(\overline{Ω_{i}(t)}) = \overline{Ω_{i}(s)}, i = g, m. Also there exist positive constants α_{1} and α_{2} such that

|det.J_{Θ_{t,s}}(x) − 1| ≤ α_{1}|t − s|, for all s, t ∈ [0, T], and x ∈ Ω,

for each η ∈ V

|η ∘ Θ_{t,s} − η|_{L^{2}(Ω)} ≤ α_{2}|t − s||η|_{V} , for all s, t ∈ [0, T],

(cf. Kenmochi [8; Lemma 3.2.2]), where J_{Θ_{t,s}} is the Jacobian of Θ_{t,s} and det.J_{Θ_{t,s}} is its determinant.

Now given z ∈ D(ϕ^{s}) = L^{2}(Ω) we take as z_{1} the function z ∘ Θ_{t,s} ∈ L^{2}(Ω) = D(ϕ^{t}). Then for each η ∈ V we have:

⟨z_{1} − z, η⟩_{V^{∗},V} = (z_{1} − z, η)_{L^{2}(Ω)}
= ∫_{Ω} z_{1}(x)η(x)dx − ∫_{Ω} z(x)η(x)dx
= ∫_{Ω} z(Θ_{t,s}(x))η(x)dx − ∫_{Ω} z(x)η(x)dx
= ∫_{Ω} z(x)η(Θ_{t,s}^{−1}(x))det.J_{Θ_{t,s}}(x)dx − ∫_{Ω} z(x)η(x)dx
≤ ∫_{Ω} |z(x)||η(Θ_{s,t}(x))||det.J_{Θ_{s,t}}(x) − 1|dx + ∫_{Ω} |z(x)||η(Θ_{s,t}(x)) − η(x)|dx
≤ α_{1}|t − s| ∫_{Ω} |z(x)||η(Θ_{s,t}(x))|dx + |z|_{L^{2}(Ω)}||η ∘ Θ_{s,t} − η|_{L^{2}(Ω)}
≤ α_{1}|t − s||z|_{L^{2}(Ω)}|η ∘ Θ_{s,t}|_{L^{2}(Ω)} + α_{2}|t − s||z|_{L^{2}(Ω)}|η|_{V} .

Since |det.J_{Θ_{s,t}}| ≤ 1 + α_{1}T, we have |η ∘ Θ_{s,t}|_{L^{2}(Ω)} ≤ α_{3}|η|_{L^{2}(Ω)}, where α_{3} := (1 + α_{1}T)^{\frac{1}{2}} . Therefore it follows from the above inequalities that

⟨z_{1} − z, η⟩_{V^{∗},V} ≤ α_{4}|t − s||z|_{L^{2}(Ω)}|η|_{V} ,  (4.6)

where α_{4} := α_{1}α_{3} + α_{2}. By the definition of β̂_{i} there are positive constant c_{β} and c′_{β} such that

β̂_{i}(r) ≥ c_{β}|r|^{2} − c′_{β} for all r ∈ R and i = 1, 2,

so that it is easy to find a positive constant c′′_{β} such that

|w|_{L^{2}(Ω)} ≤ c′′_{β}(1 + |ϕ^{s}(w)|^{\frac{1}{2}} ) for all w ∈ L^{2}(Ω) and s ∈ [0, T].  (4.7)

Combining this with (4.6) we immediately get (4.3) with any positive constant α_{0} with α_{0} ≥ α_{4}c′′_{β}.

Next, we recall

ϕ^{t}(z_{1}) − ϕ^{s}(z)

= ∫_{Ω_{m}(t)} β̂_{1}(z_{1}(x))dx + ∫_{Ω_{g}(t)} β̂_{2}(z_{1}(x))dx − ∫_{Ω_{m}(s)} β̂_{1}(z(x))dx − ∫_{Ω_{g}(s)} β̂_{2}(z(x))dx.

