Inhomogeneous boundary value problems in spaces of higher regularity

Universität Konstanz

Inhomogeneous boundary value problems in spaces of higher regularity

Robert Denk Tim Seger

Konstanzer Schriften in Mathematik Nr. 319, September 2013

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-243834

SPACES OF HIGHER REGULARITY

ROBERT DENK AND TIM SEGER

Dedicated to Yoshihiro Shibata on occasion of his 60^th birthday

Abstract. Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equalsL^p(Ω). However, much less is known for theW_p^s(Ω)-realization,s > 0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the right- hand side satisfies certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the general situation where no analytic semigroup is generated.

1. Introduction

For the treatment of nonlinear parabolic equations, a priori estimates in L^p- Sobolev spaces are an important step. Based on the theory of parameter-ellipticity, resolvent estimates have been established for a large class of equations, implying sectoriality of the corresponding operator or even maximal regularity for the non- stationary problem. For a boundary value problem in a domain Ω⊂Rⁿ, the basic space is usually L^p(Ω). This leads to a solution inW_p^m(Ω) wherem denotes the order of the differential operator. For the boundary traces, one obtains non-integer Besov spaces.

The situation becomes more complicated and much less investigated if one is interested in spaces of higher regularity. Here we start with W_p^s(Ω),s >0, as the basic space and expect the solution to be inW_p^m+s(Ω). Apart from its own inter- est, spaces of higher regularity naturally appear in mixed-order systems (Douglis- Nirenberg systems). Inhomogeneous boundary conditions and non-standard bound- ary spaces also appear in transmission problems and coupled systems. As an exam- ple, we mention the two-phase Stokes equations where the normal component of the velocity jumps across the interface. In the paper [22], Y. Shibata and S. Shimizu have shown maximal L^p-L^q-regularity for this system, introducing a special func- tion space adapted to the inhomogeneous jump conditions. The proofs in this and many other papers in fluid mechanics (see, e.g., Shibata [21]) are based on partial Fourier transform and careful estimates of the solution operators. In the present text, we essentially follow the same approach, however, aiming at uniform a priori

Date: September 6, 2013.

1991Mathematics Subject Classification. Primary 35B45; Secondary 35J40, 47D06.

Key words and phrases. Boundary value problems, higher order Sobolev spaces, parameter- ellipticity, resolvent estimates.

1

estimates where the basic space is W_p^s(Ω) instead of L^p(Ω). We will restrict our- selves to scalar parameter-elliptic equations which can be seen as a first step in the direction of the Stokes system and general mixed-order systems.

Let us consider the boundary value problem

(1.1) (A−λ)u=f in Ω,

Bu=g on∂Ω

in a bounded sufficiently smooth domain Ω ⊂ Rⁿ. Here A is a scalar differen- tial operator of order m ∈ 2N, and B is a column of ^m₂ boundary operators, B = (B₁, . . . , B_m/2)^T, with ordB_j = m_j < m. Classical parameter-elliptic the- ory states that, under suitable ellipticity and smoothness conditions, a uniform a priori estimate for the solutionuholds. More precisely, we have

(1.2) |||u|||m,p,Ω≤C

kfkL^p(Ω)+

m/2

X

j=1

|||gj|||_m−mj−1 p,p,∂Ω

.

Here fors >0 the parameter-dependent norms||| · |||are defined by

|||u|||s,p,Ω:=kukW_p^s(Ω)+|λ|^s/mkukL^p(Ω) (s≥0)

(analogously for ||| · |||s,p,∂Ω). For s≥ 0, W_p^s(Ω) stands for the standard Sobolev- Slobedeckii space. From the a priori estimate (1.2), we immediately obtain the resolvent estimate

(1.3) kλ(λ−A_B)⁻¹k_L(Lp(Ω))≤C

for theL^p-realizationAB of the boundary value problem (A, B). This unbounded operator in L^p(Ω) is defined byD(AB) := {u∈ W_p^m(Ω) : Bu= 0} and ABu:=

A(D)u(u∈D(A_B)). In particular, under suitable parabolicity assumptions,A_B is sectorial and generates an analytic semigroup. In fact, A_B is evenR-sectorial and therefore admits maximal L^p-regularity. A priori estimates of the form (1.2) are known since long; we refer to the classical works Agmon [1], Agranovich-Vishik [4], Geymonat-Grisvard [13], and Roitberg-Sheftel [19]. ConcerningR-sectoriality and maximal regularity, we mention Denk-Hieber-Pr¨uss [10] and the references therein.

