Resolvent Estimates for Elliptic Systems in Function Spaces of Higher Regularity

Resolvent Estimates for Elliptic Systems in Function Spaces of Higher Regularity

Robert Denk Michael Dreher

Konstanzer Schriften in Mathematik Nr. 265, Mai 2010

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-119237

URL: http://kops.ub.uni-konstanz.de/volltexte/2010/11923/

Function Spaces of Higher Regularity

Robert Denk and Michael Dreher

Abstract. We consider parameter-elliptic boundary value problems and uniforma prioriestimates in L^p- Sobolev spaces of Bessel potential and Besov type. The problems considered are systems of uniform order and mixed-order systems (Douglis-Nirenberg systems). It is shown that compatibility conditions on the data are necessary for such estimates to hold. In particular, we consider the realization of the boundary value problem as an unbounded operator with the ground space being a closed subspace of a Sobolev space and give necessary and sufficient conditions for the realization to generate an analytic semigroup.

MSC2010 classification. 35G45, 47D06.

Keywords. Parameter-ellipticity, Douglis-Nirenberg systems, analytic semigroups.

1 Introduction

The aim of this paper is to establish resolvent estimates for parameter-elliptic boundary value problems in L^p-Sobolev spaces of higher order. A priori estimates involving parameter-dependent norms for parameter- elliptic or parabolic systems are known since long; classical works are, e.g., Agmon [1], Agranovich-Vishik [2] for scalar equations, and Geymonat-Grisvard [9], Roitberg-Sheftel [13] for systems. Further results on theL^p-theory for mixed-order systems were obtained, e.g., by Faierman [7]. For pseudodifferential boundary value problems, we refer to the parameter-dependent calculus developed by Grubb [10].

Parameter-dependenta prioriestimates are motivated by their connection to operator theory: In the ground spaceL^p, the estimate immediately implies a uniform resolvent estimate for theL^p-realization of the boundary value problem. In particular, if the sector of parameter-ellipticity is large enough, i.e., if the problem is parabolic in the sense of Petrovskii, then the operator generates an analytic semigroup in L^p. Moreover, spectral properties and completeness of eigenfunctions can be obtained, see Denk-Faierman-M¨oller [4] and Faierman-M¨oller [8]. If the equation is given in the whole space, we obtain the generation of an analytic semigroup in the whole scale of Sobolev spaces. In fact, the operator even admits a bounded H^∞-calculus which was shown for general mixed-order systems of pseudodifferential operators in Denk-Saal-Seiler [5].

Consider the boundary value problem ((A−λ)u=f, in Ω,

Bju=gj, on∂Ω, j= 1, . . . , M, (1.1)

in a bounded smooth domain Ω ⊂ R^d. Here A is a system of differential operators, and B_j is a vector of differential operators, and the number M of boundary conditions is determined by the order and the dimension of the systemA (see below for details). In the present paper we study the question under which additional (compatibility) assumptions on the right-hand side this boundary value problem has a unique solution satisfying uniform (inλ)a priori estimates. In particular, fors≥0 and 1< p <∞let us consider a closed linear subspaceY of the Sobolev spaceW_p^s(Ω) as a ground space and define the realization of (1.1) as an unbounded operator A in Y with domain D(A) := {v ∈ Y:Av ∈ Y, Bjv = 0, j = 1, . . . , M}. In the particular cases= 0, the parameter-elliptic theory mentioned above yields the generation of an analytic semigroup in L^p(Ω), provided the sector of parameter-ellipticity is large enough. For s >0, however, the

1

2

R. Denk, M. Dreher

situation is more complicated. As an example, one may consider the Dirichlet-Laplacian4_Din Y =W_p¹(Ω) with domainD(4_D) ={u∈W_p³(Ω) : u|∂Ω = 0}. This operator does not generate an analytic semigroup in Y; in fact, its resolvent decays as|λ|^{−1/2−1/2p}as|λ| → ∞(see Nesensohn [12]). Roughly speaking, additional compatibility conditions have to be incorporated into the basic spaceY in order to obtain a decay of|λ|⁻¹. Therefore, the question is to find equivalent conditions onY for whichAgenerates an analytic semigroup on Y. This question is fully answered by Theorem 3.5 below. We also study compatibility conditions for which the problem with inhomogeneous boundary data (1.1) is uniquely solvable with appropriatea prioriestimate for the solution. As a ground space, we consider subspaces of integer or non-integer Sobolev spaces both of Besov type and of Bessel potential type.

