Parabolic boundary value problems connected with Newton's polygon and some problems of crystallization

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Universität Konstanz

Parabolic boundary value problems connected with Newton's polygon and some problems of crystallization

Robert Denk Leonid R. Volevich

Konstanzer Schriften in Mathematik und Informatik Nr. 243, Februar 2008

ISSN 1430-3558

Fach D 188, 78457 Konstanz, Germany E-Mail: preprints@informatik.uni-konstanz.de

WWW: http://www.informatik.uni-konstanz.de/Schriften/

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/4912/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-49127

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PARABOLIC BOUNDARY VALUE PROBLEMS CONNECTED WITH NEWTON’S POLYGON AND SOME PROBLEMS OF

CRYSTALLIZATION

R. DENK, L. R. VOLEVICH¹

Abstract. A new class of boundary value problems for parabolic operators is introduced which is based on the Newton polygon method. We show unique solvability and a priori estimates in corresponding L2-Sobolev spaces. As an application, we discuss some linearized free boundary problems arising in crystallization theory which do not satisfy the classical parabolicity condition.

It is shown that these belong to the new class of parabolic boundary value problems, and two-sided estimates for their solutions are obtained.

1. Introduction

This paper is motivated by several linearized boundary value problems from mathematical physics arising, e.g., in crystallization theory. In particular, we will consider the Stefan problem with Gibbs-Thomson correction and the Cahn-Hilliard equation with dynamics boundary conditions. These two problems have common specific features: the equation in the interior of the domain is of a rather simple structure and is parabolic in the sense of Petrovskii, but the boundary operators do not satisfy the condition of Shapiro-Lopatinskii type.

In fact, in these and similar examples there are several reasons why the Shapiro- Lopatinskii conditions cannot by satisfied:

• The boundary conditions aredynamic, i.e. time derivatives appear in the boundary conditions.

• In the case of afree boundary value problem, the free boundary is usually transformed by the Hanzawa transform to a fixed boundary. This leads to an additional unknown function on the boundary.

• The boundary operators have aninherent inhomogeneity. In the examples mentioned above, the equation in the interior of the domain is 2b-parabolic in the sense of Petrovskii, so the time derivative has weight 2b with re- spect to space derivatives. In the boundary operators, however, the time derivative appears again but with a different weight 2b⁰6= 2b.

The third reason is the most serious one. Due to the missing (quasi-)homogeneity of the corresponding symbols, uniform a priori estimates cannot be obtained by usual arguments.

The goal of this paper is to include these two problems in a rather general class of parabolic problems in which also “lower-order” terms in the boundary conditions play an important role.

Date: November 13, 2007.

1 The second author was supported by Russian Foundation of Basic Research, grant 06–01–

00096.

1

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The main idea of our approach is the following. As it is well known, even for scalar operators the theory of parabolic (as well as elliptic and parameter- elliptic) problems is deeply connected with the theory of mixed order systems of pseudodifferential operators acting on the boundary. The matrix-symbol of this system is the so-called Lopatinskii matrix. In standard parabolic problems, this system is parabolic in the sense of Solonnikov [10]. We study a more general class of parabolic boundary value problems replacing systems parabolic in the sense of [10] by a more general class of N-parabolic systems which were studied by the second author in [12]. In the definition and analysis of these systems the notion of the Newton polygon plays a crucial role.

We now come to the formulation of the class of boundary value problems under consideration. To simplify the presentation, we will restrict ourselves to the model problem in half-spaceRⁿ+:={x= (x⁰, xn)∈Rⁿ :xn>0} with boundaryRⁿ⁻¹.

In the interior of the domain we will consider an equation of the form A(Dx, Dt)u(x, t) =f(x, t) (x∈Rⁿ+, t∈R),

u(x, t) =f(x, t) = 0 (x∈Rⁿ+, t <0). (1-1) HereDx=−i(_∂x^∂

1, . . . ,_∂x^∂

n) andDt=−i_∂t^∂. We will assume thatAis 2b-parabolic in the sense of Petrovskii and we will denote the order ofAby 2m. On the boundary we haveκadditional functionsσ₁, . . . , σ_κ. Consequently, we needm+κboundary conditions

Bj(Dx, Dt)u(x⁰, t) +

κ

X

k=1

Cjk(Dx⁰, Dt)σk(x⁰, t) =gj(x⁰, t) (j= 1, . . . , m+κ, x⁰ ∈Rⁿ⁻¹, t∈R).

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Again we assumeσ_k(x⁰, t) =g_j(x⁰, t) = 0 fort <0. The 2b-parabolicity condition onAmeans that

A₀(ξ, τ)6= 0 ((ξ, τ)∈Rⁿ×Cwith|ξ|^2b+|τ|= 1, Imτ≤0), (1-3) where A0 is the principal part of A. Here for the definition of the principal part we have to assign the weight 2bto the co-variableτ. For simplicity we will suppose that our operators have constant coefficients and have no lower terms, so thatA₀ coincides withA.

The two main examples of boundary value problems of the form (1-1)–(1-2) are the Stefan problem and the Cahn-Hilliard equation. First, the linearized Stefan problem with Gibbs-Thomson correction is given by

∂tu(x, t)−∆u(x, t) =f(x, t) (t >0, x∈Rⁿ+), u(x⁰,0, t) + ∆⁰σ(x⁰, t) =g(x⁰, t) (t >0, x⁰∈Rⁿ⁻¹),

∂nu(x⁰,0, t)−∂tσ(x⁰, t) =h(x⁰, t) (t >0, x⁰∈Rⁿ⁻¹), u(x,0) =u₀ (x∈Rⁿ),

σ(x⁰,0) =σ₀ (x⁰∈Rⁿ⁻¹).

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Here ∆⁰is the Laplace operator on the boundaryRⁿ⁻¹and∂_nstands for the normal derivative.

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In the case of zero initial conditions we can consider the problem on the whole time axis, supposing that

u(x, t) =f(x, t) = 0 (x∈Rⁿ+, t <0), σ(x⁰, t) =g(x⁰, t) =h(x⁰, t) = 0 (x⁰ ∈Rⁿ⁻¹, t <0).

The Stefan problem (1-4) was studied in a number of papers. Recently it was treated in detail by Escher-Pr¨uss-Simonett in [7]. It was shown that in appropriate solution spaces this equation is uniquely solvable. Whereas in [7] the approach is semigroup based, we will see below that we can understand the structure of this boundary value problem and of the solution spaces in terms of the Newton polygon.