In spaces of higher regularity, however, the resolvent estimate (1.3) in general does not hold. In fact, it is easily seen (cf. Nesensohn [18]) that the Dirichlet- Laplacian ∆D in W_p¹(Ω) with domain D(∆D) := {u ∈ W_p³(Ω) : u|∂Ω = 0} does not generate an analytic semigroup; its resolvent decays like|λ|−1/2−1/(2p)as|λ| →

∞. The paper Denk-Dreher [8] deals with resolvent estimates for mixed-order systems. Here conditions on the basic space Y ⊂ W_p^s(Ω) were formulated which are necessary and sufficient for a generation of an analytic semigroup. It was shown that additional conditions have to be included in the basic space; these conditions can be seen as compatibility relations. For scalar equations or systems with the same order in each component, the method of Banach scales developed by Amann in [5] can be applied and gives a rather complete answer to the question of generation of an analytic semigroup. We will comment on this in Section 2 below. Generation of analytic semigroups for parabolic equations was also studied by Guidetti in [17] (see also [16]). Here in particular mixed-order systems were studied which arise by the reduction of a higher-order system (in time) to a first- order system. A priori estimates in parameter-dependent norms have been studied, e.g., by Faierman and his coauthors in [2], [9], [12]. We also remark that a particular

case of an a priori estimate inW_p^s(Ω) was used in the second author’s thesis [20] to obtain a compactness property needed for a Schauder-type fixed-point argument in the context of a nonlinear elliptic-parabolic system.

In the present paper, we will discuss uniform a priori estimates for the boundary value problem (1.1) with inhomogeneous boundary conditions. To avoid technicali- ties, we will restrict ourselves to the model problems in the whole spaceRⁿ and the half-space Rⁿ+. Here the operators are assumed to have constant coefficients and no lower-order terms. The generalization to bounded sufficiently smooth domains and to variable coefficients by localization and partition of unity is quite standard, and we will not dwell on this.

In Section 2, we will study the whole space case and the case of homogeneous boundary conditions. Whereas in the whole space the a priori estimates leading to the generation of an analytic semigroup follows quite directly from Michlin’s theo- rem, the case of homogeneous boundary conditions can be treated by the Banach scale method. In Section 3, we will consider the case of inhomogeneous boundary conditions and derive the main a priori estimates of the present text.

2. The whole space case and the case of homogeneous boundary conditions

Let A(D) = P

|α|=ma_αD^α be a linear differential operator in Rⁿ, n ≥ 2, of orderm∈2Nwith constant coefficients a_α∈C. Here and in the following, we set D:=−i∂ and use the standard multi-index notationD^α:= (−i)^|α|∂_x^α₁¹. . . ∂_x^α_nⁿ. Let L ⊂Cbe a closed sector in the complex plane with vertex at the origin. Without loss of generality, we may assume thatL= Σθwith Σθ:={z∈C\{0}:|argz|< θ}

for someθ∈(0, π]. The operatorA(D) is called parameter-elliptic in Σ_θ (see [4]) ifA(ξ)−λ6= 0 holds for all (ξ, λ)∈(Rⁿ×Σθ)\ {0}. HereA(ξ) :=P

|α|=maαξ^αis the symbol ofA(D). If the latter condition is satisfied withθ≥ ^π₂, the operatorA is called parabolic.

We will consider the realization of the operator A(D) in different scales of Sobolev spaces. For s∈Randp∈(1,∞), we denote byH_p^s(Rⁿ) andB^s_pp(Rⁿ) the standard Bessel potential and Besov space, respectively. The Sobolev-Slobodeckii space W_p^s(Rⁿ), s ≥ 0, coincides with H_p^s(Rⁿ) for s ∈ N0 and with B_pp^s (Rⁿ) for s ∈ (0,∞)\N. We recall that a closed linear operator A: X ⊃ D(A) → X in a complex Banach space X is called sectorial if the domain and the range of A are dense in X and if there exists φ ∈ (0, π) such that ρ(A) ⊃ Σφ and the set {λ(λ−A)⁻¹ : λ∈Σφ} is bounded in L(X). In this case, the supremum over all angles satisfying this condition is called the spectral angleφ_A ofA.

In the following,Cstands for a generic constant which may vary from inequality to inequality but is independent of the variables appearing in the inequality (and in particular independent ofλ).

In the whole space, it is easily seen that the realization of the operatorA(D) is sectorial:

Lemma 2.1. Let A(D)be parameter-elliptic inΣθ, and lets∈Randp∈(1,∞).

Then for everyλ∈Σ_θ\ {0}and every f ∈H_p^s(Rⁿ), the equation(A(D)−λ)u=f has a unique solutionu∈H_p^m+s(Rⁿ), and the a priori estimate

kuk_Hm+s

p (Rⁿ)+|λ| kukH_p^s(Rⁿ)≤CkfkH_p^s(Rⁿ)

holds. In particular, ifθ≥ ^π₂, the operatorA^(s) in H_p^s(Rⁿ), defined byD(A^(s)) :=

H_p^m+s(Rⁿ), A^(s)u:= A(D)u(u∈ D(A^(s))), is sectorial with spectral angle larger than ^π₂ and therefore generates an analytic semigroup.

The analog results hold whenH_p^s(Rⁿ)is replaced by the Besov spaceB_pp^s (Rⁿ).

Proof. This essentially follows from more general results on the existence of a bounded H^∞-calculus, see, e.g., Denk-Saal-Seiler [11]. However, the result can also easily be seen by an application of the Michlin’s theorem. In fact, for each λ∈ Σθ\ {0}, the unique solution ucan be written as u=F⁻¹(A(ξ)−λ)⁻¹Ff whereF stands for the Fourier transform inRⁿ. Now it is immediately seen that

m(ξ) := (1 +|ξ|²)^m/2(A(ξ)−λ)⁻¹

satisfies the conditions of Michlin’s theorem (see [23], Section 2.2.4). By this and the definition of the Bessel potential spaces, the results inH_p^s(Rⁿ) follow.