The question of generation of an analytic semigroup for parabolic equations was also studied by Guidetti [11] where higher order scalar equations are considered. Writing such an equation as a first order system, in [11] necessary and sufficient conditions for the unique solvability of the non-stationary problem are given.

Roughly speaking, in [11] the author observes that the order of the boundary operators has to be sufficiently large. This coincides with our conditions as in this case the trace conditions given in Theorem 3.5 are empty.

Whereas the equations in [11] have more general coefficients, the mixed-order system is of special structure (arising from a higher order equation), and the basic space is fixed. Our paper considers general mixed-order systems and the whole scale of Sobolev spaces.

2 Notation and Auxiliary Results

Let Ω be a bounded domain in R^d, d ≥2, with boundary Γ = ∂Ω∈ C^∞. The Besov spaces are denoted by B_p,q^s (Ω), for s∈ Rand 1 ≤p, q ≤ ∞, and the Bessel potential spaces are called H_p^s(Ω), for s ∈R and 1< p <∞. Then the Sobolev(–Slobodecky) spaces are

W_p^s=

(H_p^s :s∈N0, B_pp^s :s6∈N0, withs∈[0,∞) and 1< p <∞.

In this paper, K^s_p(Ω) shall mean everywhere either the Bessel potential space H_p^s(Ω), or one of the Besov spacesB^s_p,q(Ω), 1< q <∞. Heres∈Rand 1< p <∞. Fors >1/p, we define the space K^s−1/p_p;Γ of traces of functions fromK_p^s(Ω) at the boundary Γ =∂Ω:

K^s−1/p_p,Γ :=

(Bp,q^s−1/p(∂Ω) :K^s_p(Ω) =B_p,q^s (Ω), Bp,p^s−1/p(∂Ω) :K_p^s(Ω) =H_p^s(Ω).

To simplify later formulae, we set K⁰_p,Γ :=L^p(Ω), although this is not the space of traces of functions from Hp^1/p(Ω) orBp,q^1/p(Ω), except whenq= 1. The trace operator on∂Ω, mapping functions fromC^∞(Ω) to their boundary values, is called γ₀.

We will write [·,·]_θfor the complex interpolation method, and (·,·)_θ,qfor the real interpolation method, where 0 ≤θ ≤1 and 1 ≤q≤ ∞. Then ∂_x^α maps continuously fromK^s_p(Ω) intoK^s−|α|p (Ω), for all s∈ Rand all p∈(1,∞), and{K^s_p(Ω)}s∈R forms an interpolation scale with respect to the complex interpolation method:

K^s_p⁰(Ω),K^s_p¹(Ω)

θ=K^s_p^θ(Ω), sθ= (1−θ)s0+θs1, 0≤θ≤1.

We will also make free use of the following: if a Banach spaceX_θis an interpolation space of the pair (X₀, X₁) of orderθ, then

%^1−θkfk_θ≤C(%kfk₀)^1−θkfk^θ₁≤C(kfk₁+%kfk₀), %∈R+, f ∈X0∩X1. For detailed representations of the theory of function spaces, we refer the reader to [3] and [14].

