The Schauder approach to degenerate elliptic equations with homogeneous Neumann boundary condition II

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The Schauder approach to degenerate elliptic equations

with homogeneous Neumann boundary condition II

TOSHIO HORIUCHI*

§1. Introduction

Let Ω be a bounded domain in Rn(n〓2) with ∂Ω ∈C∞. Let φ be a given nonnegative smooth function on Ω and equivalent to a distance to the boundary:

(1-1) Ω={φ(x)>0}, ∂Ω={φ(x)=0}, dφ ≠0 on ∂Ω.

For a nonnegative integer k and a μ ∈(0,1), by Ck+μ(Ω) we denote the set of all functions on Ω whose k-th order partial derivatives are uniformly Holder continuous with exponent μ. Let α(x) be an arbitrary real-valued function of class C1+μ(Ω) and let c(x) be of class Cμ(Ω). Then we set

(1-2) Pu=-div(φ(x)α(x)▽u)+φ(x)α(x)c(x)u.

Here and in particular if α(x)≡0, then Pu≡

-Δu+c(x)u _.

The main purpose of this paper is to study the homogeneous Neumann boundary value problem defined by

(N-P)

in Ω

on ∂Ω,

where by ν we denote the unit inward normal to ∂Ω.

Under some assumptions on the operators P, the existence and uniqueness of the classical solution of (N-P) will be shown for any given f belonging to a certain Schauder space. Obviously (N-P) is a generalization of the classical Neumann problem for the uniformly elliptic operators to the wider class of operators P with a non-constant parameter α(x). Namely, the operators P may degenerate finitely in all directions on the set of the boundary points where α(x)>0, and the degeneracy of P varies in accordance with the values of α(x). Furthermore, if α(x)<0 on some subset of the boundary, then the coefficients of P do not remain bounded. As a result, it will be shown in §2 that (N-P) admits unbounded

Received February 6, 1996.

1991 Mathematics Subject Classification. Primary 35J70 Secondary 35J60, 35J20.

This research was partially supported by Grant-in-Aid for Scientific Research (08640163), Ministry of Education, Science, Sports and Culture.

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classical solutions, even if f is regular and α is nonnegative. So, in order to do with these unbounded solutions we are forced to modify the classical Schauder spaces in a proper way (See §2, §6 and Appendix A).

There are already authors who have studied the degenarate elliptic

equa-tions

in the Holder

framework. Goulaouic-Shimakura

[2]

studied

the equations

degenerated

in all

derections

at the boundary. In [5]

the author studied

the

equations degenerated

only in normal direction at each point of the boundary.

In these works, no boundary

condition was imposed because the so called

"en-trance property"

of the boundary

with respect to the operator was assumed.

And Graham studied

in

[3]

and [4]

the Dirichlet

problems

for

Bergman Laplacian

in some Holder spaces.

Our interest

is

not only to study the operators

P

un-der the homogeneous Neumann boundary condition

in

some modified

Schauder

spaces,

but also

to indicate

that

the classical

theory

for

uniformly

elliptic

differ-ential

operators

can be extended

to the

degenerate

operators

P in

a natural

way.

Our main method is,

as in [2],

[3],

[4]

and [5],

to make use of the fundamental

solution

for

some simplest

model of

our operators

(See

also

[6]

and [7]).

Although our aim is

to study the Neumann problem (N-P),

here

we breafly

remark about the homogeneous Dirichlet

boundary problem with respect

to a

similar

operator,

which will

be our starting

point

in this

paper.

Let us consider

(D-P)

in Ω

on ∂Ω.

In (D-P), we impose auxiliarily on α the following condition (D).

(D)

on ∂Ω.

Then we have

PROPOSITION. 1-1. Suppose that α(x) is of cass C2+μ(Ω) and satisfies the condition (D). Moreover we suppose that α<1 on ∂Ω. Then there is a positive number M such that if inf c(x)Ω 〓M then for every f∈Cμ(Ω), there exists one and only one solution u to (D-P) which is written as u=φ1-αv with a function v belonging to C1+μ(Ω) such that φv∈C2+μ(Ω).