Below we will show that the classical parabolicity condition is not satisfied for the Stefan problem (1-4).

Now let us come to the linearized Cahn-Hilliard equation with dynamic boundary conditions. It is given by

∂tu(x, t) + ∆²u(x, t) =f(x, t) (t >0, x∈Rⁿ+),

∂n∆u(x⁰,0, t) =g(x⁰, t) (t >0, x⁰∈Rⁿ⁻¹),

∂tu(x⁰,0, t) +∂nu(x⁰,0, t)−∆⁰u(x⁰,0, t) =h(x⁰, t) (t >0, x⁰∈Rⁿ⁻¹), u(x,0) =u0 (t= 0, x∈Rⁿ+).

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The solvability of this problem in appropriate Sobolev spaces was investigated by Pr¨uss-Racke-Zheng in [8]. Again the method was based on a semigroup approach.

We will see below that also the Cahn-Hilliard equation (1-5) fits into the context of parabolic problems connected with the Newton polygon.

The paper is organized in the following way. In Section 2, we will discuss the definition of standard 2b–parabolic problems and the Lopatinskii matrix. In particular, we will show that the linearized problems of crystallization are not 2b–parabolic.

In Section 3, the notion of N-parabolic problems will be defined which is connected with the Newton polygon. We will see that both examples belong to this larger class of parabolic boundary value problems. The last three sections of the paper are devoted to solvability results. After defining the Sobolev spaces connected with (1-4)–(1-5) in Section 4, we will prove a general result on unique solvability and a priori estimates in Section 5. The application to the two examples can be found in Section 6.

In subsequent publications we plan to treat the problems with variable coefficients using the construction of an exact parametrix of the Dirichlet problem for parabolic operators. In the case of problems (1-4) and (1-5) estimates in scales of Besov spaces will be given.

Below we will denote the co-variable tot byτ and the co-variables to x∈Rⁿ andx⁰∈Rⁿ⁻¹byξ∈Rⁿandξ⁰:= (ξ₁, . . . , ξ_n−1). In the following,Cstands for an unspecified constant which may vary from one appearance to the other but which is independent of the free variables. The notionf ≈g means that there exists a positive constantCfor whichC⁻¹f ≤g≤Cf.

2. 2b–parabolic problems. The Lopatinskii matrix

In this section we introduce the main object of the further investigation – the Lopatinskii matrix of the boundary value problem (1-1)–(1-2). We define standard

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2b–parabolic problems and show that the problems (1-4) and (1-5) do not belong to this class.

2.1. Remarks about the factorization. We assume that the operatorA(Dx, Dτ) is 2b-parabolic. We will need the factorization of the symbolA(ξ, τ), i, e. its representation in the form

A(ξ⁰, ξn, τ) =A⁺(ξ⁰, ξn, τ)A⁻(ξ⁰, ξn, τ) with

A⁺(ξ⁰, ξn, τ) :=

m

Y

j=1

(z−z⁺_j(ξ⁰, τ)) =z^m+

m

X

`=1

a`(ξ⁰, τ)z^m−`. (2-1) Herez_j⁺(ξ⁰, τ) are the roots ofA(ξ⁰,·, τ) with positive imaginary part.

In the following, the (quasi-)homogeneities of the symbols will be important, so we want to defineρ-homogeneity. For a weightρ >0, a function F:Rⁿ×C→C will be calledρ-homogeneous of degreed∈Rif

F(ηξ, η^ρτ) =η^dF(ξ, τ) (ξ∈Rⁿ\ {0}, τ ∈C\ {0}, η >0).

In this case we will calldtheρ-order of the symbolF and write ordρF(ξ, τ) :=d.

Hereρdenotes the weight of the co-variableτ when ξhas weight 1. IfF is a sum of quasi-homogeneous terms theρ-order is defined in an obvious way where again τ has weightρ.

As the symbolA(ξ⁰, z, τ) is 2b-homogeneous in (ξ⁰, z, τ), the same is true for its roots. Therefore, we have

|z_j⁺(ξ⁰, τ)| ≤C(|τ|^2b¹ +|ξ⁰|), Imz_j⁺(ξ⁰, τ)≥C(|τ|^2b¹ +|ξ⁰|) forj= 1, . . . , m. We can representz_j⁺ in the form

z_j⁺(ξ⁰, τ) = (|τ|^2b¹ +|ξ⁰|)˜z_j⁺(ξ⁰, τ), where ˜z_j⁺ is a 2b-homogeneous function of (ξ⁰, τ) of degree zero.

From this we obtain the analogous representation of the coefficientsa_`(ξ⁰, τ):

a`(ξ⁰, τ) = (|τ|^2b¹ +|ξ⁰|)^`˜a`(ξ⁰, τ)

where ˜a_` is a 2b-homogeneous function of (ξ⁰, τ) of degree 0. We will not suppose, in principle, that the polynomialsB_j(ξ, τ) andC_jk(ξ⁰, τ) are quasi-homogeneous.

2.2. The Lopatinskii matrix of the problem (1-1)–(1-2). As in the classical (quasi-homogeneous) situation, the Lopatinskii matrix of (1-1)–(1-2) is constructed using the remainders of the symbols of the boundary operators moduloA⁺(ξ⁰, z, τ).

For`≥mwe have z^`≡

m

X

k=1

γ`k(ξ⁰, τ)z^k−1 modA⁺(ξ⁰, z, τ).

From the rule of division of polynomials it follows that the coefficientsγ_`k(ξ⁰, τ) are 2b-homogeneous polynomials ofz₁⁺, . . . , z_m⁺of degree`+1−kand can be represented in the form

γ_`k(ξ⁰, τ) = (|τ|^2b¹ +|ξ⁰|)^`+1−k ˜γ_`k(τ, ξ⁰),

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where ˜γ`k are 2b-homogeneous of degree 0. Further we replace in the symbols Bj(ξ⁰, z, τ) every powerz^`with`≥mby their remainder moduloA⁺(ξ⁰, z, τ). We get

Bj(ξ⁰, z, τ)≡B˜j(ξ⁰, z, τ) modA⁺(ξ⁰, z, τ) with

B˜_j(ξ⁰, z, τ) =

m

X

k=1

b_jk(ξ⁰, τ)z^k−1.