Lemma 2.1. Suppose0≤σ0<1/p < σ1. Then we have the estimates

%^1−θkγ0uk_Lp(∂Ω)≤C kuk_K^σ1

p (Ω)+%kuk_K^σ0 p (Ω)

, θ= 1/p−σ₀ σ1−σ0

, (2.1)

%^1−θkγ0uk_Lp(∂Ω)≤C kuk_B^σ1

p,q(Ω)+%kuk_Lp(Ω)

, θ= 1

pσ1

, (2.2)

for allu∈ K^σ_p¹(Ω) and all %∈[1,∞).

Proof. In [14, Theorem 4.7.1], we findγ₀∈ L(B_p,1^1/p(Ω), L^p(∂Ω)), hence we conclude that kγ0uk_Lp(∂Ω) ≤Ckuk_B1/p

p,1(Ω). Now we have, for the aboveσ0,σ1,

B_p,q^σ0(Ω), B_p,q^σ1(Ω)

θ,1= H_p^σ0(Ω), H_p^σ1(Ω)

θ,1=B_p,1^1/p(Ω), θ= 1/p−σ0

σ₁−σ₀ ,

which brings us (2.1). And (2.2) follows from B_p,min(2,p)⁰ (Ω),→L^p(Ω),→B_p,max(2,p)⁰ (Ω)

and the interpolation identity (B_p,r⁰ (Ω), B_p,q^σ1(Ω))_θ,1=B_p,1^1/p(Ω) forr∈ {2, p}.

3 Main Results

3.1 Systems of Uniform Order

First, let A = (ajk(x, Dx))j,k=1,...,N be a matrix differential operator with ordajk ≤ m for all j, k. The coefficients ofajkare smooth on a neighborhood of Ω. IfAis a parameter-elliptic matrix differential operator in Ω, thenmN ∈2N, see [2]. Then let us be given differential operatorsBj=Bj(x, Dx) forj= 1, . . . , mN/2, with ordBj =rj ≤m−1. For λfrom a sectorL ⊂C with vertex at the origin, we consider the boundary value problem

((A−λ)u=f, in Ω,

γ₀B_ju=g_j, on∂Ω, j= 1, . . . , mN/2, (3.1)

and its variant with homogeneous boundary data:

((A−λ)u=f, in Ω,

γ0Bju= 0, on∂Ω, j= 1, . . . , mN/2. (3.2)

We suppose that the operators (A, B₁, . . . , B_mN/2) constitute a parameter-elliptic boundary value problem on Ω in an open sectorL ofC.

Proposition 3.1. Let u be any function from K^s+m_p (Ω) with s ∈[0,∞) but s 6∈N+ 1/p, and take λ∈C arbitrarily. Definef andgj by the right-hand sides of (3.1).

Then we have the inequality kfk_Ks

p(Ω)+

mN/2

X

j=1

kgjk

K^s+m−_p,Γ ^rj−1/p+|λ|^{1+m1 min(s−rj−1/p,0)}kgjk

K^max(s−_p,Γ ^rj−1/p,0)

≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks p(Ω)

,

with some constantC independent ofuandλ.

4

R. Denk, M. Dreher

Proof. It suffices to establish the inequalities kfk_Ks

p(Ω)≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks p(Ω)

, kgjk

K^s+m−_p,Γ ^rj−1/p ≤Ckuk_Ks+m p (Ω),

|λ| kg_jk

K^s−_p,Γ^rj−1/p ≤C|λ| kuk_Ks

p(Ω), (s−r_j−1/p >0), (3.3)

|λ|^{1+m1(s−rj−1/p)}kg_jk_Lp(∂Ω)≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks p(Ω)

, (s−r_j−1/p <0), (3.4) of which we will only discuss the last two. For 0< s−rj−1/p6∈N, we have

kgjk

K^s−_p,Γ^rj−1/p=kγ0Bjuk

K^s−_p,Γ^rj−1/p≤CkBjuk

K^s−p ^rj(Ω)≤Ckuk_Ks p(Ω), as claimed in (3.3). Concerning (3.4) in the case ofs≤r_j, we write

