Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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648 S. Shimizu

The initial domain Ω(0) = Ω

(0) ∪ Ω

−

(0) occupied by the ﬂuid is deﬁned by

(t)={x =(x



) ∈ R

| x

>α(x



),x



∈ R

n−1

−

(t)={x =(x



) ∈ R

| x

<α(x



),x



∈ R

n−1

under the assumption ∇



α

∞

n−1

)

≤ K

for some K

with 0 <K

≤ 1. Such a

domain is called a bent space.

Let the initial velocity v

and the initial domain Ω(0) be given. Our problem

is to ﬁnd the domain Ω

(t), the velocity vector ﬁeld v(x, t)=

,...,v

), and

the scalar pressure θ(x, t), x ∈ Ω(t) satisfying the Navier-Stokes equations:

ρ(∂

v +(v ·∇)v) − Div S(v, θ)=0 inΩ(t),t>0,

div v =0 inΩ(t),t>0,

[[ S(v, θ)ν

]] = σHν

+[[ρ]] c

on Γ(t),t>0,

V = v · ν

on Γ(t),t>0,

t=0

= v

in Ω(0). (1.1)

Here ρ and μ are given by

ρ =



in Ω

(t)

−

in Ω

−

(t)

,μ=



in Ω

(t)

−

in Ω

−

(t)

where positive constants ρ

and μ

denote the densities and viscosities, respec-

tively, which may have a jump across Γ(t). For given functions v

(t) deﬁned on

(t), we set

v(x, t)=



(x, t) x ∈ Ω

(t),

−

(x, t) x ∈ Ω

−

(t).

On the other hand, given function v(t)deﬁnedonΩ(t), v

(t) denotes the restric-

tion of v(t)toΩ

(t).

[[ v(t)]] = (v

(t) − v

−

(t))|

Γ(t)

denotes the jump of v on the interface. Positive constants σ and c

denote the

coeﬃcient of surface tension and the acceleration of gravity, respectively. ν

is the

unit outward normal to Γ(t)ofΩ

(t), S(v, θ)=μD(v) − θI is the stress tensor,

D(v)=(D

(v)) = (∂v

/∂x

+ ∂v

/∂x

) is the deformation tensor, H is the mean

curvature given by Hν

=Δ

Γ(t)

x,whereΔ

Γ(t)

is the Laplace-Beltrami operator

on Γ(t), and V is the velocity of the evolution of Γ(t) in the direction of ν

.For

diﬀerentiation, we use the symbol: div v =

j=1

∂

(v ·∇)v =



j=1

∂

,...,

j=1

∂



, Div S =



j=1

∂

,...,

j=1

∂



where ∂

= ∂/∂x

M denotes the transposed M,andS =(S

)(n ×n matrix).

Now, we shall discuss some known results of the unique existence theorem

of problem (1.1). Let W

2,1

q,p

(Ω × (0,T)) = L

((0,T),W

(Ω)) ∩ W

((0,T),L

(Ω)),

and for simplicity we write W

2,1

q,p

= W

2,1

q,p

(Ω × (0,T)) for some T>0andW

2,1

Local Solvability of Free Boundary Problems 649

2,1

p,p

. We state only the case for the incompressible Navier-Stokes equations for

unbounded domain and the case where the surface tension is taken into account.

For one-phase ocean problems, Allain [2] proved local unique solvability when

n = 2. Tani [25] proved the local unique solvability in W

α,

with α ∈ (5/2, 3) when

n = 3. Beale [5] proved the global unique solvability in W

α,

with α ∈ (3, 7/2)

when n = 3, provided that the initial data are suﬃciently small. Beale and Nishida

[6] obtained the asymptotic power-like in time decay of the global solutions. Tani

and Tanaka [26] proved the global solvability in W

α,

with α ∈ (5/2, 3) when

n = 3, provided that initial data are suﬃciently small.

