https://doi.org/10.1007/s00208-020-02007-3
Mathematische Annalen
Well-posedness and qualitative behaviour of the Mullins-Sekerka problem with ninety-degree angle boundary contact
Helmut Abels1 ·Maximilian Rauchecker2·Mathias Wilke3
Received: 12 March 2019 / Revised: 24 March 2020
© The Author(s) 2020
Abstract
We show local well-posedness for a Mullins-Sekerka system with ninety degree angle boundary contact. We will describe the motion of the moving interface by a height function over a fixed reference surface. Using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well- posedness of the full nonlinear problem via the contraction mapping principle. Here one difficulty lies in choosing the right space for the Neumann trace of the height function and showing maximalLp−Lq-regularity for the linear problem. In the second part we show that solutions starting close to certain equilibria exist globally in time, are stable, and converge to an equilibrium solution at an exponential rate.
1 Introduction
In this article we study the Mullins-Sekerka problem inside a bounded, smooth domain ⊂Rn,n=2,3, where the interface separating the two materials meets the boundary ofat a constant ninety degree angle. This leads to a free boundary problem involving a contact angle problem as well.
We assume that the domaincan be decomposed as =+(t)˙∪(t˚ )˙∪−(t), where ˚(t)denotes the interior of(t), an (n −1)-dimensional submanifold with boundary. We interpret (t) to be the interface separating the two phases,±(t), which will be assumed to be connected. The boundary of(t)will be denoted by
Communicated by Y. Giga.
B
Helmut Abelshelmut.abels@mathematik.uni-regensburg.de
1 Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany 2 Institut für Angewandte Analysis, Universität Ulm, 89081 Ulm, Germany
3 Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany
∂(t). Furthermore we assume(t)to be orientable, the unit normal vector field on (t)pointing from+(t)to−(t)will be denoted byn(t).
The precise model we study reads as
V(t)= −n(t)· ∇μ, on(t), (1.1a)
μ|(t)=H(t), on(t), (1.1b)
μ=0, in\(t), (1.1c)
n∂· ∇μ|∂=0, on∂, (1.1d)
(t)˚ ⊆, (1.1e)
∂(t)⊆∂, (1.1f)
∠((t), ∂)=π/2, on∂(t), (1.1g)
subject to the initial condition
|t=0=0. (1.1h)
Here V(t) denotes the normal velocity and H(t) the mean curvature of the free interface(t), which is given by the sum of the principal curvatures. Note that for our choice of orientation the curvature of(t)is negative if+(t)is convex. By·we denote the jump of a quantity across(t)in direction ofn(t), that is,
f(x):= lim
ε→0+
f(x+εn(t))− f(x−εn(t))
, x∈(t).
Equation (1.1g) prescribes the angle at which the interface(t)has contact with the fixed boundary∂, which will be a constant ninety degree angle during the evolution.
We can alternatively write (1.1g) as the condition that the normals are perpendicular on the boundary of the interface,
n(t)·n∂=0, on∂(t). (1.2) For the physical origin of the Mullins-Sekerka problem we refer to [12,18]. For a discussion of the Mullins-Sekerka problem in the context of gradient flows we refer to [16,17,28]. Existence of classical solutions locally in time was shown in [9] for two-dimensional domains and in [14] for general dimensions. Stability of spheres and exponential convergence to some sphere was shown in [8] for two-dimensional domains and in [15]. Existence of weak solutions globally in time was shown in [33]. In Alikakos et al. [3] consider the case of ninety degree contact in two space dimensions in the case where the initial interface is assumed to be smooth and close to a part of a circle. They discuss the qualitative behaviour for large times and sufficiently small droplets. They obtain stability and instability results in dependence of the curvature of the boundary. Their arguments rely on an harmonic extension of the curvature and finding explicit formulas in complex variables.
Let us first state some simple properties of this evolution. Note that we obtain the compatibility condition
∠(0, ∂)=π/2 on∂0.