1 + 1 m

s−rj−1 p

=s+m−rj

m ·

1− 1

p(s+m−r_j)

, %:=|λ|

s+m−rj

m ,

and make use of Lemma 2.1:

|λ|^{1+m1(s−rj−1/p)}kgjk_Lp(∂Ω)=%^{1−(p(s+m−rj))−1}kγ0B_juk_Lp(∂Ω)

≤C kBjuk

K^s+m−_p ^rj(Ω)+%kBjuk_K0 p(Ω)

≤C

kuk_Ks+m

p (Ω)+|λ|

s+m−rj m kuk_Krj

p(Ω)

.

Exploiting nows≤rj, we can interpolate:

|λ|

s+m−rj m kuk_Krj

p (Ω)≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks p(Ω)

,

which is what we wanted to show. And for (3.4) in the case ofrj < s < rj+ 1/p, we takeσ1 =s+m−rj, σ0=s−rj <1/p,%=|λ|, and thenθ from (2.1) becomesθ=−(s−rj−1/p)/m, which brings us to

|λ|^1−θkγ0Bjuk_Lp(∂Ω)≤C kBjuk

K^s+m−_p ^rj(Ω)+|λ| kBjuk

K^s−_p ^rj(Ω)

.

Then (3.4) quickly follows.

Consequently, the norms of the given functions f andgj appearing in the next result are the natural ones, and also the exponents of|λ|are natural.

Theorem 3.2. Let (3.1)be parameter-elliptic inL, and suppose thatf and theg_j are such that all solutions uto (3.1)enjoy the following estimate for allλ∈ L with large|λ|:

kuk_Ks+m

p (Ω)+|λ| kuk_Ks

p(Ω)≤Ckfk_Ks p(Ω)

+C

mN/2

X

j=1

kgjk

K_p,Γ^{s+m−rj−1/p}+|λ|^{1+m1 min(s−rj−1/p,0)}kgjk

K^max(s−_p,Γ ^rj−1/p,0)

,

with some s∈[0,∞) and1< p <∞.

Theng_j ≡0for all j withr_j < s−1/p.

Proof. From u ∈ K^s+m_p (Ω) we get BjAu ∈ K^s−rp ^j(Ω), which admits traces on ∂Ω. We then have from Lemma 2.1

|λ| · |λ|

s−rj

m (1−_p(s−¹_rj))

kgjk_Lp(∂Ω)=|λ|

s−rj

m (1−_p(s−¹_rj))

kγ0Bj(Au−f)k_Lp(∂Ω)

≤C

kBj(Au−f)k

K^s−_p ^rj(Ω)+|λ|

s−rj

m kBj(Au−f)k_K0 p(Ω)

≤C

kAu−fk_Ks

p(Ω)+|λ|^s−mrj kAu−fk_Krj p(Ω)

≤C

kuk_K^s+m

p (Ω)+|λ|

s−rj m kuk

K^m+_p ^rj(Ω)+kfk_Ks

p(Ω)+|λ|

s−rj m kfk_Krj

p (Ω)

≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks

p(Ω)+kfk_Ks

p(Ω)+|λ|^s−mrj kfk_Krj p (Ω)

,

the last step by interpolation. Then we can bring the assumed inequality into play:

|λ| · |λ|

s−rj

m (1−_p(s−¹_rj))

kg_jk_Lp(∂Ω)

≤C kfk_Ks

p(Ω)+|λ|^s−mrj kfk_Krj p(Ω)

+C

mN/2

X

l=1

kg_lk

K^s+m−_p,Γ ^rl−1/p+|λ|^{1+m1 min(s−rl−1/p,0)}kg_lk

K^max(s−_p,Γ ^rl−1/p,0)

.

Ifgj6≡0 then the exponent of|λ|on the left-hand side is greater than each exponent of|λ|on the right-hand side, giving a contradiction for large |λ|.