From the construction we see that the functionsbjk(ξ⁰, τ) are polynomials in ξ⁰, τ and in the coefficientsγ`k(ξ⁰, τ) which themselves are polynomials in|τ|^2b¹ and|ξ⁰| with coefficients depending onτ(|τ|^1/2b+|ξ⁰|)^−2b andξ⁰(|τ|^1/2b+|ξ⁰|)⁻¹. Therefore, we can define the 2b-order ofb_jk. We have

ord2bbjk ≤mj+ 1−k (j= 1, . . . , m+κ, k= 1, . . . , m).

Here we have setm_j := ord_2bB_j(ξ, τ).

The matrix-function

L(ξ⁰, τ) = Ljk(ξ⁰, τ)

j,k=1,...,m+κ

with

Ljk(ξ⁰, τ) :=

(bjk(ξ⁰, τ) (j= 1, . . . , m+κ, k= 1, . . . , m),

C_j,k−m(ξ⁰, τ) (j= 1, . . . , m+κ, k=m+ 1, . . . , m+κ) is called the Lopatinskii matrix of the problem (1-1)–(1-2).

The elements of the Lopatinskii matrix are holomorphic functions inτ.We will prove this statement at the end of this section.

2.3. The2b–parabolicity condition for the boundary value problem(1-1)–

(1-2). As it was mentioned above, although the elements of the Lopatinskii matrix are algebraic functions, their orders. Then we can define the 2b-principal part L⁰(ξ⁰, τ) ofL(ξ⁰, τ). If we pose

s_j:=m_j+ 1−m (j= 1, . . . , m+κ), t_k:=m−k (k= 1, . . . , m) we have ord2bbjk≤sj+tk. For the operatorsCjk we set

tk := max{ord2bC_j,k−m−sj:j= 1, . . . , m+κ} (k=m+ 1, . . . , m+κ).

In this way we obtain

ordLjk(ξ⁰, τ)≤sj+tk (j, k= 1, . . . , m+κ).

Let π_2bL_jk(ξ⁰, τ) be the principal part of L_jk(ξ⁰, τ) with the weight of τ being 2b. Then the principal part of the Lopatinskii matrix is defined as L⁰(ξ⁰, τ) = (L⁰_jk(ξ⁰, τ))j,k=1,...,m+κ with

L⁰_jk(ξ⁰, τ) :=

(π_2bL_jk(ξ⁰, τ) if ord_2bL_jk=s_j+t_k, 0 if ord2bLjk< sj+tk.

Definition 2.1. We say that problem (1-1)–(1-2) satisfies the 2b–parabolicity condition if the following conditions hold.

(i)A(D_x, D_t) is 2b-parabolic.

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(ii) detL⁰(ξ⁰, τ) =π2bdetL(ξ⁰, τ), i.e. we have ord2bdetL(ξ⁰, τ) =

m+κ

X

j=1

(sj+tj).

(iii) For allξ⁰∈Rⁿ⁻¹and τ∈Cwith|τ|+|ξ⁰|>0 and Imτ≤0 we have detL⁰(ξ⁰, τ)6= 0.

Conditions (ii),(iii) mean that the matrix L(ξ⁰, τ) is parabolic in the sense of Solonnikov [10].

Remark 2.2. For parabolic problems the analog of the Shapiro–Lopatinskii condition from elliptic theory was introduced in the papers of Eidelman (see [4]) and Solonnikov (see [10]). Agranovich and Vishik [3] studied elliptic problems with parameter and introduced the parameter-ellipticity condition which formally coincides with Eidelman-Solonnikov condition. The parameter-ellipticity condition was independently introduced by Agmon [1]. In the context of problems with parameter we shall call it Agmon–Agranovich–Vishik (AAV) condition.

2.4. The problems(1-4)and (1-5)are not 2b-parabolic. We will see now that neither the Stefan problem with Gibbs-Thomson correction nor the Cahn-Hilliard equation with dynamic boundary condition satisfy condition (iii) of Definition 2.1.

Let us start with the Stefan problem (1-4). Using the factorization of the symbol of the heat operator

iτ+|ξ⁰|²+ξ²_n= (ξ_n−ip

|ξ⁰|²+iτ)(ξ_n+ip

|ξ⁰|²+iτ), we calculate the Lopatinskii matrix

L(τ, ξ⁰) =

1 −|ξ⁰|²

−p

|ξ⁰|²+iτ −iτ

. (2-2)

We can take s1 = 0, s2 = 1, t1 = 0, t2 = 2, and the principal part of L(τ, ξ⁰) is given by

L⁰(τ, ξ⁰) =

1 −|ξ⁰|²

−p

|ξ⁰|²+iτ 0

.

Condition (ii) of Definition 2.1 is satisfied, but condition (iii) is violated because detL⁰(τ, ξ⁰) =−|ξ⁰|²p

|ξ⁰|²+iτ = 0 for |ξ⁰|= 0, |τ|>0.

Now let us consider the Cahn-Hilliard problem (1-5). We factorize iτ+ (|ξ⁰|²+ξ_n²)²=A⁺(ξ⁰, ξn, τ)·A⁻(ξ⁰, ξn, τ) with

A⁺(ξ⁰, ξn, τ) = (ξn−z1(ξ⁰, τ))·(ξn−z2(ξ⁰, τ)).

The Lopatinskii matrix is given by L(ξ⁰, τ) =

iz₁z₂(z₁+z₂) −i(z²₁+z₂²+z₁z₂+|ξ⁰|²)

iτ+|ξ⁰|² i

. (2-3)

In this caseτhas weight 4 andm1= 3, m2= 4. Thuss1= 2, s2= 3, t1= 1, t2= 0, and the principal part ofLis equal to

L⁰(ξ⁰, τ) =

iz1z2(z1+z2) −i(z₁²+z²₂+z1z2+|ξ⁰|²)

iτ 0

.

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Obviously, we have detL⁰(ξ⁰, τ) = 0 for τ = 0 and arbitrary ξ⁰. Again the 2b- parabolicity condition is not satisfied.

2.5. The Lopatinskii matrix is holomorphic in τ. According to the definition, the elements of the Lopatinskii matrixL(ξ⁰, τ) are remainders after division of polynomials byA⁺(z, ξ⁰, τ) defined in (2-1). From the elementary algorithm of division with remainder easily follows that we only need to check that the coeffi- cientsa_`(ξ⁰, τ) appearing in (2-1) are holomorphic inτ.