Furthermore, the volume of each of the two phases is conserved, d
dt|±(t)| =0, t ∈R+. (1.3)
Here,±(t)denote the two different phases separated by the sharp interface,= +(t)∪(t˚ )∪−(t). Then (1.3) stems from
d
dt|+(t)| =
(t)V(t)dHn−1= −
(t)n(t)· ∇μdHn−1
=
+(t)μd x=0.
However, the energy given by the surface area of the free interface(t)satisfies d
dt|(t)| ≤0, t ∈R+. Indeed, an integration by parts readily gives
d
dt|(t)| = −
(t)H(t)V(t)dHn−1=
(t)μ|(t)n(t)· ∇μdHn−1
= −
|∇μ|2d x≤0.
In this article we are concerned with existence of strong solutions of the Mullins- Sekerka problem (1.1) locally in time and stability of certain equilibria. In comparison to [3] we do not restrict ourselves to a two-dimensional situation and sufficiently small droplets. To this end we will later pick some reference surface inside the domain , also intersecting the boundary with a constant ninety degree angle, and write the moving interface as a graph over by a height functionh, depending on x ∈ and time t ≥ 0. Pulling back the equations to the time-independent domain\ we reduce the problem to a nonlinear evolution equation forh. The corresponding linearization for the spatial differential operator forhthen turns out to be a nonlocal pseudo-differential operator of order three, cf. [15]. We also refer to the introduction of Escher and Simonett [15] for further properties of the Mullins-Sekerka problem.
In the following, we will be interested in height functionshwith regularity h∈W1p
0,T;Wq1−1/q()
∩Lp(0,T;Wq4−1/q()),
where pandq are different in general. We will chooseq <2 and pfinite but large, to ensure that the real interpolation space
Xγ :=
Wq1−1/q(),Wq4−1/q()
1−1/p,p=Bq p4−1/q−3/p() (1.4) continuously embeds intoC2(), cf. e.g. Sections 3.3.1 and 3.3.6 in [35] . By an ansatz where p =q <2, this is not achievable. We need however the restrictionq <2 to avoid additional compatibility conditions for the elliptic problem, cf. also Sect.4.2.
This however requires anLp−Lqmaximal regularity result of the underlying linear problem, which we will also show in this article.
Outline of this paper In Sect. 2 we will briefly introduce function spaces and techniques we work with and give references for further discussion. In Sect.3 we rewrite the free boundary problem of the moving interface as a nonlinear problem for the height function parametrizing the interface. Section4is devoted to the analysis of the underlying linear problem, where an extensive analysis is made on the half-space model problems. This is needed since these model problems at the contact line are not well-understood until now. The main result of this section is Lp−Lq maximal regularity for the linear problem. Section5contains that the full nonlinear problem is well-posed and Sect.6is concerned with the stability properties of solutions starting close to certain equilibria. These results are part of the second author’s PhD thesis [32].
2 Preliminaries and function spaces
In this section we give a very brief introduction to the function spaces we use and techniques we employ in this article. For a more detailed approach we refer the reader to the books of Triebel [35] and Prüss and Simonett [30].
2.1 Bessel-potential, Besov and Triebel-Lizorkin spaces
As usual, we will denote the classicalLp-Sobolev spaces onRnbyWkp(Rn), where kis a natural number and 1≤ p ≤ ∞. The Bessel-potential spaces will be denoted byHps(Rn)fors ∈Rand the Sobolev-Slobodeckij spaces byWps(Rn). We will also denote the usual Besov spaces byBspr(Rn), wheres ∈ R,1≤ p,r ≤ ∞. Lastly, as usual the Triebel-Lizorkin spaces are denoted byFprs (Rn).
These function spaces on a domain⊂Rnare defined in a usual way by restric- tion. The Banach space-valued versions of these spaces are denoted by Lp(;X), Wpk(;X),Hsp(;X),Wps(;X),Bspr(;X),Fprs (;X), respectively. For precise definitions we refer to [27].
For results on embeddings, traces, interpolation and extension operators we refer to [1,30,34,35].