Theorem 3.3. Let (3.2)be parameter-elliptic in L. Fix p∈(1,∞) ands∈[0, m]. Then the following two statements are equivalent forf ∈ K^s_p(Ω).

1. there are positive constantsλ0 andC0 such that all solutionsuto(3.2)withλ∈ L,|λ| ≥λ0 enjoy the following estimate:

kuk_Ks+m

p (Ω)+|λ| kuk_Ks

p(Ω)≤C₀kfk_Ks p(Ω), 2. γ0Bjf ≡0 for allj withs−rj>1/p.

Proof. Suppose statement no.1 and choosej withs−rj>1/p. ThenBjf ∈ Kp^s−rj(Ω), which admits traces on∂Ω, and we can argue as in the proof of Theorem 3.2:

|λ|

s−rj m (1−_p(s−¹

rj))

kγ0Bjfk_Lp(∂Ω)=|λ|

s−rj m (1−_p(s−¹

rj))

kγ0Bj(Au−λu)k_Lp(∂Ω)

=|λ|

s−rj m (1−_p(s−¹

rj))

kγ0BjAuk_Lp(∂Ω)

≤C

kBjAuk

K^s−p ^rj(Ω)+|λ|

s−rj

m kBjAuk_K0 p(Ω)

≤C

kuk_Ks+m

p (Ω)+|λ|

s−rj m kuk

K^m+_p ^rj(Ω)

≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks p(Ω)

≤Ckfk_Ks p(Ω). Now sendλ→ ∞in L.

Conversely, suppose statement no.2. DefineX =L^p(Ω) and an operatorA:D(A)→X by D(A) :={u∈W_p^m(Ω) : γ₀B_ju= 0, j= 1, . . . , mN/2},

Au:=Au, and set

Ys:=

([L^p(Ω), D(A)]_s/m :K^•_p(Ω) =H_p^•(Ω), (L^p(Ω), D(A))_s/m,q :K^•_p(Ω) =B_p,q^• (Ω)

=

u∈ K^s_p(Ω) :γ₀B_ju≡0, ∀j withs−r_j >1/p

ThenD(A),→Ys,→X with dense embeddings. From [9] we quote the estimate kuk_Wm

p (Ω)+|λ| kuk_Lp(Ω)≤Ckfk_Lp(Ω), f ∈X,

foru= (A −λ)⁻¹f andλ∈ L, |λ| ≥λ₀. And forf ∈D(A), we have u= (A −λ)⁻¹f ∈D(A²), hence kuk_W2m

p (Ω)+|λ| kuk_Wm

p (Ω)≤Ckfk_Wm

p (Ω), f ∈D(A).

Interpolating between these two estimates then implies kuk_Ks+m

p (Ω)+|λ| kuk_Ks

p(Ω)≤Ckfk_Ks

p(Ω), f ∈Ys, fors∈[0, m].

6

R. Denk, M. Dreher

Theorem 3.4. Let (3.2)be parameter-elliptic in the sector L, and fix p∈(1,∞)and smax∈[0,∞). Then the following two statements are equivalent, forf ∈ K^s_p^max(Ω).

1. there are positive constants λ0 and C0 such that all solutions uto (3.2)with λ∈ L,|λ| ≥λ0 satisfy the collection of estimates

kuk_Ks+m

p (Ω)+|λ| kuk_Ks

p(Ω)≤C0kfk_Ks p(Ω), for alls∈[0, smax].

2. for each pair(j, k)∈ {1,2, . . . , m} ×N0 with s_max−r_j> mk+ 1/p, we haveγ₀B_jA^kf ≡0.

Proof. A proof for the casesmax ∈[0, m] was given in Theorem 3.3, whose notations we adopt here. And the proof of the first statement from the second is very similar to the proof of Theorem 3.3, so we skip it.

Therefore we may assumes_max≥m. We suppose now the statement no.1, and proceed by induction ons_max of step sizem.