For two-phase problems, Denisova [9, 10] proved the local unique solvability

for arbitrary initial data in W

α,

with α ∈ (5/2, 3) when n =3.Denisovaand

Solonnikov [11, 12] proved the local unique solvability for arbitrary initial data in

the H¨older spaces when n = 3. Tanaka [27] obtained the global unique solvability

in W

α,

with α ∈ (5/2, 3) for small initial data when n = 3. Abels [1] considered

varifold and measure-valued varifold solutions for singular free interfaces. All of

these results except for [1] were obtained in the L

framework or in the H¨older

space setting.

Pr¨uss and Simonett [17, 18, 19] proved local unique solvability in W

2,1

(p>

n+2) for two-phase free boundary problems with surface tension and gravity. They

proved the maximal regularity theorem relying on the H

∞

-calculus for linearized

problems and using the Hanzawa transform connecting a free boundary with a

ﬁxed boundary.

In this paper, we prove a local unique existence theorem of (1.1) in the space

2,1

q,p

(2 <p<∞ and n<q<∞) for any initial data that satisfy certain

compatibility conditions and for an initial domain as a bent space. Our approach

is completely diﬀerent from Pr¨uss and Simonett [17, 18, 19]; indeed, we prove

a maximal regularity theorem relying on the operator-valued Fourier multiplier

theorem from Weis [28] and Denk, Hieber and Pr¨uss [13] for linearized problems

and using the Lagrangean transform connecting a free boundary with a ﬁxed

boundary. We believe that solving the problem (1.1) in the space W

2,1

q,p

is important

not only from the viewpoint of a lower regularity condition for the initial data but

also from the viewpoint of the scaling argument. When we consider the scaling for

positive parameter λ,

v(x, t) → v

(x, t)=λv(λx, λ

t),

θ(x, t) → θ

(x, t)=λ

θ(λx, λ

t),

which maintains the problem (1.1) invariant. Since it holds that

v



((0,∞),L

))

= λ

1−

−

v

((0,∞),L

))

=1 (1.2)

is the critical scale under the scaling.

650 S. Shimizu

Before stating our main results, we ﬁrst discuss the formulation of the prob-

lem (1.1) by the Lagrange coordinate, instead of the Euler coordinate. Aside from

the dynamical boundary condition, a further kinematic condition for Γ(t)issat-

isﬁed that gives Γ(t)asasetofpointsx = x(ξ,t), ξ ∈ Γ(0), where x(ξ, t)isthe

solution of the Cauchy problem:

= v(x, t),x|

t=0

= ξ. (1.3)

This expresses the fact that the free surface Γ(t) exists for all t>0ofthesame

ﬂuid particles, which do not leave it and are not incident on it from Ω(t).

From here, we write Ω = Ω(0) and Γ = Γ(0). The problem (1.1) can therefore

be written as an initial boundary value problem in the given region Ω if we go over

from Euler coordinates x ∈ Ω(t) to the Lagrange coordinates ξ ∈ Ω connected with

x by (1.3). If a velocity vector ﬁeld u(ξ,t)=

,...,u

) is known as a function

of the Lagrange coordinates ξ, then this connection can be written in the form:

x = ξ +



u(ξ,τ) dτ ≡ X

(ξ,t). (1.4)

Passing to the Lagrange coordinates in (1.1) and setting θ(X

(ξ,t),t)=π(ξ,t),

we obtain

ρ∂

u − Div S(u, π)=DivQ(u)+R(u)∇π in Ω,t>0,

div u = E(u)=div

E(u)inΩ,t>0,

[[ ( S(u, π)+Q(u))ν

]] − σHν

−[[ ρ]] c

u,n

=0 onΓ,t>0,

[[ u]] = 0 o n Γ ,t>0,

t=0

= u

(ξ)inΩ, (1.5)

where u

(ξ)=v

(x)andX

u,n

stands for the n-th component of X

.Moreover,ν

stands for the unit outer normal to Γ(t)givenbyν

−1

ν/|

−1

ν|,whereν

denotes the unit outer normal to Γ of Ω

,namely

ν =(

√

)

−1

(∇



α, −1), g

=1+|∇



α|

A is the matrix whose element {a

} is the Jacobian of (1.4):

∂x

∂ξ

= δ



∂u

∂ξ

dτ,

and Q(u), R(π), E(u)and

E(u) are nonlinear terms of the following form:

Q(u)=μV





∇udτ



∇u, R(u)=V





∇udτ



E(u)=V





∇udτ



∇u,

E(u)=V





∇udτ



u, (1.6)

with some polynomials V

(·)of



∇udτ, j =1, 2, 3, 4, such as V

(0) = 0 (cf.