According to the classical Vieta formula the coefficients a_`(ξ⁰, τ) = (−1)^` X

i₁,...,im−`

z⁺_i

1(ξ⁰, τ). . . z⁺_i

m−`(ξ⁰, τ), `= 1, . . . , m

are symmetric functions of the rootsz₁⁺, . . . , z_m⁺ and can be expressed by means of

m

X

j=1

(z⁺_j(ξ⁰, τ))^r, r= 1, . . . , m−1. (2-4) To prove that sums (2-4) are holomorphic in τ we use the standard argument of the Weierstrass preparation theorem. We choose a contour γ+ in the half-plane Imz >0 enveloping all the rootsz₁⁺, . . . , z_m⁺.Then

m

X

j=1

(z_j⁺(ξ⁰, τ))^r= 1 2πi

Z

γ+

z^r∂zA(ξ⁰, z, τ)

A(ξ⁰, z, τ) dz. (2-5)

The expression under the sign of the integral is a rational function of τ and has no poles for Imτ < 0 and z ∈ γ+. From this follows that the left-hand side of (2-5) is holomorphic inτ and, consequently, the same holds for the elements of the Lopatinskii matrix.

3. N-parabolic boundary value problems

In this section we introduce a more general class of parabolic problems such that the examples above belong to it. The main idea is to replace the traditional principal part of the Lopatinskii matrix by a principal part connected with the Newton polygon of detL. We shall need some definitions and results from [5] and [12].

3.1. Newton’s polygon and N-parabolic polynomials. Consider a polynomial P(ξ, τ) =X

α,j

p_αjξ^ατ^j

in the variables ξ ∈ Rⁿ and τ ∈ C. Then the Newton polygon N(P) of the polynomial P is defined as the convex hull of all points (|α|, j) for which pαj 6= 0, the projections of all these points to the coordinate axes and the origin. The following definition is taken from [5].

Definition 3.1. The polynomialP(ξ, τ) is called N-parabolic if the following conditions hold.

(i) The Newton polygon N(P) has no edges parallel to the coordinate axes (except the trivial ones).

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(ii) There exists aτ0<0 such that the estimate

|P(ξ, τ)| ≥C X

(i,j)∈N(P)∩Z²

|ξ|ⁱ|τ|^j (ξ∈Rⁿ,Imτ≤τ₀)

holds. Here the sum runs over all integer points belonging to the Newton polygon.

3.2. N-parabolic polynomial matrices. N-parabolic polynomials can be in- cluded in the class of so-called N-parabolic matrices (see [12]). For this, we consider a polynomial matrix

P(ξ, τ) = P_jk(ξ, τ)

j,k=1,...,N

and write the determinant ofP in the form detP(ξ, τ) =X

γ

±P_1,γ(1). . . P_N,γ(N),

whereγruns through all permutations of the set{1,2, . . . , N}. Assigning weightρ to the variableτ and weight 1 to the variablesξwe define

R(ρ) := max

γ ordρP_1,γ(1)+· · ·+ ordρP_N,γ(N₎ .

Definition 3.2. a) The matrixP(ξ, τ) is called totally non-degenerate if for every ρ >0 we have

R(ρ) = ordρdetP(ξ, τ).

b) The matrixP(ξ, τ) is called N-parabolic if the following conditions hold.

(i)P(ξ, τ) is totally non-degenerate.

(ii) The determinant detP(ξ, τ) is N-parabolic.

For N-parabolic matricesP(ξ, τ) it is possible to define the principal partPρ(ξ, τ) for every fixed weightρ >0. As it was noted in [12], according to [11] for everyρ there exist real numberssj(ρ) andtk(ρ) (j, k= 1, . . . , N) for which

ord_ρP_jk(ξ, τ)≤s_j(ρ) +t_k(ρ) (j, k= 1, . . . , N), ordρdetP =

N

X

j=1

(sj(ρ) +tj(ρ)).

The principal partP_ρ(ξ, τ) is defined in the standard way.

The matrixP(ξ, τ) is calledpositively totally non-degenerateif the above functions si(ρ), ti(ρ), i = 1, . . . , N, can be chosen nonnegative. Replacing in Defini- tion 3.2 b) totally non-degenerate matrices by positively totally non-degenerate, we define positively N-parabolic matrices.

3.3. N-parabolic boundary value problems. We return to problem (1-1)–(1-2).

Formally the above definitions cannot be applied to the Lopatinskii matrix of this problem, because its elements are not polynomials but, in principal, algebraic functions. In fact, Definition 3.1 and Definition 3.2 use only the possibility to calculate ordρPij for each elementPij and eachρ >0.

The special structure of the elements of the Lopatinskii matrix permits to calculate these orders. Indeed, as it was mentioned above, the elements of this matrix are polynomials in (ξ⁰, τ) and the rootsz₁⁺, . . . , z_m⁺. Setting ordρz_j⁺=ρ/2b forρ≥2b and ord_ρz⁺_j = 1 for ρ ≤ 2b we define ord_ρ for each element of the Lopatinskii matrix. So we can give

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Definition 3.3. The problem (1-1)–(1-2) is called N-parabolic if the following conditions hold.

(i)A(Dx, Dt) is 2b-parabolic.

(ii) The Lopatinskii matrixL(ξ⁰, τ) is N-parabolic in the sense of Definition 3.2.

3.4. Reduction of an N-parabolic boundary value problem to an N-parabolic system on the boundary. For simplicity we start from problem (1-1)–(1-2) withf ≡0 and without additional functions at the boundary conditions,

A(Dx, Dt)u(x, t) = 0 (x∈Rⁿ+, t∈R), (3-1) u(x, t) = 0 (x∈Rⁿ+, t <0),

B_j(D_x, D_t)u(x⁰, t) =g_j(x⁰, t) (j= 1, . . . , m, x⁰∈Rⁿ⁻¹, t∈R). (3-2) Here A(Dx, Dt) is, as above, a 2b-parabolic operator with constant coefficients of order 2m, and Bj(Dx, Dt) are operators with constant coefficients. We do not assume that the boundary operators are quasi-homogeneous.