Here

∫_{Ω_{i}(t)} β̂_{j}(z_{1}(x))dx − ∫_{Ω_{i}(s)} β̂_{j}(z(x))dx ≤ ∫_{Ω_{i}(t)} β̂_{j}(z(Θ_{t,s}(x))dx − ∫_{Ω_{i}(s)} β̂_{j}(z(x))dx
≤ ∫_{Ω_{i}(s)} {β̂_{j}(z(x))det.J_{Θ_{t,s}}(x) − β̂_{j}(z(x))} dx
≤ ∫_{Ω_{i}(s)} β̂_{j}(z(x))|det.J_{Θ_{t,s}}(x) − 1|dx
≤ α_{1}|t − s| ∫_{Ω_{i}(s)} β̂_{j}(z(x))dx,

for (i, j) = (m, 1) and (g, 2). Now it is easy to see that (4.4) holds by taking α_{0} so that α_{0} > α_{1}. Also, for each r ≥ 0 the boundedness of L_{r} in L^{2}(Ω) is immediately obtained from (4.7).

All assertions in (ii) are easily seen from the definition of g(t, ·). □

Now, we define f^{∗} ∈ L^{2}(0, T; V^{∗}) by

⟨f^{∗}(t), η⟩_{V^{∗},V} := ∫_{Ω} f(t, x)η(x)dx + ∫_{Γ} p(t, x)η(x)dΓ for all η ∈ V,

and consider the evolution equation (K). Applying an existence result of [16] (or see [7]), we see by Lemma 4.2 that there exists at least one solution u of (K) in W^{1,2}(0, T; V^{∗}) ∩ L^{∞}(0, T; L^{2}(Ω)). Moreover we see that this solution u is a weak solution of (E), too. In fact, since

f^{∗}(t) − u_{t}(t) − g(t, u(t)) ∈ ∂ϕ^{t}(u(t)) in V^{∗} for a.e. t ∈ [0, T],

it follows from Lemma 4.1 that

F^{−1}(f^{∗} − u_{t} − g(t, u)) = β(u) ∈ L^{2}(0, T; V),

and for each η ∈ H^{1}(Q) with η(T) = 0

⟨f^{∗}(t) − u_{t}(t) − g(t, u(t)), η(t)⟩_{V^{∗},V}
= ∫_{Ω} ∇β(u(t)) · ∇η(t)dx + n_{0} ∫_{Γ} β(u(t))η(t)dΓ.

Therefore, integrating this in time on [0, T] and using integration by parts, we see that u satisfies the variational identity (2.3). Thus u is a weak solution of (E) in the sense of Definition 3.1 and the existence proof of Theorem 3.1 has been accomplished.

5. Uniqueness proof

In this section we prove the uniqueness of a weak solution of (E) obtained in the previous section. The idea of our uniqueness proof is due to Ladyženskaja, Solonnikov and Ural’ceva [11; Chapter 3, Section 3], and this was also extensively used in Niezgódka and Pawlow [12], Rodrigues and Yi [15] and Rodrigues [14] for the uniqueness proof of generalized Stefan problems and continuous casting problems.

Let u_{1} and u_{2} be two weak solutions. Then, by taking the difference of their weak variational forms and using the Green’s formula we have

− ∫_{Q}(u_{1} − u_{2})η_{t}dxdt − ∫_{Q}(β(u_{1}) − β(u_{2}))Δηdxdt − ∫_{Q}(u_{1} − u_{2})(v · ∇η)dxdt

+ ∫_{Σ}(β(u_{1}) − β(u_{2}))\frac{∂η}{∂ν} dΓdt + n_{0} ∫_{Σ}(β(u_{1}) − β(u_{2}))ηdΓdt = 0  (5.1)
for all η ∈ W.

As usual, consider the function

b(t, x) := \{ \begin{matrix}\frac{β(u_{1}(t, x)) − β(u_{2}(t, x))}{u_{1}(t, x) − u_{2}(t, x)} & if & u_{1}(t, x) ≠ u_{2}(t, x), \\ 0 & if & u_{1}(t, x) = u_{2}(t, x), \end{matrix}

which is non-negative and bounded on Q. By virtue of (5.1)

− ∫_{Q}(u_{1} − u_{2})\{η_{t} + bΔη + v · ∇η\}dxdt + ∫_{Σ}(β(u_{1}) − β(u_{2})) \{\frac{∂η}{∂ν} + n_{0}η\} dΓdt = 0 (5.2)
for all η ∈ W.