Appendix in [20]).

Local Solvability of Free Boundary Problems 651

We discuss a local in time unique existence theorem for the problem (1.5)

instead of the problem (1.1). In order to state our main results precisely, we in-

troduce the function spaces. For the diﬀerentiations of scalar θ and n-vector of

function u =(u

,...,u

), we use the following symbol:

∇θ =(∂

θ,...,∂

θ), ∇

θ =(∂

∂

θ | i, j =1,...n),

∇u =(∂

| i, j =1,...,n), ∇

u =(∂

∂

| i, j, k =1,...,n).

Let L

(G)andW

(G) denote the usual Lebesgue space and Sobolev space on a

given domain G, while ·

(G)

and ·

(G)

denote their norms, respectively. v ∈

(

)meansthatv

∈ W

) while v

(

)

= v

)

+ v

−

)

Note that v ∈ W

)isequivalenttov ∈ W

(

)and[[v]] = 0 . G i v e n B a n a c h

space X with norm ·

, X

denotes the n-product space of X, while the norm

of X

is denoted by ·

for simplicity, that is

= {f =(f

,...,f

) | f

∈ X}, f

j=1

f



for f =(f

,...,f

) ∈ X

Set

(G)={θ ∈ L

q,loc

(G) |∇θ ∈ L

(G)

Let

−1

(G) denote the dual space of



(G), where 1/q +1/q



=1.Forθ ∈

−1

(G) ∩L

(G), we have

θ

−1

(G)

=sup





θϕ dx

| ϕ ∈



(G), ∇ϕ



(G)



For 1 ≤ p ≤∞, L

(R,X)andW

(R,X) denote the usual Lebesgue space and

Sobolev space of X-valued functions deﬁned on the whole line R,and·

(R,X)

and ·

(R,X)

denote their norms, respectively. Set

p,γ

(R,X)={f : R → X | e

−γt

f(t) ∈ L

(R,X) for any γ ≥ γ

p,γ

,(0)

(R,X)={f ∈ L

p,γ

(R,X) | f(t)=0 fort<0},

p,γ

(R,X)={f ∈ L

p,γ

(R,X) | e

−γt

∂

f(t) ∈ L

(R,X)

for j =1,...,m and γ ≥ γ

p,γ

,(0)

(R,X)=W

p,γ

(R,X) ∩ L

p,γ

,(0)

(R,X),

p,(0)

(R,X)=L

p,0,(0)

(R,X),W

p,(0)

(R,X)=W

p,0,(0)

(R,X).

Let L and L

−1

denote the Laplace transform and its inverse transform, that is

L[f](λ)=



∞

−∞

−λt

f(t) dt, L

−1

[g](t)=

2π



∞

−∞

λt

g(λ) dτ, (1.7)

where λ = γ + iτ.Givens ∈ R and X-valued function f(t), we set

f(t)=L

−1

[|λ|

L[f](λ)](t).

652 S. Shimizu

We introduce the Bessel potential space of X-valued functions of order s as follows:

p,γ

(R,X)={f : R → X | e

−γt

f(t) ∈ L

(R,X) for any γ ≥ γ

p,γ

,(0)

(R,X)={f ∈ H

p,γ

(R,X) | f(t)=0 fort<0},

p,(0)

(R,X)=H

p,0,(0)

(R,X).