The following lemma is very well known and can easily be shown by using para- product estimates, see [7].
Lemma 2.1 For any s>0,1<p1,r <∞,
|vw|Bsp
1r(Rn)|v|Bsp
1r(Rn)|w|L∞(Rn)+ |v|L∞(Rn)|w|Bsp
1r(Rn) (2.1) for allv, w∈ Bsp1r(Rn)∩L∞(Rn). In particular, the space Bsp1r(Rn)∩L∞(Rn)is an algebra.
Proof See Corollary 2.86 in [7].
2.2R-boundedness,R-sectoriality andH∞-calculus
We first define the notion of sectorial operators as in Definition 3.1.1 in [30].
Definition 2.2 Let X be a complex Banach space and Abe a closed linear operator on X. Then Ais said to besectorial, if both domain and range of A are dense in X, the resolvent set of A contains(−∞,0), and there is some C > 0 such that
|t(t+A)−1|L(X)≤Cfor allt>0.
The concept ofR-bounded families of operators is next. We refer to Definition 4.1.1 in [30].
Definition 2.3 Let X andY be Banach spaces andT ⊆ L(X,Y). We say thatT is R-bounded, if there is someC >0 andp ∈ [1,∞), such that for eachN ∈N,{Tj : j =1, . . . ,N} ⊆T,{xj : j =1, . . . ,N} ⊆ X and for all independent, symmetric,
±1-valued random variablesεjon a probability space(,A, μ)the inequality
N
j=1
εjTjxj
Lp(;Y)
≤C
N j=1
εjxj
Lp(;X)
(2.2)
is valid. The smallestC>0 such that (2.2) holds is calledR-bound ofT and denote it byR(T).
We can now defineR-sectoriality of an operator as is done in Definition 4.4.1 in [30].
Definition 2.4 LetXbe a Banach space andAa sectorial operator onX. It is then said to beR-sectorial, ifRA(0):=R{t(t+A)−1: t >0}is finite. We can then define theR-angle of Aby means ofϕAR := inf{θ ∈ (0, π) : RA(π−θ) < ∞}.Here, RA(θ):=R λ(λ+A)−1: |argλ| ≤θ
.
We now define the important class of operators which admit a boundedH∞-calculus as in Definition 3.3.12 in [30]. For the well known Dunford functional calculus and an extension of which we refer to Sections 3.1.4 and 3.3.2 in [30]. Let 0< ϕ≤πand ϕ := {z ∈C : |argz| < ϕ}be the open sector with opening angleϕ. LetH(ϕ) be the set of all holomorphic functions f :ϕ → CandH∞(ϕ)the subset of all bounded functions ofH(ϕ).The norm inH∞(ϕ)is given by
|f|H∞(ϕ):=sup |f(z)| :z∈ϕ .
Furthermore let
H0(ϕ):=
α,β<0
Hα,β(ϕ),
whereHα,β(ϕ):= {f ∈ H(ϕ): |f|ϕα,β<∞}, and|f|ϕα,β:=sup{|zαf(z)| : |z| ≤ 1} +sup{|z−βf(z)| : |z| ≥1}.
Definition 2.5 Let X be a Banach space and A a sectorial operator on X. Then A admits a boundedH∞-calculus, if there areϕ > ϕAand a constantKϕ <∞, such that
|f(A)|L(X) ≤Kϕ|f|H∞(ϕ) (2.3) for all f ∈ H0(ϕ). The class of operators admitting a bounded H∞-calculus on X will be denoted byH∞(X). TheH∞-angle ofAis defined by the infimum of all ϕ > ϕA, such that (2.3) is valid,ϕ∞A :=inf{ϕ > ϕA:(2.3)holds}.
2.3 Maximal regularity
Let us recall the property of an operator having maximalLp-regularity as is done in Definition 3.5.1 in [30].