Leth_j(x⁰, t),j = 1, . . . , m, be functions with h₁(x⁰, t) =· · ·=h_m(x⁰, t) = 0 for t <0. Then we denote by U(h₁, . . . , h_m, x, t) the solution of the Dirichlet problem A(Dx, Dt)U(h1, . . . , hm, x, t) = 0 (x∈Rⁿ+, t∈R), (3-3)

U(h1, . . . , hm, x, t) = 0 (x∈Rⁿ+, t <0),

D^j−1_n U(h₁, . . . , h_m, x⁰, t) =h_j(x⁰, t) (j= 1, . . . , m, x⁰ ∈Rⁿ⁻¹, t∈R). (3-4) We shall seek the solution of (3-1)–(3-2) in the formu(x, t) = U(h₁, . . . , h_m, x, t) with functionsh₁, . . . , h_mto be found.

Lemma 3.4. Forj= 1, . . . , m we have B_j(D_x, D_t)U(h₁, . . . , h_m, x, t)

_x

n=0

=

m

X

k=1

L_jk(D_x⁰, D_t)h_k(x⁰, t) (3-5) hold.

Proof. Starting from the relation A⁺(ξ⁰, z, τ) =z^m+

m

X

`=1

a`(ξ⁰, τ)z^m−`=:

m

X

`=0

a`(ξ⁰, τ)z^m−`, we define

Mk(ξ⁰, z, τ) :=

k

X

`=0

a`(ξ⁰, τ)z^k−` (k= 0, . . . , m−1).

As it was noted in [2] (see Chapter 1, Section 1), 1

2πi Z

γ₊

z^`−1Mm−k(ξ⁰, z, τ)

A⁺(ξ⁰, z, τ) dz=δk`, (k, `= 1, . . . , m), (3-6) where the contourγ+ belongs to C+ and contains all the zeros ofA⁺(ξ⁰,·, τ). We define the symbol

Gk(ξ⁰, τ, xn) := 1 2πi

Z

γ+

M_m−k(ξ⁰, z, τ)

A⁺(ξ⁰, z, τ) e^izxⁿdz (k= 1, . . . , m). (3-7)

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According to (3-6)

U(h1, . . . , hm, x, t) =

m

X

k=1

Gk(Dx⁰, Dt, xn)hk(x⁰, t),

here G_k(D_x⁰, D_t, x_n) is the pseudodifferential operator in the variables (x⁰, t) with symbol (3-7) depending on the parameterx_n.

Now the left-hand side of (3-5) will be equal to

m

X

k=1

Bj(Dx, Dt)Gk(Dx⁰, Dt, xn)|xn=0hk(x⁰, t) . Further

m

X

k=1

Bj(Dx, Dt)Gk(Dx⁰, Dt, xn)|x_n=0= 1 2πi

Z

γ₊

Bj(ξ⁰, z, τ)Mm−k(ξ⁰, z, τ) A⁺(ξ⁰, z, τ) dz.

The integral on the right-hand side does not change if we replace B_j(ξ⁰, z, τ) by B˜j(ξ⁰, z, τ) =Pm

`=1bj`(ξ⁰, τ)z^`−1. Now we obtain forj, k= 1, . . . , m

m

X

`=1

bj`(ξ⁰, τ) 1 2πi

Z

γ₊

z^`−1Mm−k(ξ⁰, z, τ) A⁺(ξ⁰, z, τ) dz=

m

X

`=1

bj`(ξ⁰, τ)δ`k=bjk(ξ⁰, τ).

From the lemma we see that problem (3-1)–(3-2) is reduced to the system of pseudodifferential equations

L(D_x⁰, D_t,)(h₁, . . . , h_m)^T = (g₁, . . . , g_m)^T. (3-8) The approach described above can trivially be extended to the case of boundary conditions containing additional functions on the boundary of the form

Bj(Dx, Dt)u(x⁰, t) +

κ

X

k=1

Cjk(Dx⁰, Dt)σ(x⁰, t) =gj(x⁰, t) (j= 1, . . . , m+κ). (3-9) Substituting in the boundary conditions the functionU(h₁, . . . , h_m, x, t) and using the lemma we obtain the following system forh₁, . . . , h_m, σ₁, . . . , σ_κ.

m

X

k=1

Ljk(Dx⁰, Dt)hk+

κ

X

k=1

Cjk(Dx⁰, Dt)σk =gj, (j= 1, . . . , m+κ). (3-10) 3.5. N-parabolicity of the Stefan problem with Gibbs-Thomson correction. For the Stefan problem (1-4) the Lopatinskii matrix is given by (2-2), and its determinant equals to

detL(ξ⁰, τ) =−

|ξ|²p

|ξ⁰|²+iτ+iτ) . Note that for Imτ≤0 we have|p

|ξ⁰|²+iτ| ≈ |ξ⁰|+|τ|^1/2. Therefore, the Newton polygon of detL(ξ⁰, τ) has the vertices (0,0),(0,1),(2,¹₂) and (3,0) (see Figure 1).

For every ρ >0, we definesj(ρ) andtk(ρ) (j, k= 1,2) bys1(ρ) := 0,t1(ρ) := 0 and

s2(ρ) :=

(_ρ

2, ρ≥2,

1, ρ≤2, t2(ρ) :=

(_ρ

2, ρ≥4, 2, ρ≤4.

Then

ord_ρL_jk(τ, ξ⁰)≤s_j(ρ) +t_k(ρ) (j, k= 1,2)

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- 6

i k

1

1/2 HH

HH HH

HH

@

•

2 3

N(detL)

Figure 1. The Newton polygon for the Stefan problem and

2

X

j=1

(sj(ρ) +tj(ρ)) =







ρ, ρ≥4,

2 +^ρ₂, 4≥ρ≥2, 3, ρ≤2.

For all ρ, the right-hand side of the last equality coincides with ord_ρdetL(τ, ξ⁰).

From this we see that the matrix (2-2) satisfies all conditions of Definition 3.2 and the Stefan problem (1-4) is N-parabolic.

3.6. N-parabolicity of the Cahn-Hilliard equation with dynamic boundary conditions. Let us consider the Cahn-Hilliard equation with dynamic boundary conditions (1-5). The Lopatinskii matrix for this problem is given by (2-3) and its determinant equals to

detL(ξ⁰, τ) =i(iτ+|ξ⁰|²)(z₁²+z²₂+z1z2+|ξ⁰|²)−z1z2(z1+z2).

We will need an elementary lemma.