If X is a UMD space and 1 <p<∞, then replacing the Fourier multiplier theorem

of S.G. Mihlin [15, 16] by that of J. Bourgain [7] in the paper due to A.P. Calder´on

[8] about the Bessel potential space, we see that H

p,γ

(R,X) is continuously imbed-

ded into H

p,γ

(R,X)whens>r≥ 0andH

p,γ

(R,X)=W

p,γ

(R,X)whens is a

non-negative integer. Moreover, the Sobolev imbedding theorem holds, that is if

1 <p<q<∞, s>r≥ 0ands −r =1/p −1/q,thenH

p,γ

(R,X) is continuously

imbedded into H

q,γ

(R,X), and if 0 <s− 1/p < 1, then every function in H

p,γ

coincides almost everywhere with a Lipschitz continuous function of order s −1/p.

For notational simplicity, we write

2,1

q,p,γ

(G × R)=L

p,γ

(R,W

(G)) ∩ W

p,γ

(R,L

(G)),

2,1

q,p,γ

,(0)

(G × R)=L

p,γ

,(0)

(R,W

(G)) ∩ W

p,γ

,(0)

(R,L

(G)),

1,1/2

q,p,γ

(G × R)=L

p,γ

(R,W

(G)) ∩ H

p,γ

(R,L

(G)),

1,1/2

q,p,γ

,(0)

(G × R)=L

p,γ

,(0)

(R,W

(G)) ∩ H

p,γ

,(0)

(R,L

(G)),

2,1

q,p,(0)

(G × R)=W

2,1

q,p,0,(0)

(G ×R),H

1,1/2

q,p,(0)

(G ×R)=H

1,1/2

q,p,0,(0)

(G × R).

The Bessel potential space H

p,γ

(R,W

(G)) is obtained by complex interpolation:

p,γ

(R,W

(G)) = [L

p,γ

(R,W

(G)),W

p,γ

(R,L

(G))]

1/2

where [·, ·]

denotes the complex interpolation functor (cf. Shibata-Shimizu [21]),

and therefore,

p,γ

(R,W

(G)) ⊃ W

2,1

q,p,γ

(G × R). (1.8)

The following theorem is the main result in this paper.

Theorem 1.1. Let α ∈ W

3−1/q

n−1

), 2 <p<∞ and n<q<∞. Assume that

there exists constant K

with 0 <K

≤ 1 such that ∇



α

∞

n−1

)

≤ K

.Then

for any initial data u

∈ (L

(Ω),W

(Ω))

1−1/p,p

= B

2(1−1/p)

q,p

(Ω) which satisﬁes the

compatibility conditions:

div u

=0 in Ω, [[ D(u

)ν −(D(u

)ν, ν)ν]] = 0 on Γ, [[ u

]] = 0 on Γ, (1.9)

there exists T>0 depending on u



2(1−1/p)

q,p

(Ω)

and α

3−1/q

n−1

)

such that

the problem (1.5) admits a unique solution

(u, π) ∈ W

2,1

q,p

(Ω × (0,T)) ×L

((0,T),

(Ω)).

Here, (·, ·)

1−1/p,p

denotes the real interpolation functor.

Local Solvability of Free Boundary Problems 653

To prove Theorem 1.1, we use the contraction mapping principle based on

the L

-L

maximal regularity theorem of the following linearized problem:

ρ∂

u − Div S(u, π)=ρf in Ω,t>0,

div u = f

in Ω,t>0,

∂

η − ν · u = d on Γ,t>0,

[[ S(u, π)ν]] − (σΔ

+[[ρ]] c

)ην=[[h]] o n Γ ,t>0,

[[ u]] = 0 o n Γ ,t>0,

t=0

=0 inΩ,

η|

t=0

=0 onΓ. (1.10)

Theorem 1.2. Let 1 <p,q<∞, n<r<∞ and q ≤r. Assume that α

3−1/r

n−1

)

≤

M. Then there exist K

with 0 <K

≤ 1 and γ

> 1 depending on M, p, q and n

such that if ∇



α

∞

n−1

)

≤ K

, then the following assertion holds: For f , f

, d

and h of (1.10) satisfying the conditions

f ∈ L

p,γ

,(0)

(R,L

(Ω))

∈ L

p,γ

,(0)