PROOF OF PROPOSITION 1-1. From the theory of ordinary differential equa-tions, we see that (D-P) has no solutions vanishing at xn=0 if α 〓1. So the con-dition α<1 is necessary for us. Now we set u=φ1-αv in (D-P). Then, by virtue of the condition (D), this proposition follows as a corollary to the theorem due to C. Goulaouic-N. Shimakura [2] in which the equation -φ Δv-z∂nv+F=0 (F∈ Cμ(Ω), Rz>0) were studied. We can apply their result to (D-P) putting z=2-α and F=-f+[error terms arising from the change of unknown functions].

Here we give an important remark: If α<0, then the solution u to the problem (D-P) obviously satisfies the homogeneous Neumann condition as well. But this u is excluded from our framework for the problem (N-P) (See also Example in the next section).

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This paper is organized in the following way: Our main results concerning the Neumann problem (N-P) will be stated in §2. We also give the sketch of the

proofs and some examples which indicate "the sharpness" of the results. In §3 we shall establish Theorem 2-1 admitting an a priori estimate of solutions in the

half space in §5 and §6. Using Theorem 2-1. we give a proof to Theorem 2-2 in §4. In §5 we define the Green function for the Neumann problem (N-P') and describe some fundamental properties. The sections 6 is devoted to establish the a priori estimate of solutions in the half space Rn+, which plays an essential role in the present paper. Various properties of the modified Schauder spaces, which are needed in the proofs of the theorems, will be proved in the Appendix.

§2 Main Results

In the Neumann problem, we have to deal with unbounded solutions as we shall show in the end of this section. So we are obliged to modify the classi-cal Schauder spaces to admit unbounded solutions. To this end we define the function spaces represented by Λμτ(Ω) (See also Definition 6-1 in §6).

DEFINITION 2.1. Let 0<μ<1, τ ≠0. A function u is said to belong to the class Λ μτ(Ω) if

(2-1)

is

finite,

where

(2-2)

Here, τ+=max(τ,0) and τ-=min(τ,0). Also, u is said to belong to the class Λ2+μ2+τ(Ω) if Dγu∈ Λ μτ(Ω) for any γ with ￨γ￨〓2 and we denote

Here we note that if τ 〓 μ, then the class Λμτ(Ω) is reduced to the class Cμ(Ω). As for these modified Schauder spaces, it holds that the interporlation inequalities of the following type (For the proof, see §Appendix):

PROPOSITION A.

(1) Assume that u∈ Λ2+μ2+τ(Ω), τ>-1+μ. Then

(2) Let p∈ ∂Ω and wR=Ω ∩{x:￨x-p￨<R} for a small R>0. Assume that u∈ Λ2+μ2+τ(Ω), 0>τ>-1+μ and supp u⊂ ωR∪(∂ ωR∩ ∂Ω). Then _{, for} _any ε ∈(0,-τ), there is a positive number Cε such that we have

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(3) Assume that u∈ Λμτ(Ω), τ>-1+μ, τ ≠0. Then, for any ε>0, there is a positive number Cε such that

Our goal is the theorem of existence and uniqueness of the solution to (N-P) in the function space Λ2+μ2+τ (Theorem 2-2 below). And this is relied upon an a priori inequality for solutions (Theorem 2-1 below) and the method of conti-nuity. To obtain an a priori inequality, we proceed as follows. By a partition of unity of the closure Ω, the question is essentially reduced to prove an inequality for u with small support in a neighborhood of a boundary point. This is because the operator is not degenerate in the interior of the domain so that a nice inequal-ity in the interior is guaranteed by a classical result of J. Schauder. Let x0 be any boundary point. Then, there exists a neighborhood U of x0 and diffeomorphism X from U∩ Ω in the x-space to a semi-ball {lyl<r, yn〓0} in the y-space such that X(x0)=0, X(U∩ ∂Ω)⊂{yn=0} and that φ(X-1(y))=yn+o(yn).

Clearly, α(x) is close to the constant function α(x0) in U. Rewriting the new coordinates y again by x, we can locally approximate (N-P) by a problem

(N-P')

in Rn+,

on {x:xn=0}

for u with small support in a neighborhood of the origin. Here α is a constant satisfying α>-1 and f on the right hand side contains an error term arising from the localization.

THEOREM 2-1. Assume that α(x)>-1 on ∂Ω. Let τ, μ and α satisfy

(2-3) max(-1,-1+μ-α(x0)<τ<0 on ∂Ω.