The analog results for Besov spaces are obtained by real interpolation.

We will now consider boundary value problems where we will again restrict ourselves to the model problem in the half-space Rⁿ+ :={x∈ Rⁿ : xn > 0}. As before, letA(D) =P

|α|=maαD^α,m∈2N, be a differential operator with constant coefficients and let B_j(D) =P

|β|=m_jb_jβD^β, j= 1, . . . ,^m₂, be boundary operators with constant coefficientsbjβ ∈C. We setB:= (B1, . . . , B_m/2)^T and consider the boundary value problem

(2.1)

(A(D)−λ)u=f in Rⁿ+, γ₀B(D)u=g onRⁿ⁻¹.

Here γ₀:u 7→ u|_Rn−1 denotes the boundary trace operator. The boundary value problem (A(D), B(D)) is called parameter-elliptic in Σθ if A(D) is parameter- elliptic in Σθ and if the following Shapiro-Lopatinskii condition holds:

For all (ξ⁰, λ) ∈ (Rⁿ⁻¹×Σθ)\ {0} and all h = (h1, . . . , h_m/2)^T ∈ C^m/2, the ordinary differential equation

(A(ξ⁰, Dn)−λ)v(xn) = 0 (xn>0), B(ξ⁰, Dn)v(0) =h,

v(x_n)→0 (x_n→ ∞)

has a unique solution. If these conditions are satisfied with θ ≥ ^π₂, the boundary value problem (A(D), B(D)) is called parabolic. Throughout the following, we assume that the boundary value problem (A(D), B(D)) is parameter-elliptic in Σθ. We first discuss the case of homogeneous boundary conditions, i.e., we assume g= 0 in (2.1), so we discuss theL^p(Rⁿ+)-realization of (A(D), B(D)) which is given by D(A_B) :={u∈ W_p^m(Rⁿ+) :γ₀B(D)u= 0} and A_Bu:= A(D)u (u ∈D(A_B)).

For this, we apply the method of Banach scales (see [5], Chapter V). We recall the main definitions and results. LetX be a Banach space,{·,·}an exact interpolation functor, and A:X ⊃D(A) → X be the generator of a C0-semigroup. Then for k∈N0, the spaceEk is defined byEk :=D(A^k), and theEk-realizationAkofAk−1

is iteratively defined by

D(Ak) :={u∈Ek∩D(Ak−1) :Ak−1u∈Ek}, Aku:=Ak−1u(u∈D(Ak)).

Here we have setE₀:=X andA₀:=A. Fors∈(0,∞)\N, we writes=k+θwith k∈N0 andθ∈(0,1) and define the spaceE_s:={Ek, E_k+1}θ and the operatorA_s

as theEs-realization ofAk, i.e.,

D(As) :={u∈Es∩D(Ak) :Aku∈Es}, Asu:=Aku(u∈D(Ak)).

Remark 2.2. In the above situation, [(E_s, A_s) : s ≥ 0] defines a scale of Ba- nach spaces in the sense of [5], Definition V.1.1. Moreover, the operator A_s is again the generator of a C0-semigroup inEs for all s≥0. This follows from [5], Theorem V.2.1.3 and Corollary V.2.1.4, after introducing a shift, i.e. considering Ae:=A−ω withω >0 sufficiently large such that 0∈ ρ(A). Moreover, we havee ρ(As) = ρ(A) for all s≥0, and the resolvent estimate carries over fromAto As, see [5], inequality (2.1.16) in Theorem 2.1.3. In particular, ifA:E0⊃E1→E0 is sectorial with angleφthen the same holds forAs:Es⊃Es+1→Es.

To apply the above abstract definitions to the boundary value problem, we in- troduce the following spaces (see Amann [6], Section 4.9):

Definition 2.3. Assume that theL^p-realization AB of (A(D), B(D)) generates a C0-semigroup. Forp∈(1,∞) ands∈[0,∞)\{km+mj+¹_p :k∈N0, j = 1, . . . ,^m₂}, we define the space W_p;(A,B)^s (Rⁿ+) as the set of all u ∈ W_p^s(Rⁿ+) which satisfy γ₀B_jA^ku= 0 for allk∈N0andj= 1, . . . ,^m₂ withs−mk−m_j >¹_p.

The Banach scale method gives the following result.

Theorem 2.4. Let the boundary value problem(A(D), B(D))be parabolic, and let p∈(1,∞) ands ∈[0,∞)\ {km+m_j+ ¹_p :k ∈N0, j = 1, . . . ,^m₂}. Then for all f ∈W_p;(A,B)^s (Rⁿ+)and allλ∈Σ^π

2\{0}the problem(A(D)−λ)u=f,γ₀B(D)u= 0, has a unique solutionu∈W_p;(A,B)^m+s (Rⁿ+), and the a priori estimate

(2.2) kuk_Wm+s

p (Rⁿ₊)+|λ| kuk_Ws

p(Rⁿ₊)≤Ckfk_Ws

p(Rⁿ₊) (u∈W_p;(A,B)^m+s (Rⁿ+)) holds. In particular, the W_p;(A,B)^s (Rⁿ+)-realizationA^(s)_B given by

D(A^(s)_B ) :=W_p;(A,B)^m+s (Rⁿ+), A^(s)_B u:=A(D)u(u∈D(A^(s)_B ))

is sectorial with angle larger than ^π₂ and therefore generates an analytic semigroup inW_p;(A,B)^s (Rⁿ+).