(R,W

(Ω)) ∩ W

p,γ

,(0)

(R,L

(Ω)),

d ∈ L

p,γ

,(0)

(R,W

2−1/q

n−1

)),h∈ H

1,1/2

q,p,γ

,(0)

(Ω × R)

the problem (1.10) admits a unique solution (u, π, η) such that

u ∈ W

2,1

q,p,γ

,(0)

(Ω × R)

,π∈ L

p,γ

,(0)

(R,

(Ω)),

η ∈ W

p,γ

,(0)

(R,W

2−1/q

n−1

)) ∩ L

p,γ

,(0)

(R,W

3−1/q

n−1

)).

Moreover there exists an extension of the pressure ¯π such that [[ ¯π]] = [[ π]] and

¯π ∈ H

1,1/2

q,p,γ

,(0)

(Ω × R). The solution satisﬁes the estimate

e

−γt

u

2,1

q,p

(Ω×R)

+ e

−γt

∇π

(R,L

(Ω))

+ e

−γt

¯π

1,1/2

q,p

(Ω×R)

+ e

−γt

(∂

η, ∇



η)

(R,W

2−1/q

n−1

))

≤ C



e

−γt

f

(R,L

(Ω))

+ e

−γt

d

(R,W

2−1/q

n−1

))

+ e

−γt



(R,W

(Ω))

+ e

−γt

∂



(R,L

(Ω))

+ e

−γt

h

1,1/2

q,p

(Ω×R)



for all γ ≥ γ

, where positive constant C depends on M , γ

, p, q,andn.

In §2and§3 we prove Theorem 1.2. In §2 we consider the Neumann problem

in a bent space and in §3 consider a problem with surface tension and gravity in

a bent space. In §4, we reduce the boundary condition to a linearized problem. §5

is devoted to initial ﬂow. In §6, we solve the nonlinear problem by the contraction

mapping principle based on Theorem 1.2.

654 S. Shimizu

2. Analysis in a bent space for the Neumann problem

In this section, we consider the Neumann problem in a bent space:

ρ∂

u − Div S(u, π)=ρf in Ω,t>0,

div u = f

in Ω,t>0,

[[ S(u, π)ν]] = [[ h]] o n Γ ,t>0,

[[ u]] = 0 o n Γ ,t>0,

t=0

=0 inΩ. (2.1)

We set

p,q,γ

(Ω × R)={(f,f

,h) ∈ L

p,γ

,(0)

(R,L

(Ω))

×(L

p,γ

,(0)

(R,W

(Ω))

∩W

p,γ

,(0)

(R,

−1

))) × H

1,1/2

q,p,γ

,(0)

(Ω × R)

p,q,γ

(Ω × R)={(u, π, ¯π) ∈ W

2,1

q,p,γ

,(0)

(Ω × R)

× L

p,γ

,(0)

(R,

(Ω))

× H

1,1/2

q,p,γ

,(0)

(Ω × R) | [[ ¯π]] = [[ π]] }.

The following theorem is the main result in this section.

Theorem 2.1. Let 1 <p,q<∞, n<r<∞ and q ≤ r. Assume that

α

2−1/r

n−1

)

≤ M . Then there exist constants K

with 0 <K

≤ 1 and γ

> 0

depending on M , p, q and n such that if ∇



α

∞

n−1

)

≤ K

, then the following

assertion holds: For (f,f

,h) ∈I

p,q,γ

(Ω ×R), the problem (2.1) admits a unique

solution (u, π, ¯π) ∈D

p,q,γ

(Ω × R) satisfying an estimate:

e

−γt

u

2,1

q,p

(Ω×R)

+ e

−γt

∇π

(R,L

(Ω))

+ e

−γt

¯π

1,1/2

q,p

(Ω×R)

≤ C



e

−γt

f

(R,L

(Ω))

+ e

−γt



(R,W

(Ω))

+ e

−γt

(∂

,γf

)

(R,

−1

))

+ e

−γt

h

1,1/2

q,p

(Ω×R)



for any γ ≥ γ

,whereC is a positive constant depending on M , p, q and n.