We now take smooth and strictly positive approximations b_{ε} of b such that

b ≤ b_{ε} a.e. on Q, ε ≤ b_{ε} ≤ B_{0} a.e. on Q,  (5.3)
b_{ε} → b a.e. on Q as ε → 0,

where B_{0} is a positive constant, and consider the following auxiliary linear parabolic equation for any given ℓ ∈ D(Q):

\{ \begin{matrix}η_{ε,t} + b_{ε}Δη_{ε} + v · ∇η_{ε} = ℓ in Q, \\ \frac{∂η_{ε}}{∂ν} + n_{0}η_{ε} = 0 on Σ, \\ η_{ε}(T, ·) = 0 on Ω. \end{matrix}  (5.4)

By the general theory (cf. [11;Chapter 3]) of linear parabolic equations this problem has a unique solution η_{ε} ∈ H^{2+α,1+α/2}(Q̄) for a certain number 0 < α < 1.

Now we are going to show that {η_{ε}} is bounded in L^{∞}(0, T; V) and {b_{ε}^{1/2}Δη_{ε}} is bounded in L^{2}(Q). To this end we prove: Lemma 5.1

|η_{ε}(s)|_{V}^{2} + 2 ∫_{s}^{T} ∫_{Ω} b_{ε}|Δη_{ε}|^{2}dxdt
≤ \{6|v|_{C^{1}(\overline{Q})^{3}} + 1\} ∫_{s}^{T} |η_{ε}(t)|_{V}^{2} dxdt + n_{0} ∫_{s}^{T} ∫_{Γ} v · ∇(η_{ε}^{2})dΓdt + ∫_{0}^{T} |ℓ(t)|_{V}^{2} dt  (5.5)
for all s ∈ [0, T] and ε ∈ (0, 1].

Proof. Multiplying the first equation in (5.4) by Δη_{ε} and integrating over Ω with respect to x, we get for any fixed time t

∫_{Ω} \frac{∂η_{ε}}{∂t} Δη_{ε}dx + ∫_{Ω} |b_{ε}||Δη_{ε}|^{2}dx + ∫_{Ω}(v · ∇η_{ε})Δη_{ε}dx = − ∫_{Ω} ∇ℓ · ∇η_{ε}dx.  (5.6)

Here

∫_{Ω} \frac{∂η_{ε}}{∂t} Δη_{ε}dx = − ∫_{Ω} (∇\frac{∂η_{ε}}{∂t} · ∇η_{ε}) dx + ∫_{Γ} \frac{∂η_{ε}}{∂t} \frac{∂η_{ε}}{∂ν} dΓ
= −\frac{1}{2} \frac{d}{dt} ∫_{Ω} |∇η_{ε}|^{2}dx − \frac{n_{0}}{2} \frac{d}{dt} ∫_{Γ} |η_{ε}|^{2}dΓ
= −\frac{1}{2} \frac{d}{dt}|η_{ε}(t)|_{V}^{2} .

and

− ∫_{Ω}(v · ∇η_{ε})Δη_{ε}dx
= ∫_{Ω} ∇(v · ∇η_{ε}) · ∇η_{ε}dx − ∫_{Γ}(v · ∇η_{ε})\frac{∂η_{ε}}{∂ν} dΓ
= ∫_{Ω} ∑_{i,j=1}^{3} \{\frac{∂v_{j}}{∂x_{i}} \frac{∂η_{ε}}{∂x_{j}} \frac{∂η_{ε}}{∂x_{i}} + v_{j} \frac{∂^{2}η_{ε}}{∂x_{j}∂x_{i}} \frac{∂η_{ε}}{∂x_{i}} \} dx + n_{0} ∫_{Γ}(v · ∇η_{ε})η_{ε}dΓ
≤ 3|v|_{C^{1}(\overline{Q})^{3}} ∫_{Ω} |∇η_{ε}|^{2}dx + ∫_{Ω} v · ∇ (\frac{1}{2}|∇η_{ε}|^{2}) dx + \frac{n_{0}}{2} ∫_{Γ} v · ∇(η_{ε}^{2})dΓ.

Now, by using conditions (1.1) and (1.3) we see that

∫_{Ω} v · ∇ (\frac{1}{2}|∇η_{ε}|^{2}) dx = ∫_{Ω} div (\frac{1}{2}|∇η_{ε}|^{2}v) dx
= ∫_{Γ} \frac{1}{2}|∇η_{ε}|^{2}(v · ν)dΓ
= 0.