Proof. We will apply the method of Banach scales as introduced above. We consider theL^p-realization AB and remark that it is well known that AB is sectorial with angle larger than ^π₂ (see, e.g., [2]).

(i) First we show that for each k ∈ N0, we have W_p;(A,B)^km (Rⁿ+) = D(A^k_B). By definition, the inclusion “⊂” is obvious. To show the converse inclusion, we have to prove thatE_k :=D(A^k_B) :={u∈D(A_B) :A<sup>`</sup>u∈D(A<sub>B</sub>) (`= 0, . . . , k−1)} is contained inW_p;(A,B)^km (Rⁿ+).

Due to the definition ofD(A_B), we haveγ₀B_jA<sup>`</sup>u= 0 for all`= 0, . . . , k−1 and allj= 1, . . . ,^m₂. Therefore, we only have to show that D(A^k_B)⊂W_p^km(Rⁿ+). This is done iteratively. As u∈ D(A²_B) and γ₀B_ju= 0 for all j = 1, . . . ,^m₂, we have u∈W_p^m(Rⁿ+) andAu∈W_p^m(Rⁿ+). For λ₀ ∈ρ(A_B), the boundary value problem (A−λ0, B) is regular elliptic in the sense of Triebel [23], Def. 5.2.1/4. By elliptic regularity, we obtainu∈W_p^2m(Rⁿ+). ReplacingubyAuand usingA²u∈W_p^m(Rⁿ+), we can now proveAu∈W_p^2m(Rⁿ+). An iteration givesu∈W_p^km(Rⁿ+).

(ii) We consider (AB)s/m and the scale generated byAB. By real interpolation, we have fork∈N0andθ∈(0,1) the identities

Ek+θ= (Ek, Ek+1)θ,p=

W_p;(A,B)^km (Rⁿ+), W_p;(A,B)^(k+1)m(Rⁿ+)

θ,p=W_p;(A,B)^(k+θ)m(Rⁿ+).

Here the last equality was shown in Amann [6], Corollary 4.9.2, in a more general setting. Due to Amann [5], Theorem 2.1.3, we see that (AB)_s/m generates an analytic semigroup and that D((AB)_s/m) = E^s

m+1 = W_p;(A,B)^m+s (Rⁿ+). Therefore, (AB)s/m coincides with the W_p;(A,B)^s -realization A^(s)_B . In particular, ρ(A^(s)_B ) = ρ(AB)⊃Σ^π

2 \ {0}, and the resolvent estimate (2.2) holds.

We remark that the exceptional cases s =km+mj +¹_p arise due to the real interpolation results, see the discussion in Amann [6], Amann [7].

3. The case of inhomogeneous boundary conditions

Now we consider the boundary value problem (2.1) for g6= 0, again restricting ourselves to the model problem inRⁿ+. For this, we will use an explicit representa- tion of the solution. We start with the definition of the basic solutions. Throughout this section, we assume that (A(D), B(D)) is parameter-elliptic in a fixed sector Σθ.

Lemma 3.1. For each (ξ⁰, λ)∈ (Rⁿ⁻¹×Σ_θ)\ {0} and j = 1, . . . ,^m₂, we define the basic solution w_j =w_j(ξ⁰, x_n, λ) as the unique stable solution of the ordinary differential equation

(A(ξ⁰, Dn)−λ)wj(xn) = 0 (xn>0), B_k(ξ⁰, D_n)w_j(0) =δ_jk (k= 1, . . . ,m

2).

Thenwj can be written in the form w_j(ξ⁰, x_n, λ) =

Z

γ(ξ⁰,λ)

e^ixnτ(A(ξ⁰, τ)−λ)⁻¹N_j(ξ⁰, τ, λ)dτ

where γ(ξ⁰, λ)is a smooth contour in the upper complex half-plane which encloses all roots of the polynomial τ 7→ A(ξ⁰, τ)−λ with positive imaginary part. The functionsNj andwj are smooth with respect to their arguments and continuous for all(ξ⁰, λ)∈(Rⁿ⁻¹×Σθ)\ {0} , and we have the quasi-homogeneities

N_j(ρξ⁰, ρτ, ρ^mλ) =ρ^m−mj−1N_j(ξ⁰, τ, λ), wj(ρξ⁰,^x_ρⁿ, ρ^mλ) =ρ^−mjwj(ξ⁰, xn, λ) forρ >0,ξ⁰∈Rⁿ⁻¹\ {0} andλ∈Σ_θ.

Proof. These assertions are stated in Denk-Faierman-M¨oller [9], Lemma 2.5. See also Volevich [24] for an explicit construction ofNj. To define the solution operators, we will use a parameter-dependent extension operatorEλ:W^k−

1 p

p (Rⁿ⁻¹)→W_p^k(Rⁿ+) given by

(E_λg)(x⁰, x_n) := (F⁰)⁻¹exp −(|ξ⁰|+|λ|^1/m)x_n

(F⁰g)(ξ⁰).