Definition 2.6 LetXbe a Banach space, J=(0,T),0<T <∞or J =R+andA a closed, densely defined operator on X with domainD(A)⊆X.Then the operator Ais said to have maximalLp-regularity onJ, if and only if for every f ∈Lp(J;X) there is a uniqueu ∈Wp1(J;X)∩Lp(J;D(A))solving
d
dtu(t)+Au(t)= f(t), t ∈ J, u|t=0=0, in an almost-everywhere sense inLp(J;X).
There is a wide class of results on operators having maximal regularity, we refer to Section 3.5 and Chapter 4 in [30] for further discussion. For results onR-boundedness and interpolation we refer to [22].
3 Reduction to a fixed reference surface
In this section we transform the problem (1.1a)–(1.1h) to a fixed reference configura- tion. To this end we construct a suitable Hanzawa transform, taking into account the possibly curved boundary of∂, by locally introducing curvilinear coordinates. We discuss the casen=3 in detail, in the case wheren=2 one has to replace the normal to∂with the conormal to, since∂consists of two isolated points only.
Let ⊂be a smooth reference surface and∂be smooth at least in a neigh- bourhood of∂. Furthermore, let∠(, ∂) = π/2 on∂. From Proposition 3.1
in [36] we get the existence of so called curvilinear coordinates at least in a small neighbourhood of, that is, there is some possibly smalla > 0 depending on the curvature ofand∂, such that
X :×(−a,a)→Rn, (p, w)→X(p, w),
is a smooth diffeomorphism onto its image and X(., .) is a curvilinear coordinate system. This means in particular that points on the boundary∂only get transported along the boundary, X(p, w)∈ ∂for allp ∈ ∂, w∈(−a,a). We need to make use of these coordinates since the boundary∂may be curved. Therefore a transport only in normal direction ofnis not sufficient here.
More precisely, the curvilinear coordinatesXare of form
X =X(s,r)=s+r n(s)+τ(s,r)T(s), s∈,r ∈(−a,a), (3.1) where the tangential correctionτTis as in [36]. Herebyndenotes the unit normal vector field ofwith fixed orientation,Tis a smooth vector field defined on the closure of with the following properties: it is tangent to, normal to∂, of unit length on∂ and vanishing outside a neighbourhood of ∂. In particular,T is bounded.
Furthermore, τ = τ(s,r)is a smooth scalar function such that X(s,r)lies on∂ whenevers ∈∂. It satisfiesτ(s,0)=0 for alls ∈. Moreover, since and∂ have a ninety degree contact angle, we have that
∂rτ(s,0)=0, s∈∂. (3.2)
Hence we may chooseτ in [36] to satisfy (3.2) for alls∈.
With the help of these coordinates we may parametrize the free interface as follows.
We assume that at timet ≥0, the free interface is given as a graph over the reference surface, that is, there is someh:× [0,T] →(−a,a), such that
(t)=h(t):= {X(p,h(p,t)): p∈}, t ∈ [0,T],
for smallT >0, at least. With the help of this coordinate system we may construct a Hanzawa-type transform as follows.
Letχ ∈ C0∞(R)be a fixed function satisfyingχ(s)=1 for|s| ≤1/3,χ(s)=0 for|s| ≥2/3 and|χ(s)| ≤ 4 for alls ∈Randa := X(×(−a,a)). Then for a given height functionh :→(−a,a)describing an interfacehwe define
h(x):=
x, x∈/a, (X◦Fh◦X−1)(x), x∈a, where
Fh(p, w):=(p, w−χ((w−h(p))/a)h(p)) , p∈, w∈(−a,a).
Recall that by properties of the curvilinear coordinate system, we have = {x ∈Rn: x=X(p,0),p∈}.Let
U := h ∈Xγ : |h|L∞()<a . Then we have the following result.
Theorem 3.1 For fixed h ∈ U, the transformation h : → is a C1- diffeomorphism satisfyingh(h)=.
Proof The proof is straightforward. It is easy to check that forx ∈ h we have that h(x)=X(p,0), wherep∈is determined by the identityx=X(p,h(p)). Hence h(h)=. Furthermore it is easy to see thatD Fhand hence Dhis invertible in every point. This concludes the proof sinceXγ →C2().