Lemma 3.5. Let z₁=z₁(ξ⁰, τ)andz₂=z₂(ξ⁰, τ)be the roots of the equation (z²+|ξ⁰|²)²+iτ = 0

withImz1>0,Imz2>0 whereImτ≤0. Then

z₁²+z₂²+z1z2+|ξ⁰|²

≈ |ξ⁰|²+|τ|^1/2. Proof. The proof is based on two elementary observations.

(i) With appropriate numbering, we have z₁²+|ξ⁰|²=√

−iτ , z₂²+|ξ⁰|²=−√

−iτ . (ii) The product ofz₁andz₂ is equal toz₁z₂=−p

|ξ⁰|⁴+iτ. From (i) we directly obtain

z₁²+z²₂+|ξ⁰|²=−|ξ⁰|².

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Now we apply (ii) to get

|z₁²+z₂²+z1z2+|ξ⁰|²|=

− |ξ|²−p

|ξ|⁴+iτ

≤C2(|ξ|²+|τ|^1/2) for Imτ ≤0. In the same way we have

|ξ|²+p

|ξ|⁴+iτ

≥C1(|ξ|²+|τ|^1/2).

We still have to prove (i) and (ii). For this we use the explicit formula for the root of a complex numberA+iB

√A+iB=± s

A+√

A²+B²

2 +i

s

−A+√

A²+B² 2

.

According to this formula, the sign “+” selects the root with positive imaginary part, and the sign “−” selects the root with negative imaginary part.

The equation (z²+|ξ⁰|²)²+iτ = 0 has the four roots z1,2,3,4=±

q

−|ξ⁰|²±√

−iτ . The roots with positive imaginary part are given by

z1= q

−|ξ⁰|²+√

−iτ , z2= q

−|ξ⁰|²−√

−iτ . From this we get (i). To see (ii), we write

z₁=i q

|ξ⁰|²−√

−iτ and z₂=i q

|ξ⁰|²+√

−iτ and obtainz1z2=−p

|ξ⁰|⁴+iτ.

From this lemma we obtain for large−Imτ

|detL(τ, ξ⁰)| ≥C1(1 +|τ|^1/2+|ξ⁰|²)(1 +|τ|^1/2+|ξ⁰|²|)−C2(|τ|^3/4+|ξ⁰|³)

≥C3(1 +|τ|^3/2+|ξ⁰|²|τ|+|ξ⁰|⁴) and

|detL(ξ⁰, τ)| ≈1 +|τ|^3/2+|ξ⁰|²|τ|+|ξ⁰|⁴.

Therefore, the Newton polygon of detL(ξ⁰, τ) has the vertices (0,0),(0,³₂),(2,1) and (4,0) (see Figure 2).

For everyρ >0, we defines_j(ρ) and t_k(ρ) (j, k= 1,2) by s₂(ρ) = 0, t₂(ρ) = 0, and

s₁(ρ) = 2, t₁(ρ) = 2 ifρ≤2, s1(ρ) = 2, t1(ρ) =ρ if 2≤ρ≤4, s₁(ρ) = ρ

2, t₁(ρ) =ρ ifρ≥4.

Then it is easy to check that

ord_ρL_jk(ξ⁰, τ)≤s_j(ρ) +t_k(ρ) (j, k= 1,2) and

2

X

j=1

(sj(ρ) +tj(ρ)) = ordρL(ξ⁰, τ).

From this we see that the Cahn-Hilliard problem (1-5) is N-parabolic.

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- 6

i k

3 2

1 aa

aa aa

a c

c c

•

• •

2 4

N(detL)

Figure 2. The Newton polygon for the Cahn-Hilliard equation 4. Functional spaces

In this short section we will introduce the functional spaces in which we will prove unique solvability of N-parabolic boundary value problems. For nonstation- ary problems, as a rule, the corresponding Sobolev spaces are defined by pseudodifferential operators with symbols which are analytic in the co-variableτ dual to the time variable. However, the main construction of [12] used below does not permit to construct symbols holomorphic in τ. In the situation considered in the present paper, the systems are defined by the Lopatinskii matrix which is holomorphic inτ.

Thus we will be able to prove unique solvability on the half-line in the framework of spaces defined below.

4.1. Functional spacesH^χ. At first we recall the known definition of functional spaces defined by means of pseudodifferential operators (see [13]).

Let C− := {z ∈ C : Imz < 0}. Denote by F(Rⁿ×C−) = F the set of all functionsχ: Rⁿ×C−→Cwhich do not vanish and for which

χ(ξ¹, σ¹) χ(ξ², σ²)

≤C(1 +|ξ¹−ξ²|+|σ¹−σ²|)^M^(χ) (ξ^1,2∈Rⁿ, σ^1,2∈C−) holds with some positive constantM(χ). We set

H^χ(Rⁿ⁺¹) :=

u(x, t)∈H^−∞(Rⁿ⁺¹)|χ(D_x, D_t)u∈L₂(Rⁿ⁺¹) . This space is endowed with the norm

u, H^χ(Rⁿ⁺¹) :=

χ(Dx, Dt)u, L2(Rⁿ⁺¹) .

Replacing the symbolχ(ξ, σ) by an equivalent symbolχ⁰(ξ, σ)≈χ(ξ, σ), we obtain an equivalent norm in H^χ(Rⁿ⁺¹). In the caseχ(ξ, σ)≈1 +|ξ|^r+|σ|^s we will also writeH^r,s(Rⁿ⁺¹) instead ofH^χ(Rⁿ⁺¹).

In a standard way the space H^χ(Rⁿ+ ×R) is defined as the factor space of H^χ(Rⁿ⁺¹) modulo the subspace of all distributions in H^χ(Rⁿ⁺¹) whose support

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belongs toRⁿ−×R:={(x, t)∈Rⁿ×R:xn<0}. This space is endowed with the standard factor-norm.

Forχ∈ F andγ <0 we setχγ(ξ, σ) :=χ(ξ, σ+iγ). Forγ, γ⁰ ∈ F we have

|χ_γ(ξ, σ)/χ_γ⁰(ξ, σ)| ≤C(1 +|γ−γ⁰|)^M(χ).

Varying the parameterγ, we obtain a family of norms||χγ(Dx, Dt)u, L2(Rⁿ⁺¹)||in the spaceH^χ(Rⁿ⁺¹) which are equivalent.