Here F⁰ stands for the partial Fourier transform with respect to the firstn−1 variables. This operator was studied in Grubb-Kokholm [15] and in Agranovich- Denk-Faierman [2] in connection with the parameter-dependent norms above. It was shown that for allk∈N, the trace operator

γ0: W_p^k(Rⁿ+),||| · |||k,p,Rⁿ₊

→ W^k−

1 p

p (Rⁿ⁻¹),||| · |||_k−1 p,p,Rⁿ⁻¹

is continuous andEλis a continuous right-inverse toγ0. Here and in the following, we call a linear operator continuous with respect to the parameter-dependent norms if for each λ₀ >0 the norm of this operator can be estimated by a constantC = C(λ₀) which does not depend onλfor eachλ∈Σ_θ with|λ| ≥λ₀.

Let us now consider the boundary value problem (3.1)

(A(D)−λ)u= 0 inRⁿ+, γ0B(D)u=g onRⁿ⁻¹ withg∈Qm/2

j=1W^m+k−mj−

1

p p(Rⁿ⁻¹). Following an idea from Volevich [24], we write F⁰uin the form

(F⁰u)(ξ⁰, xn) =

m/2

X

j=1

wj(ξ⁰, xn, λ)(F⁰gj)(ξ⁰)

=−

m/2

X

j=1

Z ∞

0

∂

∂y_n h

wj(ξ⁰, xn+yn, λ)(F⁰Eλgj)(ξ⁰, yn)i dyn

=

m/2

X

j=1

T_j⁽¹⁾(λ)E_λg_j+T_j⁽²⁾(λ)(∂_nE_λg_j) .

Here the solution operatorsT_j⁽¹⁾, T_j⁽²⁾ are given by (T_j⁽¹⁾(λ)ϕ)(x) :=−

Z ∞

0

(F⁰)⁻¹(∂nwj)(ξ⁰, xn+yn, λ)(F⁰ϕ)(ξ⁰, yn)dyn, (T_j⁽²⁾(λ)ϕ)(x) :=−

Z ∞

0

(F⁰)⁻¹w_j(ξ⁰, x_n+y_n, λ)(F⁰ϕ)(ξ⁰, y_n)dy_n. Lemma 3.2. Let k∈N0 andp∈(1,∞). Then the operators

T_j⁽¹⁾(λ) : W_p^m+k−mj(Rⁿ+),||| · |||_m+k−mj,p,Rⁿ₊

→ W_p^m+k(Rⁿ+),||| · |||m+k,p,Rⁿ₊

, T_j⁽²⁾(λ) : W_p^m+k−mj−1(Rⁿ+),||| · |||m+k−mj−1,p,Rⁿ+

→ W_p^m+k(Rⁿ+),||| · |||m+k,p,Rⁿ+

, j= 1, . . . ,^m₂, are continuous with respect to the parameter-dependent norms.

Proof. We only consider T_j⁽¹⁾, the proof for T_j⁽²⁾ essentially being the same. Let λ0>0. We make use of the equivalence of norms

|||u|||m+k,p,Rⁿ₊≈

m+k

X

`=0

X

|α|=`

λ^m+k−`m (D⁰)^α0∂_n^αnu _Lp(

Rⁿ+).

Therefore, we have to estimate

|||T_j⁽¹⁾(λ)ϕ|||^p_m+k,p,Rn +

≤C

m+k

X

`=0

X

|α|=`

Z ∞

0

Z ∞

0

(F⁰)⁻¹λ^m+k−`m (ξ⁰)^α0(∂_n^αn+1w_j)(ξ⁰, x_n+y_n, λ) (F⁰ϕ)(ξ⁰, yn)dyn

p

L^p(Rⁿ⁻¹)dxn

≤C

m+k

X

`=0

X

|α|=`

Z ∞

0

Z ∞

0

(F⁰)⁻¹M_j,`,α(ξ⁰, x_n+y_n, λ) (F⁰ϕ)(ξe ⁰, yn)

L^p(Rⁿ⁻¹)

dyn

p

dxn.

Here we have defined

ϕe:= (F⁰)⁻¹(|ξ⁰|+|λ|^1/m)^m+k−mjF⁰ϕ∈L^p(Rⁿ+) and

Mj,`,α(ξ<sup>0</sup>, xn, λ) := (|ξ<sup>0</sup>|+|λ|<sup>1/m</sup>)<sup>−m−k+mj</sup>λ<sup>m+k−`m</sup> (ξ⁰)^α0(∂_n^αn+1wj)(ξ⁰, xn, λ).