Let u∈ Λ2+μ2+τ(Ω) be a solution of (N-P) for f/φ(x)α(x)∈ Λμτ(Ω). Then it holds that

(2-4)

Here C and C' are positive numbers independent of f and u.

We get first the inequality for solutions to (N-P') with small support with the aid of Theorem 6-1 and estimate the error terms arising from the partition of unity. We need Proposition A for an interpolation argument and to treat the terms involving logφ. Finally, to prove Theorem 2-1, we sum up the both sides of a finite number of inequalities in the local coordinate neighborhoods.

The existence and uniqueness of the solution to (N-P), which is the main theorem in this article, is stated as follows:

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THEOREM 2-2. Assume that α(x)>-1 on ∂Ω, c(x)>0 on Ω and that

(2.5) max

on ∂Ω.

Then, (N-P) has a unique solution in the space Λ2+μ2+τ(Ω) for every f such that f/φ α ∈ Λμτ(Ω).

Theorem 2-2 is proved in the following way: First, we prove the uniqueness of the solution with the aid of a function space

(2-6)

Note that f belongs to the dual space of W(Ω) provided that τ>-(α(x)+1)/2 and α(x)>-1 on ∂Ω. Since the weak solution is unique in the space W(Ω), the classical solution belonging to Λ2+μ2+τ(Ω) is also unique if there exists any. Second, if α(x) is identically equal to 0, then (N-P) is a variant of the classical Neumann problem for the Laplacian in a bounded domain. Lastly, for general α(x), we make use of the method of continuity. In each step of the reasoning, we need some restrictions on the values of c(x), α(x), μ and τ (see (2-3) and (2-5)).

REMARK. Theorems 2-1 and 2-2 are also valid for the following (N-P") if τ ≠0 (in place of τ<0) and if infΩc(x) is sufficiently large.

(N-P")

in Ω

on ∂Ω.

In the rest of this section, we give an example of unbounded solution to (N-P). For simplicity we consider (N-P) locally in some neighborhood of the origin of Rn+ in place of a general domain Ω.

EXAMPLE. Let Q be the operator defined to be

(2-7)

Then, for u=x2nlogxn-2x1-x21, we have ∂u/∂xn￨xn=0=0 and

Qu=xx1n(x1-1)∈Cμ if x1>μ. However ∂2xnu=3+2log xn〓Cμ. And we see that

u∈ Λ2+μ2+τ(Rn+;loc) for any μ ∈(0,1) and any τ<0.

Note also that w=w1-x1n satisfies Lw=0 with the homogeneous Dirichret condition w￨xn=0=0 if x1<1 and the homogeneous Neumann condition

∂w/∂xn￨xn

=0=0 if x1<0 respectively. According to Proposition 1-1, we can consider w as a null-solution to the Dirichret problem (See (D-P) with c=0). Moreover, we see that w is excluded from our framework for the problem (N-P). To see this, let K be a compact set contained in Rn+∩{x:-1<x1<0}

such that K∩{x:xn=0}≠ φ and assume that w∈ Λ2+μ2+τ(K;loc) for some μ ∈(0,1) and τ<0. Then, we have

Furthermore, it follows from the assertions (1) and (3) in Lemma A-2 that τ 〓minx∈K∩{x

n=0}(-1-x1). But this contradicts to the conditions (2-3) and (2-5) in the main theorems, so that the assertion follows.

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§3. The proof of Theorem 2-1.

In this subsection we shall establish Theorem 2-1 using the results in the sub-sequent sections and Appendix. Though the function spaces and the degeneracy of the operators are rather different from those in [2] and [3], their arguements work in a similar way. Since the interpolation inequalities (Proposition A) are valid, it suffices to prove the following weak inequality in place of (2-4).

LEMMA 3-1. Assume that α(x)>-1 on ∂Ω. Let τ, μ and α satisfy

(2-3) max(-1,-1+μ-α(x))<τ<0 on ∂Ω.

Let u∈ Λ2+μ2+τ(Ω) be a solution of (N-P) for f/φ(x)α(x)∈ Λμτ(Ω). Then it holds

that

(3-1)

Here C is a positive number independent of f and u.

We begin with a localization. Let {Ωk}Nk=0 be a family of an open set with a smooth boundary such that

(3-2) Ω0⊂ Ω, Ω ⊂UNk=0Ωk.

Let {θk}Nk=1 be a partition of unity subordinate to this open covering, namely,

(3-3)

on Ω.