Let

= R

\ R

= R

∪ R

−

= {x =(x



) ∈ R

| x

=0,x



∈ R

n−1

= {x =(x



) ∈ R

|±x

> 0,x



∈ R

n−1

In order to prove Theorem 2.1, we based our argument on the maximal L

-L

regularity result of the Neumann problem with the plainer interface R

ρ∂

u − Div S(u, p)=ρf, in

,t>0,

div u = f

,t>0,

[[ S(u, p)ν

]] = [[ h]] o n R

,t>0,

[[ u]] = 0 o n R

,t>0,

t=0

=0 in

, (2.2)

Local Solvability of Free Boundary Problems 655

where ν

(0,...,0, −1). We set

p,q

(

×R)={(f,f

,h) ∈ L

p,(0)

(R,L

(

))

× (L

p,(0)

(R,W

(

))

∩ W

p,(0)

(R,

−1

))) × H

1,1/2

q,p,(0)

(

× R)

p,q

(

×R)={(u, π, ¯π) ∈ W

2,1

q,p,(0)

(

×R)

×L

p,(0)

(R,

(

))

×H

1,1/2

q,p,(0)

(

× R) | [[ ¯π]] = [[ π]] }.

We obtained the following result (cf. Theorem 1.2 in [23]).

Theorem 2.2. Let 1 <p,q<∞.For(f,f

,h) ∈I

p,q

(

× R), the problem (2.2)

admits a unique solution (u, π, ¯π) ∈D

p,q

(

× R) satisfying the estimate:

e

−γt

u

2,1

q,p

(

×R)

+ e

−γt

∇π

(R,L

(

))

+ e

−γt

¯π

1,1/2

q,p

(

×R)

+ γe

−γt

u

(R,L

(

))

≤C(e

−γt

f

(R,L

(

))

+ e

−γt



(R,W

(

))

+ e

−γt

(∂

,γf

)

(R,

−1

))

+ e

−γt

h

1,1/2

q,p

(

×R)

)

for any γ ≥ γ

,whereC is a positive constant depending on p, q and n.

For a function α(x



) deﬁned on x



∈ R

n−1

,wehaveanextensionA

(x).

Lemma 2.3. Let n<r<∞ and 0 <<1.Forα ∈ W

2−1/r

n−1

),thereexists

(x) ∈ W

) which satisﬁes A

=±0

= α, ∂



∞

)

≤  and

A



)

≤ Cα

n−1

)

∇



)

≤ Cα

k−1/r

n−1

)

,k=1, 2. (2.3)

Proof. For ±x

> 0, if we set

(x)=2F



∓

√

1+|ξ



ˆα(ξ



)](x



) −F



∓2

√

1+|ξ



ˆα(ξ



)](x



then A

satisﬁes A

=±0

= α, ∂

=±0

=0.ForA



,sx

),sincewecan

take s>0sosmallas

|∂



,sx

)|≤s∂



∞

)

≤ sA



)

≤ ,

it holds that ∂



∞

)

≤ . For the estimate (2.3), see, e.g., §2, Theorem 8.2

in [14]. 

In what follows, we prove Theorem 2.1. For A

(y) deﬁned in Lemma 2.3, we

set A(y)=A

(y), y

> 0, A(y)=A

−

(y), y

< 0. Let us consider the map x ∈

Ω →

 y deﬁned by the formula: x =



+ A(y)) with y



=(y

,...,y

If we set f(y

)=y

+ A(·,y

), then by Lemma 2.3,

∂

f =1+∂

A ≥ 1 −∂

A

∞

(

)

≥

656 S. Shimizu

it follows that f (y

) is a strictly monotone increasing function. Therefore there

exists an inverse function y

= f

−1

(·,x

) which satisﬁes

> 0 ⇒ x



, 0) = α(x



< 0 ⇒ x

−



, 0) = α(x



This shows that Ω and

are homeomorphic.