The following lemma gives a decomposition of the transformed curvature operator K(h):= Hh ◦h forh ∈ U. The result and proof are an adpation of the work in Lemma 2.1 in [2] and Lemma 3.1 in [15].
Lemma 3.2 Let n=2,3, q ∈(3/2,2),p >3/(2−3/q)andU ⊂Xγ be as before.
Then there are functions
P∈C1(U,B(Wq4−1/q(),Wq2−1/q())), Q∈C1(U,Wq2−1/q()), such that
K(h)=P(h)h+Q(h), for all h∈U∩Wq4−1/q().
Moreover,
P(0)= −,
wheredenotes the Laplace-Beltrami operator with respect to the surface. Remark 3.3 Note that the orthogonality relations (3.2) in [15] do not hold if we take Xto be curvilinear coordinates, since inXwe not only have a variation in normal but also in tangential direction. Therefore we have to modify the proofs in [2,15].
Proof We will derive a formula for the transformed mean curvature K(h)in local coordinates. We follow the arguments of [15].
The surfaceh(t)is the zero level set of the function ϕh(x,t):=
X−1
2(x)−h
X−1
1(x),t
, x∈a,t ∈R+,
whence we define
h(s,r):=ϕh(X(s,r),t)=r−h(s,t), s∈,r∈(−a,a).
SinceX :×(−a,a)→Rnis a smooth diffeomorphism onto its image it induces a Riemannian metricgX on×(−a,a). We denote the induced differential operators Gradient, Laplace-Beltrami, and Hessian with respect to(×(−a,a),gX)by∇X, X
and hessX. As in equation (3.1) in [15] we find that K(h)|s = 1
∇XhX
Xh−[hessXh](∇Xh,∇Xh)
∇Xh2X
(s,h(s)),
for alls ∈ , where∇XhX :=(gX(∇Xh,∇Xh))1/2.Note at this point that since X induces also a variation in tangential direction, the orthogonality relations (3.2) in [15] do not hold in general. However, we get in local coordinates that
(∂jX|∂nX)=(∂jX|n)+∂rτ(∂jX| T), j ∈ {1, . . . ,n−1},
and(∂nX|∂nX) =1+(∂rτ)2(T| T).In particular we see that on the surface the relations (3.2) in [15] still hold, but not away fromin general. By using well-known representation formulas for∇X,X, and hessX in local coordinates, one finds that
K(h)|s =
⎛
⎝n−1
j,k=1
aj k(h)∂j∂kh+
n−1
j=1
aj(h)∂jh+a(h)
⎞
⎠|(s,h(s)),
where
aj k(h)= 1 X(h)3
−X(h)2wj k +wj nwkn−
l=1
wjlwkn∂lh
−
n−1
l=1
wj nwkl∂lh+
n−1
l,m=1
wj mwkl∂lh∂mh
,
as well as aj(h)= 1
X(h)3
X(h)2 n l,k=1
lkjwlk−
n−1
q,k=1
n i,l=1
ilkwi qwl j∂kh∂qh
+
n−1
q=1
n i,l=1
nilwi qwl j∂qh+
n−1
k=1
n i,l=1
kilwi nwl j∂kh− n i,l=1
ilnwi nwl j
+
n−1
k=1
n i,l=1
kilwi jwln∂kh− n i,l=1
ilnwi jwln− n i,l=1
iljwi nwln
,
and
a(h)= − 1 X(h)
n j,k=1
nj kwj k+ 1 X(h)3
n i,j=1
ni jwi nwj n,
wherewi j :=(∂iX|∂jX),(wi j)=(wi j)−1,X(h):= ∇XhX, andki jdenote the Christoffel symbols with respect to(wi j). Let
P(h)|s =
⎛
⎝n−1
j,k=1
aj k(h)∂j∂k+
n−1
j=1
aj(h)∂j
⎞
⎠|(s,h(s)),
Q(h)|s =a(h)|(s,h(s)), (3.3)
in local coordinates. Mimicking the proof of Lemma 2.1 in [2],K(h)=P(h)h+Q(h) is the desired decompostion ofK, sinceXγ →C2(). The fact thatP(0)= −
follows from (3.3) and the formulas foraj k andaj. We are now able to transform the problem (1.1a)–(1.1h) to a fixed reference domain