By a techinical reason we also impose on each θk the following condition:

(3-4) (▽ φ,▽ θk)￨∂ Ω=0, for k=1,2,3,…,N.

By virtue of a localization argument and the condition (3-3), it suffices to es-tablish the inequality for each function θku with u∈ Λμτ(Ω) for k=1,2,…, N. Let U be one of the openset Ωk (k=1,2,…,N) and let x0∈U∩ ∂Ω. As has been already mensioned, by X(x)=(X1(x), X2(x),…,Xn(x)) we denote a smooth diffeomorphism such that

(3-5)

and

and X satisfies the equations on the boundary:

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(3-6) (▽ φ,▽Xj)=δn,j, on ∂Ω ∩U for j=1,2,…,n.

By f*(y), we denote f(X-1(y)) for any function f. By virtue of this X, the operator A defined by

(3-7)

is transformed to

(3-8)

where

Here we note that

(3-9)

for some number

where hk(y) is a bounded differentiable function.

Then Lemma 3-1 follows directly from the following local version on Rn+:

LEMMA 3-2. Assume that α(x)>-1 on ∂Ω. Let τ, μ and α satisfy the in-equality (2-3) in Theorem 2-1. Moreover we suppose that the operator B satisfies (3-8) and (3-9). Then, for sufficiently small r∈(0,R/2), there exists a positive number C such that

(3-10)

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Here by ￨￨・￨￨μ _{,τ;S and ￨￨・￨￨2+μ,2+τ;S,} _we _denote _{the norms} _defined _{by (6-3)} _on the spaces Λμτ(S) and Λ2+μ2+τ(S) respectively.

In order to establish this, we auxiliarily set

(3-11)

So that B=B0+B1+B2,

where

(3-12)

Since it follows from (3-5) that

(3-13) bj,n(x)￨x∈ ∂Ω=0, j=1,2,…,n-1,

by a linear transformation

in Rn-1, the operator B0 is transformed

to

(3-14)

By ωR we denote the image of WR by this transform. Then the following a priori estimate holds.

PROPOSITION 3-1. Suppose that R>0 and α*(0)>-1. Let τ, μ and α*(0) satisfy

(3-15) max(-1,-1+μ-α*(0))<τ.

Then, there is a positive constant C such that for every u∈ Λ2+μ2+τ(ωR)∩ ε'(ωR), we have

(3-16)

Here ωR=ωR∪(∂Rn+∩ ∂ωR) (See also (3-5)).

This proposition follows directly from Theorem 6-1 in§6 if n〓3. In case of n=2, we can show this proposition in the following way: In place of Lα*(0), we consider the operator Lα.*(0)-∂∂2X0, where ∂x0 is the derivative with respect to an auxiliary variable x0. Then the Green function for this operator clearly exists and has the same properties, so that we can apply Theorem 5-1 again to prove Proposition 3-1 by using a suitable cut-off function in x0-space.

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Admitting this proposition for the moment, we shall prove Theorem 2-1 in the rest of this subsection (See §6, for the proof of this proposition). From this a

priori estimate, we immediately have for any v∈ Λ2+μ2+τ(WR)∩ ε'(WR)

(3-17)

Now we show that (3-17) implies Lemma 3-2 by virtue of a partition of unity, diffeomorphism and interporlation inequalities (Theorem A-1), so that Theorem 2-1 follows. We take ζ(y)∈C∞0(Rn) such that supp ζ ⊂W2τ, ζ=1 on Wτ and ∂ynζ(y)￨y

n=0, where r is an arbitrary number satisfying 0<r<R/2. For u∈ Λ2+μ2+τ(WR)∩ ε'(WR), we put v=ζu. Then we have

(3-18) Bv=B0v+B1v+B2v=ζ(y)Bu+[B,ζ]u,

and it follows from (3-17) that

(3-19)

Since ∂ynζ(y) and the coefficients of the operator B1 are vanishing on the hyper-plane {yn=0}, there is a sufficiently small ε0>0 such that if 0<r=R/2<ε0 we have the desired estimate (3.10).

§4. The proof of Theorem 2-2.

In this subsection we shall prove Theorem 2-2, namely, the existence and the uniqueness of the solution to the homogeneous Neumann problem (N-P).