It follows from (5.6) with the above estimates that

−\frac{1}{2} \frac{d}{dt}|η_{ε}|_{V}^{2} + ∫_{Ω} |b_{ε}||Δη_{ε}|^{2}dx ≤ (3|v|_{C^{1}(\overline{Q})^{3}} + \frac{1}{2})|η_{ε}|_{V}^{2} + \frac{1}{2}|ℓ|_{V}^{2} + n_{0} ∫_{Γ} v · ∇(η_{ε}^{2})dΓ.

for all t ∈ [0, T], so that (5.5) is obtained by integrating this inequality over [s, T] in time.

□

Lemma 5.2 There is a positive constant B_{1} such that

∫_{Γ} v(t) · ∇(η_{ε}^{2}(t))dΓ ≤ B_{1}|η_{ε}(t)|_{L^{2}(Γ)}^{2} for all t ∈ [0, T] and ε ∈ (0, 1].

Proof. Our geometric condition (⋆) ensures that there is a finite open covering {U_{k}}_{k=1}^{N} of Γ such that a local coordinate transformation y := (y_{1}, y_{2}, y_{3}) = X_{k}(x) := (X_{k1}(x), X_{k2}(x), X_{k3}(x)) from U_{k} onto an open subset Ũ_{k} of the y-space satisfying properties

• X_{k}(U_{k} ∩ Ω) ⊂ Y := {y; y_{3} > 0} and X_{k}(U_{k} ∩ Γ) ⊂ ∂Y := {y; y_{3} = 0} for all k = 1, 2, ..., N.

• \frac{∂}{∂ν} = A_{k}(y′)\frac{∂}{∂y_{3}} on U_{k} ∩ Γ, where y′ = (y_{1}, y_{2}, 0) and A_{k}(y′) is a positive bounded

function of C^{1} on U_{k} ∩ Γ for all k = 1, 2, ...N.

Also we take a partition of unity {ϕ_{k}} on Γ, namely

ϕ_{k} ∈ C_{0}^{∞}(R^{3}), supp(ϕ_{k}) ⊂ U_{k}, 1 ≤ k ≤ N, ∑_{k=1}^{N} ϕ_{k} = 1 on Γ,

and use the relation dΓ = J_{k}(y′)dy′ on U_{k} ∩ Γ for k = 1, 2, ..., N, where J_{k} is a positive and bounded function of C^{1} on U_{k} ∩ Γ.

We have for any fixed t ∈ [0, T] that

∫_{Γ} v · ∇(η_{ε}^{2})dΓ = ∑_{i=1}^{3} ∫_{Γ} v_{i} \frac{∂η_{ε}^{2}}{∂x_{i}} dΓ = ∑_{i=1}^{3} \{∑_{k=1}^{N} ∫_{Γ} ϕ_{k}v_{i} \frac{∂η_{ε}^{2}}{∂x_{i}} dΓ\}

= ∑_{i,j=1}^{3} \{∑_{k=1}^{N} ∫_{R^{2}} ϕ_{k}v_{i} \frac{∂X_{kj}}{∂x_{i}} \frac{∂η_{ε}^{2}}{∂y_{j}} J_{j}dy′\} .

Therefore, for w_{ijk} := ϕ_{k}v_{i}(∂X_{kj}/∂x_{i})J_{j}, it is enough to estimate the integral

I_{ik} := ∫_{R^{2}} ∑_{j=1}^{3} w_{ijk} \frac{∂η_{ε}^{2}}{∂y_{j}} dy′.

In fact, we see that

I_{ik} = − ∫_{R^{2}} \{\frac{∂w_{i1k}}{∂y_{1}} + \frac{∂w_{i2k}}{∂y_{2}} \} η_{ε}^{2}dy′ + ∫_{R^{2}} w_{i3k} \frac{∂η_{ε}^{2}}{∂y_{3}} dy′.