\ by means of the Hanzawa transform. This however yields a highly nonlinear problem for the height function. The transformed differential operators are given by
∇h :=(Dth)∇, divhu:=Tr(∇hu), h :=divh∇h,
and the transformed normal bynh∂:=n∂◦th. This leads to the equivalent system
∂th= −nh(t)· ∇hη+(β(h)|nh(t)−n), on, (3.4a)
η|=K(h), on, (3.4b)
hη=0, in\, (3.4c)
nh∂· ∇hη|∂=0, on∂, (3.4d)
nh∂·nh(t)=0, on∂, (3.4e)
h|t=0=h0, on, (3.4f)
whereh0is a suitable description of the initial configuration such that|t=0 = 0
andβ(h) := ∂thn+∂rτT, cf. (3.1). Note that by the initial condition (1.1h) we have thatnh∂0 ·nh0 =0, which is a necessary compatibility condition for the system (3.4a)–(3.4f).
The following lemma states important differentiability properties of the transformed differential operators.
Lemma 3.4 Let n = 2,3, q ∈ (3/2,2), p > 3/(3−4/q)andU ⊂ Xγ as before.
Then
[h →h] ∈C1(U;B(Wq2(\);Lq())),
[h→ ∇h] ∈C1(U;B(Wqk(\);Wqk−1(\))), k=1,2, [h→nh], [h→nh∂] ∈C1(U;C1()).
Proof The proof follows the lines of Section 4 in [2], sinceXγ → C2() by the
choice of pandq.
4 MaximalLp−Lqregularity for linearized problem 4.1 Reflection operators
We denote the upper half space of Rn by Rn+ := {x ∈ Rn : xn > 0}. We will denote by R the even reflection of a function defined on Rn+ across the boundary ∂Rn+ in xn direction, that is, we define R as an extension operator via Ru(t,x1, . . . ,xn):=u(t,x1, . . . ,−xn)for allxn<0. Note thatRadmits a bounded extension R : Lq(Rn+)→ Lq(Rn). The following theorems state that even more is true.
Theorem 4.1 Let1<q<∞. The even reflection in xndirection R induces a bounded linear operator from Wq1+α(Rn+)to Wq1+α(Rn), whenever0≤α <1/q.
Proof It is straightforward to verify that for a givenu∈Wq1+α(Rn+),
∂jRu(x1, . . . ,xn)=∂ju(x1, . . . ,−xn), j =1, . . . ,n−1, xn <0, and∂nRu(x1, . . . ,xn)= −∂nu(x1, . . . ,−xn). Hence also R: Wq1(Rn+)→Wq1(Rn) is a bounded operator. To show the claim for the fractional order space of order 1+α, it remains to show that the odd reflection of∇u ∈ Wqα(Rn+)n, that is, say T∇u,is again inWqα(Rn)n and that the corresponding bounds hold true.
We first note thatT∇u(x1, . . . ,xn) = e0∇u(x1, . . . ,xn)−e0∇u(x1, . . . ,−xn), wheree0denotes the extension by zero to the lower half plane. Note that by the real interpolation method,
Wqα(Rn+)=
Lq(Rn+),Wq1,0(Rn+)
α,q, Wqα(Rn)=
Lq(Rn),Wq1(Rn)
α,q, since 0 < α <1/q, cf. Equation (12) in Section 2.5.7 as well as Sections 3.3.6 and 3.4.2 in [35]. HerebyWq1,0(Rn+)denotes the closure ofC0∞(Rn+)inWq1(Rn+). Now, both zero extension operators
e0:Lq(Rn+)→Lq(Rn), e0:Wq1,0(Rn+)→Wq1(Rn),
are bounded and linear. From Theorem 1.1.6 in [25] we obtain thate0is therefore also a bounded and linear operator between the corresponding interpolation spaces. Hence
the theorem is proven.