Choosing s=m, we find γ0Bjf ≡0 for all j, hence f ∈D(A), and then also Au∈D(A). By parameter- ellipticity inL, there is a numberλ_∗∈ Lwith|λ_∗| ≥λ0+ 1 such thatA −λ_∗:D(A)∩W_p^σ+m(Ω)→W_p^σ(Ω) is a continuous isomorphism, for allσ∈N0. Fix thisλ_∗. Put ˜u:= (A −λ_∗)u, ˜f := (A −λ_∗)f, and note that

((A−λ)˜u= ˜f , in Ω, γ₀B_ju˜= 0, on∂Ω,

with ˜f ∈ K^s_p^max−m(Ω). For 0≤s≤smax−mandλ∈ L,|λ| ≥λ0, we then have k˜uk_Ks+m

p (Ω)+|λ| k˜uk_Ks p(Ω)

≤C

kukK^m+(s+m)p (Ω)+|λ| kuk_Ks+m p (Ω)

≤Ckfk_Ks+m

p (Ω)=C

(A −λ)⁻¹f˜ _Ks+m

p (Ω)

≤C˜0

f˜

K_p^s(Ω).

By induction, we know thatγ0BjA^kf˜≡0 for all pairs (j, k)∈ {1, . . . , m}×N0with (smax−m)−rj > mk+1/p.

The definition of ˜f then brings us to γ₀B_jA^kf ≡0 for all (j, k) withs_max−r_j> mk+ 1/p.

Theorem 3.5. Let (3.2) be parameter-elliptic in a sector L that is greater than the right half-plane. For s≥0 and1< p <∞, let Y be a closed linear subspace of K^s_p(Ω), equipped with the norm of K^s_p(Ω). Define an operatorA in the ground spaceY byAu:=Aufor

u∈D(A) :={v∈Y:Av∈Y, γ0Bjv≡0 ∀j}. Then the following are equivalent:

1. The operator Agenerates an analytic semigroup onY,

2. The embeddingD(A),→Y is dense,(A−λ)⁻¹∈ L(Y)for allλ∈ Lof large modulus, andγ0BjA^kf ≡0 for allf ∈Y and all pairs(j, k)with s−rj> mk+ 1/p.

Proof. The domain of a generator of a C0 semigroup is always dense in the ground space. Under the assumptions onL,Y andD(A), the analyticity of the semigroup is equivalent to the resolvent estimate

(A −λ)⁻¹

_L(Y)≤ C

|λ|

for allλ∈ Lof large modulus. Now apply Theorem 3.4.

Theorem 3.6. Let (3.2)be parameter-elliptic in L. Fix p∈(1,∞), s∈[0,∞), andγ∈(−∞,1]. Choose a function f ∈ K^s_p(Ω).

Assume that there are positive constants λ0 andC0 such that all solutions uto (3.2)with λ∈ L, |λ| ≥ λ0

enjoy the following estimate:

|λ|^γkuk_Ks

p(Ω)≤C0kfk_Ks p(Ω).

Thenγ0Bjf ≡0 for allj with γ > m+r_j+ 1/p−s

m . (3.5)

Remark 3.7. In case of the Dirichlet Laplacian 4_D, considered in the space W_p¹(Ω), the condition (3.5) turns into

γ > p+ 1 2p ,

which matches the result of [12], where the resolvent estimate from below, (4_D−λ)⁻¹

_L(W1

p(Rⁿ+),W_p¹(Rⁿ+))≥ C

|λ|^(p+1)/(2p), C >0, was proved.