We will apply Michlin’s theorem to the functions Mj,`,α. For this, we abbreviate ρ:= ρ(ξ⁰, λ) := |ξ⁰|+|λ|^1/m and use the homogeneities stated in Lemma 3.1. In the integral representation for the basic solutions wj in Lemma 3.1, we make the substitutionτ7→τe=^τ_ρ and use the fact thatγ(^ξ_ρ⁰,_ρ^λm) can be replaced by a contour eγ which is independent ofξ⁰ andλ. Forβ⁰∈Nⁿ⁻¹0 , we obtain

(ξ⁰)^β0D^β_ξ0⁰Mj,`,α(ξ⁰, xn, λ)

=

(ξ⁰)^β0λ^m+k−`m Z

γ(ξ⁰,λ)

τ^αn+1e^{iτ xn}D^β_ξ0⁰

ρ^−m−k+mj(ξ⁰)^α0 (A(ξ⁰, τ)−λ)⁻¹N_j(ξ⁰, τ, λ)

dτ

=

ξ⁰ ρ

β⁰ λ ρ^m

^m+k−`_m Z

γ(ξ⁰,λ) τ ρ

αn+1

e^{iτ xn}Hj,α⁰,β⁰ ξ⁰ ρ,^τ_ρ,_ρ^λm

dτ

=

ξ⁰ ρ

^β0 λ ρ^m

^m+k−`_m Z

eγ

ρτe^αn+1e^{iρeτ xn}Hj,α⁰,β⁰ ξ⁰ ρ,eτ ,_ρ^λm

deτ

≤Cρexp(−Cρxn)≤ C x_n. Here we have set

Hj,α⁰,β⁰(ξ⁰, τ, λ) :=D^β_ξ0⁰

ρ^−m−k+mj(ξ⁰)^α0(A(ξ⁰, τ)−λ)⁻¹Nj(ξ⁰, τ, λ) and used the fact that Hj,α⁰,β⁰ is quasi-homogeneous in its arguments of degree

|α⁰| −m−k−1− |β⁰|. The two inequalities follow by a compactness argument and by the elementary inequalityte^−t≤1 fort≥0.

Now an application of Michlin’s theorem inRⁿ⁻¹ gives

|||T_j⁽¹⁾(λ)ϕ|||m+k,p,Rⁿ₊

≤C

m+k

X

`=0

X

|α|=`

Z ∞

0

Z ∞

0

kϕ(·, ye n)k_Lp(Rⁿ⁻¹)

xn+yn

dy_np

dx_n1/p

≤CZ ∞ 0

kϕ(·, ye _n)k^p_Lp(

Rⁿ⁻¹)dy_n1/p

=Ckϕke _Lp(Rⁿ+).

Here we have used the continuity of the Hilbert transform inL^p(R+) for the second inequality. Finally, we have

kϕke L^p(Rⁿ₊)=

(F⁰)⁻¹(|ξ⁰|+|λ|^1/m)^m+k−mjF⁰ϕ _Lp(

Rⁿ+)

≤C

kϕk_m+k−mj,p,Rⁿ₊+|λ|

m+k−mj

m kϕk_Lp(Rⁿ+)

≤C|||ϕ|||_m+k−mj,p,Rⁿ₊,

which shows the continuity ofT_j⁽¹⁾(λ).

The last lemma is the main step in the proof of uniform a priori estimates with respect to parameter-dependent norms. We obtain the following result, cf.

Agranovich [3], Theorem 3.2.1 for the casep= 2.

Theorem 3.3. Let s ∈ [0,∞) and p ∈ (1,∞) with m+s−m_j − ¹_p 6∈ N0 for all j = 1, . . . ,^m₂. Then for every λ ∈ Σθ \ {0}, all f ∈ W_p^s(Rⁿ+) and all g ∈ Qm/2

j=1W^m+s−mj−

1

p p(Rⁿ) there exists a unique solution u ∈ W_p^m+s(Rⁿ+) of (2.1).

Moreover, the operator

(3.2) (A(D)−λ, γ0B(D)) :W_p^m+s(Rⁿ+)→W_p^s(Rⁿ+)×

m/2

Y

j=1

W^m+s−mj−

1 p

p (Rⁿ⁻¹)

is an isomorphism of Banach spaces with respect to the parameter-dependent norms.

In particular, for everyλ0>0 there exists a constant C=C(λ0) such that (3.3) |||u|||m+s,p,Rⁿ₊≤C

|||f|||s,p,Rⁿ₊+

m/2

X

j=1

|||gj|||_m+s−mj−1 p,p,Rⁿ⁻¹

holds for allλ∈Σ_θ with |λ| ≥λ₀.

Proof. (i) First we assume s ∈ N0. The case s = 0 is well-known, see, e.g., Agranovich-Denk-Faierman [2], Theorem 2.1. In particular, we already know unique solvability of (2.1) with the solution ubeing at least in W_p^m(Rⁿ+). Moreover, the continuity of the operator in (3.2) with respect to the parameter-dependent norms is an immediate consequence of the continuity of the derivatives and of the trace operator. Therefore, we only have to show the a priori estimate (3.3) which also gives the smoothness of the solution.

Letr+ ∈L(W_p^s(Rⁿ), W_p^s(Rⁿ+)), r+f :=f|_Rⁿ

+, denote the operator of restriction ontoRⁿ+. Using the fact that there exists a coretractione₊ ofr₊ (see Amann [6],

Section 4.4) with e+ ∈L(W_p^{`</sup>(R<sup>n</sup>+), W<sub>p</sub><sup>`}(Rⁿ)) for all `∈[0, s], we see that bothr+

ande+ are continuous with respect to the parameter-dependent norms, too.