Note that the above proof makes essential use of the fact that the derivative ofu ∈ Wq1+α(Rn+)has no trace on∂Rn+sinceα <1/q. If one has a trace it needs to be zero to reflect appropriately, which is the statement of the next theorem. The proof follows similar lines, we omit it here.
Theorem 4.2 Let q and R be as above. Then R induces a bounded linear operator Wq1+β(Rn+)∩ {u:∂xnu|xn=0=0} →Wq1+β(Rn)
for allβ∈(1/q,1).
We also need a reflection argument for the initial data in Xγ. The result reads as follows.
Theorem 4.3 The even reflection R induces a bounded linear operator Wq3+α(Rn+)∩ {u:∂xnu|xn=0=0} →Wq3+α(Rn)
for allα ∈ (0,1/q),q ∈ (3/2,2). In particular, R also induces a bounded linear operator
Bq p4−1/q−3/p(Rn+)∩ {u:∂xnu|xn=0=0} →Bq p4−1/q−3/p(Rn) for all q∈(3/2,2)and p>3/(2−3/q).
Proof The second statement follows from the first one forα=1−1/q−3/p<1/q sinceq <2. The first claim is shown as in the proof of Theorem4.1, using additionally
that∂xn∂xnRu =R∂xn∂xnu.
4.2 The shifted model problem on the half space
Letn=2,3. In this section we will be concerned with the linearized problem on the whole upper half space=Rn+with a flat interface := {x ∈ Rn+ : x1 =0}. Let ± := Rn+∩ {x : x1 ≷ 0}and let the normaln point from+ to−. We will consider
∂th+ω3h+n· ∇μ=g1, on, (4.1a)
μ|+xh =g2, on, (4.1b)
ω2μ−μ=g3, onRn+\, (4.1c)
en· ∇μ|∂Rn+ =g4, on∂Rn+, (4.1d)
en· ∇h|∂ =g5, on∂, (4.1e)
h|t=0=h0, on. (4.1f)
Here,x=(x2, . . . ,xn)andω >0 is a fixed shift parameter we need to introduce to get maximal regularity results on the unbounded time-space domainR+×Rn+.
Let us discuss the optimal regularity classes for the data. We seek a solutionh of this evolution equation in the space
Wp1(R+;Wq1−1/q())∩Lp(R+;Wq4−1/q()),
where pandqare specified below. In particular,μ∈Lp(R+;Wq2(Rn+\)). Let X0:=Wq1−1/q(), X1:=Wq4−1/q(),
and the real interpolation space
Xγ :=(X0,X1)1−1/p,p=Bq p4−1/q−3/p().
By simple trace theory, we may deduce the necessary conditions
g1∈Lp(R+;X0), g2∈Lp(R+;Wq2−1/q()), (4.2) g3∈Lp(R+;Lq(Rn+)), g4∈Lp(R+;Wq1−1/q(∂Rn+)), h0∈ Xγ. (4.3) It is now a delicate matter to find the optimal regularity condition forg5, which turns out to be
g5∈Fpq1−2/(3q)(R+;Lq(∂))∩Lp(R+;Wq3−2/q(∂)), (4.4) cf. TheoremB.1in the Appendix. Note thatg5has a time trace att =0, whenever 1−2/(3q)−1/p >0. Hence there is a compatibility condition inside the system whenever this inequality is satisfied, namely
g5|t=0=n∂· ∇h0|∂ on∂, (4.5) where∇denotes the surface gradient on. Here we consider elements in the tangent space ofas vectors inRnin the natural way. In the present casen∂· ∇h0|∂=
∂nh0|∂. Note that, sinceq <2, the trace of a generalg4∈Lp(R+;Wq1−1/q(∂Rn+)) is not defined on ∂Rn+∩ = ∂. Therefore there is no compatibility condition stemming from (4.1b) and (4.1d) on∂, whenever q < 2. Moreover, q < 2 will allow for a reflection argument at∂Rn+. The following theorem now states that these conditions are also sufficient. Note that the assumptions in Theorem4.4imply that q <2 and 1−2/(3q)−1/p>0 hold.