Proof of Theorem 3.6. Choose such aj. Byγ≤1 and the condition (3.5), we gets−r_j>1/p, and therefore γ0Bjf ∈ K^s−r_p,Γ^j−1/p exists. Now we can take a frozenλ_∗ as in the proof of Theorem 3.4, and write

u= (A −λ_∗)⁻¹(f+ (λ−λ_∗)u), and consequentlykuk_Ks+m

p (Ω)≤C(kfk_Ks

p(Ω)+|λ| kuk_Ks

p(Ω))≤C|λ|^1−γkfk_Ks

p(Ω), due to|λ| ≥1. Now we can compute:

|λ|

s−rj m (1−_p(s−¹

rj))

kγ0Bjfk_Lp(∂Ω)=|λ|

s−rj m (1−_p(s−¹

rj))

kγ0BjAuk_Lp(∂Ω)

≤C

kBjAuk

K^s−p ^rj(Ω)+|λ|

s−rj

m kBjAuk_K0 p(Ω)

≤C

kuk_Ks+m

p (Ω)+|λ| kuk_Ks p(Ω)

≤C|λ|^1−γkfk_Ks p(Ω).

Per (3.5), the left-hand side has a higher power of|λ|than the right-hand side.

3.2 Systems of Mixed Order

In this section,Ashall be a matrix differential operator of mixed order:

A= (a_jk(x, D_x))j,k=1,...,N, orda_jk≤s_j+m_k,

for integerssj andmk. The orders on the diagonal ofAshall be equal, s1+m1=. . .=sN +mN =:m,

and without loss of generality, we can set minjmj= 0.

The principal parta⁰_jkofajkis that part with degree exactly equal tosj+mk(if such a part exists, otherwise a⁰_jk := 0). Then we put A⁰ := (a⁰_jk)j,k=1,...,N, and the operator Ais called parameter-elliptic in the sector

8

R. Denk, M. Dreher

L ⊂Cif det(A⁰(x, ξ)−λ)6= 0 for all (x, ξ, λ)∈Ω×R^d× Lwith|ξ|+|λ|>0. Then (see [2]) mN ∈2N, and we can consider a matrix of boundary differential operators,

B= (bj,k(x, Dx))j,k, j= 1, . . . , mN/2, k= 1, . . . , N, ordbjk≤rj+mk,

with integers r_j ≤ m−1. We define the principal partB⁰ of B in the same way as A⁰ was defined. We say that the Shapiro–Lopatinskii condition is satisfied if at eachx^∗ ∈∂Ω, after introducing a new frame of Cartesian coordinates with center at x^∗ and the xd–axis pointing along the inner normal vector at x^∗, the system of ordinary differential equations











(A⁰(x^∗, ξ⁰, Dx_d)−λ)v(xd) = 0, 0≤xd<∞, B⁰(x^∗, ξ⁰, Dx_d)v(xd) = 0, xd= 0,

x_dlim→∞v(x_d) = 0

possesses only the trivial solution, for all (ξ⁰, λ)∈Rⁿ⁻¹× Lwith|ξ⁰|+|λ|>0.

Then the system (A, B) is called a parameter-elliptic boundary value problem in the sectorL ⊂ Cif A is parameter-elliptic inL, and the Shapiro–Lopatinskii condition holds.

WriteB= (B1, . . . , B_mN/2)^> as a column of rows, and consider the boundary value problem ((A−λ)u=f, in Ω,

γ₀B_ju= 0, on∂Ω, j= 1, . . . , mN/2. (3.6)

In [7], it has been shown that a number λ0 exists such that, for all λ from L with |λ| ≥ λ0, and for all f ∈W_p^m1(Ω)×. . .×W_p^mN(Ω), a unique solutionu∈W_p^m+m1(Ω)×. . .×W_p^m+mN(Ω) to (3.6) exists, and the estimate

N

X

k=1

kukk_Wm+mk

p (Ω)+|λ|^1+mk/mkukk_Lp(Ω)

≤C

N

X

k=1

kfkk_Wmk

p (Ω)+|λ|^mk/mkfkk_Lp(Ω)

holds, with Cdepending only on (A, B).

Having secured the existence of ufor large|λ|, we can now ask under which conditions resolvent estimates forAmight exist. Sufficient conditions were presented in Dreher [6].