We writeu=r+u1+u2, whereu1is the unique solution of (A(D)−λ)u1=e+f in Rⁿ. By the explicit representation of u₁ (see the proof of Lemma 2.1), we see that

|||r+u1|||m+s,p,Rⁿ₊≤ |||u1|||m+s,p,Rⁿ≤C|||e+f|||s,p,Rⁿ≤C|||f|||s,p,Rⁿ₊. Foru₂ we obtain the boundary value problem

(A(D)−λ)u₂= 0 inRⁿ+, γ₀B(D)u₂=ge onRⁿ⁻¹

witheg:=g−γ0B(D)r+u1. By the continuity ofr+,B(D) andγ0(with respect to the parameter-dependent norms), we see that

(3.4) |||eg_j|||_m+s−m

j−¹_p,p,Rⁿ⁻¹≤C

|||g_j|||_m+s−m

j−_p¹,p,Rⁿ⁻¹+|||f|||_s,p,Rⁿ

+

. Due to Lemma 3.2, we have

u₂=

m/2

X

j=1

T_j⁽¹⁾(λ)E_λeg_j+T_j⁽²⁾(λ)∂_nE_λeg_j .

Now the continuity ofEλ, ∂n, andT_j⁽¹⁾, T_j⁽²⁾ yields

|||u2|||m+s,p,Rⁿ+≤C

m/2

X

j=1

|||eg_j|||_m+s−mj−¹

p,p,Rⁿ⁻¹

which in connection with (3.4) gives the a priori estimate (3.3) and the proof for s∈N⁰.

(ii) For s ∈ (0,∞)\N with m+s−mj −_p¹ 6∈ N0, the result follows by real interpolation. Here we use the fact that fork∈N0andθ∈(0,1) we have

W_p^k(Rⁿ+), W_p^k+1(Rⁿ+)

θ,p=W_p^k+θ(Rⁿ+)

uniformly inλwith respect to the parameter-dependent norms, see Grubb-Kokholm

[15], Theorem 1.1.

A combination of Theorem 2.4 and Theorem 3.3 immediately yields the following result.

Corollary 3.4. Let (A(D), B(D)) be parabolic, and let s ∈[0,∞)\ {km+mj−

1

p : k ∈ N, j = 1, . . . ,^m₂}. Then for all λ ∈ Σ^π

2 \ {0}, |λ| ≥ λ0 > 0, and all f ∈W_p;(A,B)^s (Rⁿ+)andg∈Qm/2

j=1W^m+s−mj−

1 p

p (Rⁿ⁻¹)there exists a unique solution u∈W_p^m+s(Rⁿ+)of (2.1), and

(3.5) kuk_Wm+s

p (Rⁿ+)+|λ| kuk_Ws

p(Rⁿ+)≤C kfk_Ws

p(Rⁿ+)+

m/2

X

j=1

|||gj|||_m+s−mj−1 p,p,Rⁿ⁻¹

.

In particular, this holds for all f ∈W_p^s(Rⁿ+)if s < m_j+¹_p for allj = 1, . . . ,^m₂.

Proof. We writeu=u1+u2, where u1 solves (A(D)−λ)u1 =f, γ0B(D)u1 = 0 and u2 solves (A(D)−λ)u2 = 0, γ0B(D)u2 = g, and apply Theorem 2.4 and Theorem 3.3, respectively. For the application of Theorem 3.3, we note that by the interpolation inequality, the left-hand side of (3.5) is not larger than a constant times|||u|||m+s,p,Rⁿ+. The last statement follows directly from the fact that for these s, the spacesW_p^s(Rⁿ+) andW_p;(A,B)^s (Rⁿ+) coincide.

Remark 3.5. On the right-hand side of (3.5) large powers ofλmay appear, although we only have |λ| on the left-hand side. The following elementary example in R¹+

shows that even in the one-dimensional case this cannot be avoided: Consider the boundary value problemλu(xn)−u⁰⁰(xn) = 0 (xn >0), u(0) =g ∈ C. Then for λ >0 the stable solution isu(xn) = exp(−√

λxn)g, and a direct calculation shows that fors∈N0 we have

kuk_W2+s

p (R+)≥C|λ|^2+s−1/p2 |g|.

The power on the right-hand side is consistent with the exponent (m+s−m_j−¹_p)/m appearing on the right-hand side of (3.5).

In some sense the parameter-dependent norms in Theorem 3.3 are natural for parameter-elliptic problems. However, they do not yield resolvent estimates as the parameter λappears on both sides. Moreover, on the left-hand side we have

|λ|^(m+s)/minstead of|λ|. We will now derive a resolvent estimate for theW_p^s(Rⁿ+)- realization. As it was shown in [8], for a decay like|λ|⁻¹ additional conditions on f are necessary. The following result gives a general resolvent estimate.

Theorem 3.6. a) In the situation of Theorem 3.3, assume that

(3.6) σ:= max

j=1,...,m/2

(s−m_j−¹_p)>0.