THE PROOF OF THEOREM 2-2. First we prove the uniqueness of the clas-sical solution belonging to Λ2+μ2+τ(Ω). Auxiliarily we define

(4-1) W(Ω)={u:u is measurable and ￨￨u￨￨w<∞},

where

Then W(Ω) becomes a Hilbert space and by [W(Ω)]' we denote the dual space of W(Ω).

LEMMA 4-1. Assume that τ>-(α(x)+1)/2, and α(x)>-1 on ∂Ω. Then f belongs to the dual space [W(Ω)]' in a canonical way, if f/φ α ∈ Λμτ(Ω).

PROOF OF LEMMA 4-1. Let f/φ α be an element of Λμτ(Ω). Then it follows

from Lemma

A-2 that

(4-2)

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Hence we see ￨f￨￨f￨2φ-α ∈L1(Ω) provided 2(-τ)+<α(x)+1 on∂ Ω. This implies the assertion.

It follows from this lemma and Riesz' theorem that for every f∈ Λμτ(Ω) there is a unique element u∈W(Ω) satisfying

(4-3)

for any v∈W(Ω).

Therefore the classical solution to (N-P) belonging to Λμτ(Ω) is also unique if there exists any.

Secondly we establish the existence of the solution. To do so we introduce a family of operators, namely,

(4-4) Lθu=-φ(x)-θ α(x)div(φ(x)θ α(x)▽u)+c(x)u,

with parameter θ ∈[0,1]. If α(x)≡0, then (N-P) is a variant of classical homogeneous Neumann problem. More precisely, it holds that

LEMMA 4-2. Assume that α(x)≡0 and max(-1+μ,-1/2)<τ<0 on ∂Ω. Then, for every f∈ Λμτ(Ω), (N-P) has a unique solution u∈ Λ2+μ2+τ(Ω).

PROOF OF LEMMA 4-2. In view of the classical theory, there is a Green function G(x,y) to (N-P) with α(x) being identically 0 such that for every f∈ Cμ(Ω). [Gf](x)≡ ∫Ωg(x,y)f(y)dy∈C2+μ(Ω) is a unique solution to (N-P). By Theorem 2-1 and Theorem 6-1, it is easy to see that the Green operator G:Cμ →C2+μ can be extended to the bounded operator G* such that G*: Λμτ(Ω)→ Λ2+μ2+τ(Ω) with the homogeneous Neumann condition. And clearly u=G*f, f∈ Λμτ(Ω) is a unique solution to (N-P), if -1/2<τ. In particular if Ω=Rn+, G(x,y) coincides with K0(x,y), where K0(x,y) is the Green function to (N-P') (the homogeneous Neumann boundary value problem for the Laplacian in Rn+ in §5).

END OF THE PROOF OF THEOREM 2-2. Let F be the set of θ such that Lθ has the continuous inverse operator under the homogeneous Neumann condition. Then, F is non-empty because 0∈F by virtue of the above lemma. And Theorem 2-1 guarantees that F is an open and closed subset of [0,1]. Therefore, F=[0,1], in particular, 1∈F. Consequently, (N-P) has a unique solution for every f satisfying the prescribed condition. Here we note that in each step of the reasoning, we need the restrictions on the values of c(x), α(x), μ and τ (see (2-3) and (2-5)).

§5. The Green kernel for the Neumann problem (N-P').

In this section we shall define in Rn+ the Green function Kα(x,y) to (N-P') in §2 and describe its fundamental properties.

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DEFINITION 5-1: GREEN FUNCTION FOR (N-P'). For α>0, let us set

(5-1)

where

F(p,q,r,ω) is a hypergeometric functions defined by

(5-2)

and F(p,q,r,ω) satisfies the following equation:

(5-3) ω(1-ω)F"+(r-(1+p+q)ω)F'-pqF=0.

The kernel Kα(x,y) is not defined so far for α 〓0. However it can be continued to an entire function with respect to α ∈(2-n,0] if n〓3. To see this, choose

and fix a nonnegative integer And we assume that

(5-4) α=2・p〓0, and -m-1<p〓-m.

Then by the analytic continuation with respect to α, we immediately have

(5-5)

Kα(x,y)=Ip(x,y)+Jp(x,y),

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For example,

(5-6)

Here C(0) and C(-2) are positive numbers independent of (x,y)∈Rn+.