4.2. Functional spaces H_[γ]^χ with exponential weight. For χ ∈ F(Rⁿ×C−) and γ < 0 we defineH_[γ]^χ (Rⁿ×R) as the space of such distributions u(x, t) that exp(γt)u(x, t)∈H^χ(Rⁿ⁺¹).We endow this space with the norm

||u, H_[γ]^χ (Rⁿ×R)||=||χ_γ(D_x, D_t)(exp(γt)u), L₂(Rⁿ⁺¹)||. (4-1) For a functionu(x, t) inRⁿ×Rdenote by ˆu(ξ, τ) the complex Fourier transform or, equivalently, the real Fourier transform of exp(γt)u(x, t) where ξ ∈ Rⁿ and τ =σ+iγ ∈C. For a weight function χ∈ F(Rⁿ×C+) andγ <0 we obtain the relation

||u, H_[γ]^χ (Rⁿ×R)||= Z

Imτ=γ

Z

Rⁿ

χ(ξ, τ)²|ˆu(ξ, τ)|²dξdτ^1/2

. (4-2)

As above, in the caseχ(ξ, τ)≈1 +|ξ|^r+|τ|^swe will write H_[γ]^r,s(Rⁿ×R).

In a standard way we denote byH_[γ]^χ (Rⁿ+×R) the factor space of H_[γ]^χ (Rⁿ×R) modulo the subspace of all distributions in H_[γ]^χ (Rⁿ×R) whose support belongs to Rⁿ− ×R. This space is endowed with standard factor-norm. We will write H_[γ]^χ (Rⁿ⁻¹×R) for the corresponding space of functions defined on the hyperplane {xn= 0,(x⁰, t)∈Rⁿ⁻¹×R}.

We denote byH_[γ]+^χ (Rⁿ×R), H_[γ]+^χ (Rⁿ+×R) etc. the subspaces of the above spaces consisting of distributions vanishing fort≤0.

Remark 4.1. a) From the definitions we see that

φ:=e^−γtχ(Dx, Dt+iγ)e^γtu∈L_2[γ](Rⁿ⁺¹) if u∈H_[γ]^χ (Rⁿ×R). (4-3) Therefore,u∈H_[γ]^χ (Rⁿ×R) can be represented as

u=e^−γtχ⁻¹(D_x, D_t+iγ)e^γtφ, φ∈L_2[γ](Rⁿ⁺¹). (4-4) In other words the map

H_[γ]^χ (Rⁿ×R)−→L_2[γ](Rⁿ⁺¹), u7−→φ:=e^−γtχ(Dx, Dt+iγ)e^γtu (4-5) is an isomorphism of Sobolev spaces. Unfortunately the restriction of the isomorphism (4-5) to the subspace of functions with support on a half-axis is not an isomorphism.

b) LetF_hol be the space of weight functions fromF which are holomorphic for Imτ <0. In the caseχ∈ F_hol all the statements of Remark a) hold if we replace H_[γ]^χ (Rⁿ×R) andL_2[γ](Rⁿ⁺¹) byH_[γ]+^χ (Rⁿ×R) andL_2[γ]+(Rⁿ⁺¹), respectively. In particular, the map

H_[γ]+^χ (Rⁿ×R)−→L2[γ]+(Rⁿ⁺¹), u7−→φ:=e^−γtχ(Dx, Dt+iγ)e^γtu (4-6) is an isomorphism. This yields the inclusionsH_[γ]+^χ ⊂H_[ρ]+^χ for allρ < γ (see the detailed exposition in [6]).

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4.3. Functional spaces H_[γ]^χ connected with the Newton polygon. Let Γ be a convex polygon in the positive quadrant with (0,0) as one vertex and the coordinate lines as two edges. Denote by (p0, q0), . . . ,(pJ+1, qJ+1) the vertices of polygon Γ, starting with (0,0) and indexed in clockwise direction. Define

ΞΓ(ξ, τ) := 1 +

J+1

X

j=1

|ξ|^p^j|τ|^q^j.

We denote byH_[γ]^(Γ) the spaceH_[γ]^χ withχ(ξ, τ)≈ΞΓ(ξ, τ) for Imτ ≤0.

Lemma 4.2. For a given polygon Γthere exists a weight function λ_Γ(ξ, τ)∈ Fhol

such that λ_Γ(ξ, τ)≈Ξ_Γ(ξ, τ)forImτ ≤0.

Proof. Using the argument of [5], Chapter I, Subsection 1.3, it is easy to prove that ΞΓ(ξ, τ)≈

J

Y

j=1

(1 +|ξ|^p^j+1^−p^j+|τ|^q^j^−q^j+1).

Now it remains to note that for Imτ≤0

1 +|ξ|^p^j+1^−p^j +|τ|^q^j^−q^j+1≈(1 +|ξ|^r^j+iτ)^q^j^−q^j+1, where

rj= pj+1−pj

qj−qj+1

. (4-7)

5. Solvability of N-parabolic boundary value problems.

In this section we discuss unique solvability of the model problem (1-1)–(1-2) in the half-space by reducing it to the system (3-8) or (3-10) on the boundary. We will obtain unique solvability and two-sided a priori estimates in the weightedL2- Sobolev spaces defined above, provided some additional conditions on the weight function hold. We will see in Section 6 that for the crystallization problems considered in this paper these additional conditions are satisfied.

We will prove the main solvability result in three steps: first we will consider the system defined by the Lopatinskii matrix, then consider (1-1)–(1-2) withf = 0, and finally we will treat the general case.

5.1. Solvability of the system inRⁿ⁻¹×Rdefined by the Lopatinskii matrix of an N-parabolic problem. We will study the systems (3-8) (or (3-10)). They have the same structure and we use the same notation. The number of unknown functions will be denoted by M, where either M =m or M =m+κ. Using the technique of [12], we can prove the following main result.

Theorem 5.1. Suppose the problem (1-1)–(1-2)is N-parabolic, and let L(ξ⁰, τ)be the Lopatinskii matrix of the problem. Then there exist weight functions µj, νj ∈ F(Rⁿ⁻¹×C−) (j= 1, . . . , M)andγ0<0such that the following statements hold.