By using

∂y

∂x

= −

∂

1+∂

∂y

∂x

1+∂

=1−

∂

1+∂

by the chain rule we have

∂

∂x

∂

∂y

−

∂

1+∂

∂

∂y

∂

∂x

∂

∂y

−

∂

1+∂

∂

∂y

. (2.4)

We set y =Φ(x)=



− A(x



−1

(x)). For (u, π, ¯π) ∈D

p,q,γ

(Ω × R)which

satisfy (2.1), we set

v(y)=u ◦Φ

−1

(y)=u(x),θ(y)=π ◦ Φ

−1

(y)=π(x).

Then by (2.4), (2.1) is equivalent to the equation:

ρ∂

v − Div S(v,θ)=ρ

f + F (v, θ)in

,t>0,

div v =(1+∂

+ F

(v)in

,t>0,

[[ S(v, θ)ν

]] = [[ H

+ H(v)]] on R

,t>0,

[[ v]] = 0 o n R

,t>0,

t=0

=0 in

, (2.5)

where

f = f ◦Φ

−1

= f

◦Φ

−1

,andF =

,...,F

), F

, H

,...,H

)

and H =

,...,H

)aregivenby

±i

(v, θ)=−μ

=1

∂





∂



1+∂

∂

1+∂

∂





−

=1

∂



1+∂

∂





∂



+ ∂



−

∂



1+∂

∂

−

∂

1+∂

∂





− δ

i



i =1,...,n,

(v)=

j=1

∂

(∂

) −

j=1

∂

(∂

±i

√

(

±i

+ ∂

±n

),i=1,...,n− 1,

= h

◦ Φ

−1

= h

◦ Φ

−1

±n

√

±n

±i

(v)=2μ

∂

A∂

−B

±i

(v) − ∂

±n

(v),i=1,...,n− 1,

±n

(v)=−B

±n

(v),

Local Solvability of Free Boundary Problems 657

±i

(v)=μ



∂

1+∂

∂

1+∂

∂



+ μ

n−1

=1

∂





∂



+ ∂



−

∂



1+∂

∂

−

∂

1+∂

∂





i =1,...,n− 1,

±n

(v)=2μ

∂

1+∂

∂

+ μ

n−1

=1

∂





∂



+ ∂



−

∂



1+∂

∂



−

∂

1+∂

∂





. (2.6)

By using Theorem 2.2, we solve (2.5) by the contraction mapping principle. Given

(v, θ,

θ) ∈D

p,q

(

× R), let (w, κ, ¯κ) be a solution to the equation:

ρ∂

w − Div S(w, κ)=ρ

f + F (v, θ)in

,t>0

div w =(1+∂

+ F

(v)in

,t>0

[[ S(w, κ)ν

]] = [[ H

+ H(v)]] on R

,t>0

[[ w]] = 0 o n R

,t>0.

t=0

=0 inR

. (2.7)

We remark that if f

∈ W

p,(0)

(R,

−1

)), then

(1 + ∂

∈ W

p,(0)

(R,

−1

)). (2.8)

Indeed, for ϕ ∈



(Ω) we set ψ(y)=ϕ ◦ Φ

−1

(y)=ϕ(x). We have



(x)ϕ(x) dx =



(y)ψ(y)(1 + ∂

A) dy

because of the Jacobian det

∂x

∂y

=1+∂

A. Therefore it holds that

|((1 + ∂

,ψ)

| = |(f

,ϕ)

|≤f



−1

)

∇ϕ



)

≤ Cf



−1

)

∇ψ



)

which shows

(1 + ∂



(R,

−1

))

≤ Cf



(R,

−1

))

. (2.9)

Also we have

(1 + ∂



(R,W

(

))

≤ Cf



(R,W

(Ω))

. (2.10)

For notational simplicity we set

= ∇A

∞

(

)

= A

(

)

We may assume that 0 <K

< 1 a priori, and therefore K



≤ K

for  ≥ 1. For

a ∈ L

(

)andb ∈ L

p,(0)

(R,W

(

)), we have

ab

(R,L

(

))

≤ Ca

(

)

b

(R,W

(

))

. (2.11)