Then one can show the following fundamental properties of Kα(x,y). First of all we set

(5-7)

Then u is a solution of the problem (N-P'). More pricisely we have

PROPOSITION 5-1. Assume that α>2-n. Then Kα(x,y) is the Green function of the problem (N-P'). Namely, u(x)=[Kαf](x) satisfies the equation

Lα=-div(xαn∇u)=f in Rn+ and ∂xnu‖_xn=0=0, _for _any _{f/xαn∈C0(Rn+)∩} ε'(Rn+).

PROOF OF PROPOSITION 5-1. We shall establish this lemma assuming that α>0. The proof in the excluded case follows in a similar way. By virtue of (5-3) we see that for any (x,y)∈Rn+×Rn+ with x≠y

(5-8)

where

(5-9)

By virtue of the homogeneity of the kernel Kα, it is not difficult to see that LαKα(x,y)=δ(x-y) (Dirac's delta function). Also from Theorem B-3, we see that u satisfies the Neumann boundary condition.

After elementary calculations we see that the kernel Kα(x,y) obeys the following estimates. We only show a sketch of proofs in the case α>0 (For the detailed proofs see [7]). Let us set

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PROPOSITION 5-2. Assume that α>2-n. Then for any multi-index γ, there is a positive number C(γ,α) such that

(5-11)

otherwise.

Here C(α,γ) is a positive number independent of each (x,y)∈Rn+×Rn+.

PROPOSITION 5-3. Assume that α>2-n. Then, for any μ ∈(0,1), there is a positive number C(n,α) such that a kernel ∂nKα(x,y) is a homogeneous function of degree 1-n-α with respect to (x,y) and satisfies the majoration:

(5-12)

Here C(n,α) is independent of each point (x,y)∈Rn+×Rn+.

PROOF OF PROPOSITION 5-2 (α>0). We use the notations in (5-6). For any x,y∈Rn+ with x≠y, we see that

(5-13)

where γ is a arbitrary multi-index and C(α,γ) is a positive number independent of each (x,y)∈Rn+×Rn+. Then, the desired estimate follows by a straightforward calculation.

PROOF OF PROPOSITION 5-3 (α>0). Integration by parts gives us

(5-14)

Combining this

with (5-11)

we get the desired

estimate.

§6. An a priori inequality in Rn+ (Proof of Proposition 3-1). We shall estimate the solution to (N-P') in this subsection. First we define the function spaces on Rn+ analogous to Λ μτ(Ω) and Λ μτ(Ω) in §2.

DEFINITION 6-1. Let 0<μ<1, τ ≠0, and let f∈C(Rn+) be compactly supported in Rn+. Then f∈ Λ μτ(Rn+) if

(6-1)

Here (μ-τ)+=max(0,μ-τ). For positive integer k, f∈ Λkτ+μ(Rn+) if Dγf∈ Λ μ_β-k(Rn+) _{for any} _{γ with} _￨γ￨=k. _We _put _{￨u￨k+μ,τ=Σ￨γ￨=k￨Dγu￨μ,τ-k.} _We also set Λ μτ(Rn+;loc)={f∈C0(Rn+);gf∈ Λ μτ(Rn+), for any g∈C∞0(Rn+)}.

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￨・￨μ,τ is then a norm on Λ μβ(Rn+), and Λ μβ(Rn+) is topologized as the inductive limit of Banach spaces exactly as Cμ(Rn+).

Auxiliarily we define

(6-2) Cτ(Rn+)={f:f∈C(Rn+) and ￨f￨∞,τ<+∞},

where ￨f￨∞,τ ≡sup￨f(x)￨x(-τ)+n.

We also use the following notations. Let S be a domain

of Rn+. Then we set

(6-3)

Let us recall that for R>0

(6-4)

and

Now we are in a position to state the main result in this section which was essentially used in §3 to prove Proposition 3-1 (a priori estimate of solutions).

THEOREM 6-1. Suppose that α>max(-1,2-n), max(-1,-1+μ-α)< τ ≠0, and f/xαn∈ Λμτ(Rn+). Let u=Kαf. Then,

(a) ∂j∂ku,∂nu/xn∈ Λ μτ(Rn+;loc) for 1〓j, k〓n, and ∂nu=0 on the boundary.