Theorem 3.8. If f is such that for all λof large modulus the inequality

N

X

j=1

kujk

W_p^m+mj(Ω)+|λ| kujk_Wmj p (Ω)

≤C

N

X

j=1

kfjk_Wmj p (Ω)

holds with a constantC independent ofλ, thenγ0Bjf ≡0 for allj with rj ≤ −1.

Proof. Fromfk∈W_p^mk(Ω) and ordbjk ≤rj+mk, we deduce thatBjf ∈Wp^−rj(Ω), and this has a trace at the boundary forrj≤ −1. Pick such an indexj.

Now we can estimate as usual:

|λ|^m1(1−1p)kγ₀B_jfk_Lp(∂Ω)=|λ|^m1(1−1p)kγ₀B_jAuk_Lp(∂Ω)

≤C

kBjAuk_W1

p(Ω)+|λ|^m1 kBjAuk_Lp(Ω)

≤C

N

X

k=1

kukk

Wp^m+mk+rj+1(Ω)+|λ|^m1 kukk

Wp^m+mk+rj(Ω)

≤C

N

X

k=1

kukk_Wm+mk

p (Ω)+|λ|^m1 kukk_Wm+mk−1

p (Ω)

≤C

N

X

k=1

kukk_Wm+mk

p (Ω)+|λ| kukk_Wmk p (Ω)

≤C

N

X

k=1

kfkk_Wmk p (Ω).

Sending λto infinity in Ω then impliesγ0Bjf ≡0.

References

[1] S. Agmon. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems.

Comm. Pure Appl. Math., 15:119–147, 1962.

[2] M. S. Agranovich and M. I. Vishik. Elliptic problems with a parameter and parabolic systems of general form. Russ. Math. Surv., 19:53–157, 1964.

[3] J. Bergh and J. L¨ofstr¨om.Interpolation Spaces. An Introduction. Grundlehren math. Wiss. 223. Springer, 1976.

[4] R. Denk, M. Faierman, and M. M¨oller. An elliptic boundary problem for a system involving a discon- tinuous weight. Manuscripta Math., 108(3):289–317, 2002.

[5] R. Denk, J. Saal, and J. Seiler. BoundedH_∞-calculus for pseudo-differential Douglis-Nirenberg systems of mild regularity. Math. Nachr., 282(3):386–407, 2009.

[6] M. Dreher. Resolvent estimates for Douglis–Nirenberg systems. Journ. Evol. Equ., 9:829–844, 2009.

[7] M. Faierman. Eigenvalue asymptotics for a boundary problem involving an elliptic system.Math. Nachr., 279(11):1159–1184, 2006.

[8] M. Faierman and M. M¨oller. Eigenvalue asymptotics for an elliptic boundary problem. Proc. Roy. Soc.

Edinburgh Sect. A, 137(2):281–302, 2007.

[9] G. Geymonat and P. Grisvard. Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici.

Rend. Sem. Mat. Univ. Padova, 38:121–173, 1967.

[10] G. Grubb. Functional calculus of pseudodifferential boundary problems, volume 65 of Progress in Math- ematics. Birkh¨auser Boston Inc., Boston, MA, second edition, 1996.

[11] D. Guidetti. On boundary value problems for parabolic equations of higher order in time.J. Differential Equations, 124(1):1–26, 1996.

[12] M. Nesensohn. Randwertprobleme in Sobolevr¨aumen h¨oherer Ordnung. Diploma Thesis, University of Konstanz, 2009.

[13] J. A. Roitberg and Z. G. Sheftel. Boundary value problems with a parameter for systems elliptic in the sense of Douglis-Nirenberg. Ukrain. Mat. ˇZ., 19(1):115–120, 1967.

[14] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. Deutscher Verlag der Wis- senschaften, 1978.

Robert Denk, Michael Dreher¹, Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

robert.denk@uni-konstanz.de, michael.dreher@uni-konstanz.de

1Corresponding author