Then for allλ0>0there exists a constant C=C(λ0)>0 such that (3.7)

kuk_m+s,p,Rⁿ

++|λ| kuk_s,p,Rⁿ

+≤C

|λ|^σ/mkfk_s,p,Rⁿ

++

m/2

X

j=1

|||g_j|||_m+s−m

j−¹_p,p,Rⁿ⁻¹

.

b) Let (A(D), B(D)) be parabolic, and define σ as in (3.6). For s ∈ [0,∞)\ {mk+m_j− ¹_p : k ∈ N0, j = 1, . . . , m/2} define the unbounded operator Ae^(s)_B in W_p^s(Rⁿ+)by

D(Ae^(s)_B ) :={u∈W_p^m+s(Rⁿ+) :γ₀B(D)u= 0}, Ae^(s)_B u:=A(D)u(u∈D(Ae^(s)_B )).

Thenρ(Ae^(s)_B )⊃Σ^π

2 \ {0}, and for all λ₀>0 there existsC=C(λ₀)>0 such that (Ae^(s)_B −λ)⁻¹

_L(Ws

p(Rⁿ+))≤C|λ|−1+max{0,σ/m} (λ∈Σ^π

2,|λ| ≥λ0).

Proof. a) We write the solutionu in the formu =u1+u2+u3. Here u1 solves (A(D)−λ)u1= 0, γ0B(D)u1=g, while u2:=r+ue2 withue2 being the solution of (A(D)−λ)ue₂=e₊f in Rⁿ. Due to Corollary 3.3, we obtain

ku₁k_m+s,p,Rⁿ

++|λ| ku₁k_s,p,Rⁿ

+≤C

m/2

X

j=1

|||g_j|||_m+s−m

j−¹_p,p,Rⁿ⁻¹.

Moreover, by Lemma 2.1, we have

(3.8) ku2km+s,p,Rⁿ₊+|λ| ku2ks,p,Rⁿ₊≤Ckfks,p,Rⁿ₊. The functionu3is the solution of the boundary value problem

(A(D)−λ)u₃= 0 inRⁿ+,

γ₀B(D)u₃=−γ₀B(D)u₂ onRⁿ⁻¹. We apply Corollary 3.3 again and get

ku3km+s,p,Rⁿ++|λ| ku3ks,p,Rⁿ+≤C

m/2

X

j=1

|||γ0B(D)u2|||_m+s−mj−¹

p,p,Rⁿ⁻¹

=C

m/2

X

j=1

kγ0B(D)u2k_m+s−mj−1 p+|λ|

m+s−mj−1/p

m kγ0B(D)u2k_Lp(Rⁿ⁻¹)

. We estimate the norms on the right-hand side for eachj. First, we have

kγ0B(D)u2k_m+s−mj−¹

p,p,Rⁿ⁻¹ ≤CkB(D)u2km+s−mj,p,Rⁿ+

≤Cku2km+s,p,Rⁿ+

for allj= 1, . . . ,^m₂.

Ifs > mj+¹_p, we can estimate

|λ|

m+s−mj−1/p

m kγ0B(D)u2k_Lp(Rⁿ⁻¹)

≤ |λ|

m+s−mj−1/p

m kγ0B(D)u2k_s−mj−1 p,p,Rⁿ⁻¹

≤C|λ|

m+s−mj−1/p

m kB(D)u2ks−m_j,p,Rⁿ+

≤C|λ|

m+s−mj−1/p

m ku2ks,p,Rⁿ+

≤C|λ|^σ/m|λ| ku2ks,p,Rⁿ+

≤C|λ|^σ/mkfk_s,p,Rⁿ

+. Here we applied (3.8) in the last step.

Ifs < mj+¹_p, we similarly write

|λ|

m+s−mj−1/p

m kγ0B(D)u2k_Lp(Rⁿ⁻¹)

≤ |λ|^{m+s−mjm−1/p}kγ₀B(D)u₂k_σ,p,Rn−1

≤C|λ|^σ/m|λ|

m+s−mj−σ−1/p

m ku2k_σ+mj+¹

p,p,Rⁿ₊

≤C|λ|^σ/m

ku₂k_m+s,p,Rⁿ

++|λ| kuk_s,p,Rⁿ

+

≤C|λ|^σ/mkfks,p,Rⁿ₊,

where we used the interpolation inequality (see Grisvard [14], Theorem 1.4.3.3) for the third inequality. This finishes the proof of the a priori estimate (3.7) and of part a).

b) In the caseσ <0 the a priori estimate (and the fact that the operatorAe^(s)_B is sectorial) follows from the last statement in Corollary 3.4. Forσ > 0, we apply part a) withg= 0. Note that the caseσ= 0 is excluded.

Remark 3.7. For the Dirichlet-Laplacian ∆D, we havem= 2,m1= 0, and therefore σ=s−¹_p. For the resolvent inW_p^s(Rⁿ+), we obtain from Theorem 3.6

k(λ−∆_D)⁻¹k_L(Ws

p(Rⁿ₊)) ≤C|λ|^{−1+s2−2p1} .

Fors= 1, we have a decay like|λ|−1/2−1/(2p). It was shown in [18] that this is the exact decay rate.

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Department of Mathematics and Statistics, University of Konstanz, D-78457 Kon- stanz, Germany

E-mail address:robert.denk@uni-konstanz.de

Department of Mathematics and Statistics, University of Konstanz, D-78457 Kon- stanz, Germany

E-mail address:tim.seger@uni-konstanz.de