Moreover we have

(b) If f/xαn∈ Λ μτ(Rn+)∩ ε'(WR/2), for some positive number R, then there is a positive number C such that for j,k=1,2,…,n, we have

(6-5)

REMARK. (1) From Lemma A-2, it holds that for any g∈ Λ μβ_-α(Rn+)∩ ε'(WR)

(6-6) ￨g￨∞,τ 〓C￨g￨μ,τRmin(τ+,μ),

where C is a positive number independent of each f and R.

(2) Since ∂xnu/xn∈ Λ μτ(Rn+;loc) by this Theorem, it follows from Lemma A-2 that ∂x_nu￨xn=0=0, _provided _τ>-1.

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(3) The condition -1+μ-α<τ is essentially needed to estimate the singular integrals.

PROOF OF THEOREM 6-1. Now we shall establish Theorem 6-1 in a chain of auxiliary lemmas, which are corresponding to the estimates of the classical singular integral concerning the Newtonian potential. First we shall deal with the singular integrals involving the kernel ∂xj∂xk[yαnKα(x,y)], for j=1,2,…,n and k=1,2,…,n-1.

LEMMA 6-1. Assume that α>max(-1,2-n). Let us set

for Then it holds that

(6-7)

Here j=1,2,…,n, k=1,2,…,n-1,v(x)=(v1(x),v2(x),…,vn(x)) is

the outward normal to ∂WR and dSx indicates (n-1)-dimensional area element in ∂WR.

PROOF OF LEMMA 6-1. It suffices to use Green's formula. Note that the kernel Kα(x,y) is symmetric with respect to x'=(x1,x2,…,xn-1) and y'=

(y1,y2,…,yn-1), and yαn∂j-Kα(x,y), j=1,2,…,n are locally integrable with respect to y∈Rn+ provided α>max(-1,2-n). For the precise proof, see Lemma 4.2 in [1; P. 55] for example.

In the representation formula (6-7), the second term is smooth in Rn+ and easily verified to satisfy the assertion. Hence it suffices to study the term of the form (6-8) for

where

(6-9)

for

To this end we choose non-negative smooth functions η ∈C∞0(WR) and ξ ∈ C∞0(Rn) so that η ≡1 on W2R/3 and ξ(x)≡1 for ￨x￨〓3/2, ξ(x)≡0 for

￨x￨〓2. Let us set forρ>0, x∈WR/2. Then we have the

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LEMMA 6-2. Assume that 0<μ<1, max(-1,-1+μ-α)<τ ≠0, α> max(-1,2-n) and F(x)∈ Λ μτ(WR/2). Then, the function v defined by (6-2) belongs to Λ μτ(Rn+;loc). More precisely, it holds that

(6-10)

Here C is positive number independent of each F.

PROOF OF LEMMA 6-2. We may assume that F∈C∞0(Rn+)∩ ε'(WR/2). First we estimate ￨v∞,τ:WR. For z=(x',R)∈ ∂WR, we have

(6-11)

(Proposition

5-1 and Lemma A-2)

(Homogeneity).

Then we have from Lemma A-2 (3)

(6-12)

Therefore all we have to do is to establish the Holder estimate of v. To this end,

we subdivide v as follows.

(6-13)

where

Clearly v3 is smooth. So we shall establish Holder estimates for v1 and v2. Take and fix a point x∈WR/2. It suffices from Lemma A-3 (k=1/4) that we estimate them in such a ball

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Choose a point a∈ ∂Bρ(x) so that ￨x-a￨=ρ<xn/4.

ESTIMATE OF v1 AND v2. From Theorem B-1 we immediately have

(6-15)

And

(6-16)

So that we have the desired estimate for v1. With somewhat more calculations we see that v2 satisfies the estimate (6-4). Hence we omit the detail.

END OF THE PROOF OF THEOREM 6-1. For F=f/xαn∈ Λμτ(Rn+), let us set u(x)=[KαF](x). In the previous lemmas, we have already shown that

∂xju,∂xj∂xku∈ Λ μτ(Rn+;loc), j=1,2,…,n, k=1,2,…,n-1. Therefore

it suffices to show that ∂x_nu/xn _and _{∂∂2xnu∈ Λ μτ(Rn+;loc)} _From _the _equation (N-P'), we have

(6-17)

where

By virtue of Proposition 5-3, we have

(6-18)

where

(6-19)

Since h(x)∈ Λ μτ(Rn+;loc), the rest of the proof follows from the next lemma ( the proof is ommited).