Here, noting that

\frac{∂η_{ε}^{2}}{∂y_{3}} = \frac{1}{A_{j}} \frac{∂η_{ε}^{2}}{∂ν} = −\frac{2n_{0}}{A_{j}} |η_{ε}|^{2},

we get from the above expression of I_{ik} that

|I_{ik}| ≤ B_{2}|η_{ε}|_{L^{2}(Γ)}^{2}

for all i, k = 1, 2, 3, and B_{2} is a positive constant independent of i, j, k. Thus the required estimate holds with B_{1} = 6B_{2}. □

Now, from Lemmas 5.1 and 5.2 it follows that

|η_{ε}(s)|_{V}^{2} + ∫_{s}^{T} ∫_{Ω} b_{ε}|Δη_{ε}|^{2}dxdt ≤ B_{3} ∫_{s}^{T} {|η_{ε}|_{V}^{2} + |ℓ|_{V}^{2} }dt  (5.7)

for all s ∈ [0, T] and all ε ∈ (0, 1]. Applying the Gronwall’s inequality to (5.7), we have the uniform estimate

sup_{0≤t≤T} |η_{ε}(t)|_{V}^{2} + ∫_{Q} b_{ε}|Δη_{ε}|^{2}dxdt ≤ B_{4},  (5.8)

where B_{4} is a positive constant independent of ε ∈ (0, 1].

By (5.8) we can find a sequence {ε_{n}} with ε_{n} → 0 as n → ∞ and a function η ∈ V such that

η_{ε_{n}} → η weakly in L^{2}(Q),
∇η_{ε_{n}} → ∇η weakly in L^{2}(Q)^{3} as n → 0.

Taking η_{ε_{n}} as a test function η in (5.2), we have

− ∫_{Q}(u_{1} − u_{2}){η_{ε_{n},t} + bΔη_{ε_{n}} + v · ∇η_{ε_{n}}}dxdt

= − ∫_{Q}(u_{1} − u_{2}){η_{ε_{n},t} + b_{ε_{n}}Δη_{ε_{n}} + v · ∇η_{ε_{n}}}dxdt + ∫_{Q}(u_{1} − u_{2})(b_{ε_{n}} − b)Δη_{ε_{n}}dxdt
= − ∫_{Q}(u_{1} − u_{2})ℓdxdt + ∫_{Q}(u_{1} − u_{2})(b_{ε_{n}} − b)Δη_{ε_{n}}dxdt.

Moreover, passing to the limit in n yields that

|∫_{Q}(u_{1} − u_{2})(b_{ε_{n}} − b)Δη_{ε_{n}}dxdt|
≤ \{∫_{Q} |u_{1} − u_{2}|^{2}|b_{ε_{n}} − b|dxdt\}^{\frac{1}{2}} \{∫_{Q} |b_{ε_{n}} − b||Δη_{ε_{n}}|^{2}dxdt\}^{\frac{1}{2}}
≤ \{∫_{Q} |u_{1} − u_{2}|^{2}|b_{ε_{n}} − b|dxdt\}^{\frac{1}{2}} B_{4}
→ 0.

Therefore

∫_{Q}(u_{1} − u_{2})ℓdxdt = 0 for all ℓ ∈ D(Q),

which implies that u_{1} = u_{2} a.e. on Q. □

References

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6. T. Fukao, N. Kenmochi and I. Pawlow, Stefan problems in non-cylindrical domains arising in Czochralski process of crystal growth, in preparation.

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Takeshi Fukao Department of Mathematics, Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-chō, Inage-ku, Chiba, 263-8522, Japan e-mail: t-fukao@math.e.chiba-u.ac.jp

Nobuyuki Kenmochi Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-chō, Inage-ku, Chiba, 263-8522, Japan e-mail: kenmochi@math.e.chiba-u.ac.jp

Irena Pawlow Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland e-mail: pawlow@ibspan.waw.pl

Technical Reports of Mathematical Sciences

Chiba University, vol. 17 (2001)

No. 1 A note on admissibility of the inverse estimator in multivariate calibration, by H. Tsukuma.

No. 2 Isomorphisms of C^{∗}-algebras after tensoring, by Y. Katabmi.

No. 3 Two-phase Stefan problems in non-cylindrical domains, by T. Fukao, N. Kenmochi and I. Pawlow.

No. 4 Transmission problems arising in Czochralski process of crystal growth, by T. Fukao, N. Kenmochi and I. Pawlow.

Correspondences concerning these Technical Reports should be addressed to Koji Nishida Graduate School of Science and Technology Chiba University Yayoi-cho 1-33, Inage-ku, Chiba-shi, 263-8522 JAPAN. e-mail: nishida@math.s.chiba-u.ac.jp