(i)Let χ∈ F(Rⁿ⁻¹×C−)andγ < γ0. For every g= (g1, . . . , gM)^T ∈

M

Y

k=1

H^χ·ν

−1 k

[γ] (Rⁿ⁻¹×R)

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the system

M

X

k=1

L_jk(D_x⁰, D_t)h_k =g_j (j= 1, . . . , M) (5-1) has a unique solutionh= (h1, . . . , hM)^T ∈QM

j=1H_[γ]^χ·µ^j(Rⁿ⁻¹×R), and the estimate C⁻¹

M

X

j=1

||hj,H_[γ]^χ·µ^j(Rⁿ⁻¹×R)|| ≤

M

X

k=1

||gk, H^χ·ν

−1 k

[γ] (Rⁿ⁻¹×R)||

≤C

M

X

j=1

||hj, H_[γ]^χ·µ^j(Rⁿ⁻¹×R)||

(5-2)

holds.

(ii)Let χ∈ F(Rⁿ⁻¹×C−)andγ < γ0. For arbitrary g= (g₁, . . . , g_M)^T ∈

M

Y

k=1

H^χ·ν

−1 k

[γ]+ (Rⁿ⁻¹×R) equation (5-1)has a unique solution h= (h₁, . . . , h_M)^T ∈QM

j=1H_[γ]+^χ·µ^j(Rⁿ⁻¹×R), and for this solution estimate (5-2)holds.

The proof of the theorem is based on a statement which was essentially proved in [12]. We will comment on the proof of the following proposition in Subsection 5.4.

Proposition 5.2. Suppose the Lopatinskii matrixL(ξ⁰, τ) = (Ljk(ξ⁰, τ))j,k=1,...,M

is N-parabolic. Then there exist weight functions µj, νj ∈ F (j = 1, . . . , M) and γ0<0 such that

|detL(ξ⁰, τ)| ≥C

M

Y

j=1

|µj(ξ⁰, τ)| · |νj(ξ⁰, τ)| (Imτ ≤γ0). (5-4) Proof of Theorem 5.1. Denote by G(ξ⁰, τ) = (Gjk(ξ⁰, τ))j,k=1,...,M the inverse of the matrixL(ξ⁰, τ).As it was remarked in [12], the estimates (5-3) and (5-4) induce estimates for the elements ofG:

|Gjk(ξ⁰, τ)| ≤C|µ⁻¹_j (ξ⁰, τ)| · |ν_k⁻¹(ξ⁰, τ)|.

Since

hj=

M

X

k=1

Gjk(Dx⁰, Dt)gk

is the solution of (5-1), these estimates easily imply statement (i).

To prove (ii) it is enough to show that the elements ofG are holomorphic inτ for Imτ <0.According to Subsection 2.5, the symbols Ljk(ξ⁰, τ) are holomorphic in τ for Imτ < 0, hence detL(ξ⁰, τ) has the same property. From (5-4) we know that the determinant does not vanish and (detL(ξ⁰, τ))⁻¹ also holomorphic in τ for Imτ <0.Since the elementsGjk(ξ⁰, τ) are polynomials inLjk(ξ⁰, τ) divided by

detL(ξ⁰, τ), they are also holomorphic.

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5.2. Solvability of problem (1-1)–(1-2) in the case f ≡ 0. Now we apply Theorem 5.1 to obtain the solvability of the parabolic problem in the half-space.

The order of the 2b-parabolic equation (2.1) is equal to 2m, so it is natural to search the solutionu(x, t) of (2.1) in the spaceH^r,

r 2b

[γ]+(Rⁿ+×R),wherer≥2mis such that boundary conditions (1-2) are defined inH^r,

r 2b

[γ] .

According to Subsection 3.4, in the case f ≡ 0 the unknown function u can be found as solution of the homogeneous parabolic equation with the Dirichlet boundary conditions

γ_j−1u(x⁰, t) :=D^j−1_n u(x, t)|_x_n₌₀=h_j(x⁰, t) (j= 1, . . . , m) (5-5) where (h1, . . . , hM)^T is the solution of the system (5-1).

If our problem is N-parabolic then there exist the weight functions from Propo- sition 5.2 such that the statement of Theorem 5.1 (ii) holds. To obtain a solution u(x, t)∈H^r,

r 2b

[γ]+(Rⁿ+×R) we must demand the Dirichlet data to belong to the trace space ofH^r,

r 2b

[γ]+(Rⁿ+×R):

h_j(x⁰, t) =γ_j−1u(x⁰, t)∈H^r−j+¹²^,^2b^r⁻

j 2b+_4b¹

[γ]+ (Rⁿ⁻¹×R) (j= 1, . . . , m). (5-6) On the other side, according to Theorem 5.1 (ii), ifg_k ∈H^χ·ν

−1 k

[γ]+ (Rⁿ⁻¹×R) fork= 1, . . . , M thenh_j ∈H_[γ]+^χ·µ^j(Rⁿ⁻¹×R) and the estimate (5-2) holds. Therefore, for conditions (5-6) to be valid we must suppose that following continuous embeddings

H_[γ]+^χ·µ^j(Rⁿ⁻¹×R)⊂H^r−j+

1

2,_2b^r−_2b^j+_4b¹

[γ]+ (Rⁿ⁻¹×R) (j= 1, . . . , m) (5-7) hold. These embeddings will take place if and only if

χ(ξ⁰, τ)≥C (1 +|ξ⁰|+|τ|^2b¹)^r−j+¹²

µj(ξ⁰, τ) (j= 1, . . . , m). (5-8) As a result we proved the

Theorem 5.3. Suppose problem(1-1)–(1-2)is N-parabolic. Letµj, νj∈ F(Rⁿ⁻¹× C−) (j = 1, . . . , M)be weight functions as in Proposition 5.2. Then there exists a γ0<0 such that for each r≥2m and eachχ∈ F(Rⁿ⁻¹×C−)satisfying (5-8)the following statement holds.

For each

(g1, . . . , gM)^T ∈

M

Y

k=1

H^χ·ν

−1 k

[γ]+ (Rⁿ⁻¹×R)

the boundary value problem (1-1)–(1-2)withf = 0 has a unique solution (u, σ1, . . . , σκ)∈H^r,

r 2b

[γ]+(Rⁿ+×R)×

κ

Y

j=1

H_[γ]+^χ·µ^j+m(Rⁿ⁻¹×R).

Moreover the tracesγ0u, . . . , γ_m−1uhave the additional smoothness γj−1u∈H_[γ]+^χ·µ^j(Rⁿ⁻¹×R) (j= 1, . . . , m),