LEMMA 6-3. Assume that α>max(-1,2-n), max(-1,-1+μ-α)<

τ ≠0, and 0<μ<1. Then if

g∈ Λ μτ(Rn+;loc). More precisely it holds that, for R>0,

(6-20) ￨G￨μ,τ;,WR〓C￨g￨μ,τ,WR, and ￨G￨∞,τ;WR〓C￨g￨∞,τ;WR.

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§Appendix (The modified Schauder space)

Our main aim is to prove Proposition A which was stated in §2 and es-sentially used in the proof of Theorem 2-1. The proof is done in the following way: By virtue of a diffeomorphism and a partition of unity, the questions are essentially reduced to prove similar inequalities for u with small support in a neghborhood of a boundary point, and this reduction can be done on the same lines as that in the classical theory. So it is easy to see that Proposition A directly follows from Lemmas below on the modified Schauder spaces on Rn+. More precisely, the assertions (1), (2) and (3) respectively follow from Lemma A-4, Lemma A-6, and Lemmas A-7 and A-8.

PROOF OF PROPOSITION A. First of all we note that the similar spaces as Aμβ(Rn+) were treated in C.R. Graham [3] and [4], and Lemmas A-1, A-2 and

A-3 below are seen there. Therefore we omit the proofs of them.

LEMMA

A-1.

(A-1)

And if τ 〓 μ, then

Here Cμ(Rn+;loc) is a local version of Cμ(Rn+) defined as

(A-2)

for any

LEMMA A-2.

(1) If f∈ Λ μτ(Rn+) and τ<0, then f∈Cτ.

(2) If τ>0 and f∈ Λ μτ(Rn+), then f(x',・)∈Cmin(τ,μ)(R1+), uniformly in x'=(x1,…xn-1)∈Rn-1.

(3) If f∈Aμ β(Rn+), then it holds that

(A-3)

where R is a diameter of the support of f, and C is a positive number independent of each f.

LEMMA A-3. Assume that 0<μ<1, 0≠ τ 〓 μ and 0<k〓1. Let f∈C(Rn+) be compactly supported in Rn+. If there is a positive number C such that

(A-4) ￨f(x)-f(y)￨〓C(xτ-μn+yγn-μ)￨x-y￨μ,

for every x, y with ￨xn-yn￨〓kmin(xn,yn), then f∈ Λ μβ(Rn+).

LEMMA A-4 (LOCAL VERSION OF PROPOSITION A (1)).

If f∈ Λ1+μ1+γ(Rn+), γ>-1+μ, τ ≠0, then f∈Cμ. Moreover we have

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where τ_=min(τ,0) and C is a positive number independent of each f with support in a fixed compact set.

PROOF OF LEMMA A-4. Take and fix a zn∈(ρ,2ρ). Then we see that

(A-6)

In a similar way we see

(A-7)

So that we get, denoting by C a positive

number independent of each point

(x,y),

(A-8)

Therefore desired estimate follows.

LEMMA A-5. logxn∈ Λ μτ(Rn+;loc) if and only if τ<0.

PROOF OF LEMMA A-5. In order to see logxn∈ Λ μβ, if and only if τ<0, it suffices to show the inequality for some positive number C:

(A-9)

for

If we put xn=tyn,

this is equivalent to

(A-10)

for any

And this is clear if and only if τ<0 for some positive number C independent of each yn.

From Lemma A-4 and A-5 we immediately have

LEMMA A-6 (LOCAL VERSION OF PROPOSITION A (2)). If f∈Aμ+1β+1(WR) with -1+μ<τ<0, then

(A-11)

for any

Here WR={x∈Rn+:￨x'￨<R,0〓xn<R} and C is a positive number

independent of each f and R.

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LEMMA A-7 (LOCAL VERSION OF PROPOSITION A (3)).

Let u∈ Λ2+μ2+τ(Rn+)∩ ε'(WR) with τ>-1+μ, β ≠0. then we have for any ε>0,

(1)

for any

(2)

(3)

(4) ￨u￨∞,τ-μ 〓Rmax[0,μ-(τ)+]￨u￨∞,τ.

where ε'(WR) is the set of distributions having compact support in WR=BR(0)∩ Rn+ and C(ε) is a positive number independent of each function f.

PROOF OF LEMMA A-7. Since these inequality can be established in a usual way. we omit the proof. For the detailed proof, see Horiuchi [7].