Theorem 4.4 Let 6 ≤ p < ∞, q ∈ (3/2,2)∩(2p/(p +1),2p) and ω > 0.
Then (4.1a)–(4.1f) has maximal Lp − Lq-regularity on R+, that is, for every (g1,g2,g3,g4,g5,h0)satisfying the regularity conditions (4.2)–(4.4)and the com- patibility condition (4.5), there is a unique solution (h, μ) ∈ (Wp1(R+;X0)∩ Lp(R+;X1))×Lp(R+;Wq2(Rn+\))of the shifted half space problem(4.1a)–(4.1f).
Furthermore,
|h|W1p(R+;X0)∩Lp(R+;X1)+ |μ|Lp(R+;Wq2(Rn+\))
is bounded by
|g1|Lp(R+;X0)+ |g2|L
p(R+;Wq2−1/q())+ |g3|Lp(R+;Lq(Rn+))
+|g4|L
p(R+;Wq1−1/q(∂Rn+))+ |g5|F1−2/(3q)
pq (R+;Lq(∂))∩Lp(R+;Wq3−2/q(∂))+ |h0|Xγ
up to a constant C=C(ω) >0which may depend onω >0.
Remark 4.5 The restriction for pcomes from the arguments to make sure that (1.4) embeds intoC2, the ones forqstem from the reflection arguments and TheoremB.1.
Proof We first reduce to a trivial initial value by extendingh0to˜ = {0} ×Rn−1 using standard extension results, cf. e.g. [35, Section 2.9] , and solving an Lp−Lq
auxiliary problem on Rn−1 using results of Section 4 in [31] to find some hS ∈ Wp1(R+;X0)∩Lp(R+;X1)such thathS|t=0=h0, cf. problem (4.9). Then we define
˜
g5:=g5−∂nhS|∂. Clearly,
˜
g5|t=0=g5|t=0−∂nh0|∂=0, on∂,
by the compatibility condition (4.5). This allows us to use TheoremB.1to find some h˜ ∈0W1p(R+;X0)∩Lp(R+;X1), where
0W1p(R+;X0):= {h ∈Wp1(R+;X0):h|t=0=0}, such that
∂nh|˜ ∂= ˜g5, on∂.
By simple trace theory we may findμ4∈ Lp(R+;Wq2(Rn+\))such that∂nμ4|∂Rn+= g4on∂Rn+. Let˜ :=R:= {x ∈Rn:x1=0}.We then solve the elliptic auxiliary problem
ω2μ˜−μ˜ = −R(ω2−)μ4, onRn\ ˜, (4.6a)
˜
μ|˜ = −Rxh˜−RxhS+Rg2−Rμ4|˜, on,˜ (4.6b) by a uniqueμ˜ ∈ Lp(R+;Wq2(Rn\ ˜)), cf. Section 4 in [4]. Note at this point that we used that due toq < 2 and Theorem4.1we have that the data in (4.6b) is in Lp(R+;Wq2−1/q())˜ . Note that by constructionμ˜ is even inxn direction since both the data in (4.6) are.
We have reduced the problem to the case where(g2,g3,g4,g5,h0)=0, that is, we are left to solve
∂th+ω3h+n· ∇μ=g1, on, (4.7a)
μ|+xh=0, on, (4.7b)
ω2μ−μ=0, onRn+\, (4.7c)
en· ∇μ|∂Rn+ =0, on∂Rn+, (4.7d)
en· ∇xh|∂=0, on∂, (4.7e)
h|t=0=0, on, (4.7f)
for possibly modifiedg1not to be relabeled in an Lp−Lq-setting. We reflect the problem once more across the boundary∂Rn+using the even reflection inxndirection
