The technique based on an inverse function theorem of Nash-Moser type is illustrated by an application in the parabolic case

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NASH-MOSER TECHNIQUES FOR NONLINEAR BOUNDARY-VALUE PROBLEMS

MARKUS POPPENBERG

Abstract. A new linearization method is introduced for smooth short-time solvability of initial boundary value problems for nonlinear evolution equa- tions. The technique based on an inverse function theorem of Nash-Moser type is illustrated by an application in the parabolic case. The equation and the boundary conditions may depend fully nonlinearly on time and space vari- ables. The necessary compatibility conditions are transformed using a Borel’s theorem. A general trace theorem for normal boundary conditions is proved in spaces of smooth functions by applying tame splitting theory in Fr´echet spaces. The linearized parabolic problem is treated using maximal regular- ity in analytic semigroup theory, higher order elliptic a priori estimates and simultaneous continuity in trace theorems in Sobolev spaces.

1. Introduction

The purpose of this paper is to introduce a new linearization method for smooth short-time solvability of initial boundary value problems for nonlinear evolution equations. The technique based on an inverse function theorem of Nash-Moser type is illustrated by an application in the parabolic case. The equation and the boundary conditions may depend fully nonlinearly on time and space variables. The general Theorem 4.3 applies to a nonlinear evolutionary boundary value problem provided that the linearized equation with linearized boundary conditions is well posed; here a loss of derivatives is allowed in the estimates of the linearized problem.

An application in the parabolic case is given in Theorem 8.1.

We mention some points of the proof which might be of independent interest. A Borel’s theorem is applied to transform the compatibility conditions. A trace the- orem is proved for normal boundary operators in spaces of smooth functions using tame splitting theory in Fr´echet spaces. Some results on simultaneous continuity in trace theorems in Sobolev spaces are proved. In the application, higher order Sobolev norm estimates including the dependence of the constants from the coef- ficients are derived for the linearized parabolic problem using analytic semigroup theory involving evolution operators and maximal regularity.

Inverse function theorems of Nash-Moser type [13, 15, 25, 34, 39] have been applied to partial differential equations in several papers, for instance, concerning

2000Mathematics Subject Classification. 35K60, 58C15, 35K30.

Key words and phrases. Nash-Moser, inverse function theorem, boundary-value problem, parabolic, analytic semigroup, evolution system, maximal regularity, trace theorem.

2003 Southwest Texas State University.c Submitted July 14, 2002. Published May 5, 2003.

1

small global solutions in [17], periodic solutions in [41, 18, 11], and local solutions in [13], III. 2.2 or [21, 35, 36, 37, 38]. Different from these articles we consider initial boundary value problems including compatibility conditions in this note. It seems that the technique introduced in this paper is the first general linearization method in the literature based on a Nash Moser type inverse function theorem which applies to smooth initial boundary value problems with loss of derivatives including compatibility conditions.

This paper continues and completes the work in [36] where the whole space case is considered. It turned out that the case of boundary value problems treated in this note is completely different from the whole space case and requires substantially other methods.

In the literature many results are known on linear and on nonlinear parabolic boundary value problems. It is beyond the scope of this paper to give a complete survey. We here only mention some articles which contain also additional references.

For the classical linear theory of parabolic equations and systems we refer to [4, 12, 49, 20, 24]. Early results on short-time solvability of nonlinear second order equations can be found in the references [1] through [8] of the survey paper [28].

Since then nonlinear parabolic problems have been studied in many papers, for instance in [16, 12, 20, 19], or, more recently, in [23, 22, 47]. Semigroup theory has been applied by many authors to the solution of linear and of nonlinear parabolic problems, we refer to [48, 50, 29, 5, 1, 26, 6].

This paper is organized as follows. Section 2 contains notations which are used throughout this article. In section 3 a smoothing property for Fr´echet spaces is recalled from [32] which is required as a formal assumption in the inverse function theorem of Nash-Moser type [34]. The spaces C₀^∞([0, T], H^∞(Ω)) are shown to enjoy this property with uniform constants for all small T >0; here C₀^∞ denotes the subspace ofC^∞containing functions vanishing with all derivatives at the origin.

In section 4 the inverse function theorem [34] is used to linearize the initial boundary value problem. Mainly due to compatibility conditions this approach is completely different from the whole space case [36]. A transformation based on a Borel’s theorem gives a reduction to zero compatibility conditions. The smallness assumptions required by the inverse function theorem can be achieved by choosing a small time interval without supposing smallness assumptions on the initial values.

This is based on a uniformity argument and on Borel’s theorem.

Using results of section 5 the linear problem is reduced to a problem with ho- mogeneous boundary conditions. The results of section 5 might be of independent interest. A trace theorem including estimates is proved for normal boundary op- erators in spaces of smooth functions by applying the tame splitting theorem [40]

in Fr´echet spaces. Note that classical right inverses for trace operators in Sobolev spaces constructed e.g. by the Fourier transform depend on the order of the Sobolev space and do not induce right inverses in spaces of smooth functions. In addition, results on simultaneous continuity are proved for trace theorems in Sobolev spaces.

Sections 6, 7, 8 contain an application in the parabolic case.

In section 6 the linearized parabolic initial boundary value problem of arbitrary order is considered. Under suitable parabolicity assumptions the necessary higher order Sobolev norm estimates are proved. In order to derive the appropriate de- pendence of the constants from the coefficients these estimates are formulated and proved by means of a symbolic calculus involving the weighted multiseminorms

[ ]m,kintroduced in [34]. The estimates are based on maximal regularity in H¨older spaces and on the results of section 5 on simultaneous continuity in trace theorems.

In section 7 we obtain sufficient conditions of elliptic type for the parabolicity assumptions of section 6. It is shown that the constants in the higher order elliptic a priori estimates due to Agmon, Douglis, Nirenberg [3] depend on the coefficients of the problem as required by the Nash-Moser technique; we note that this means more than only uniformity as stated in [3]. Furthermore, resolvent estimates due to Agmon [2] are used to establish the assumptions of section 6.

Finally, in section 8 the short-time solvability of the nonlinear parabolic problem is proved in Theorem 8.1 under general and natural assumptions. It is enough that the linearized problem together with the linearized boundary conditions is given by a regular elliptic problem in the usual sense (cf. Definition 7.5 or [24]).

The technical Theorems 4.3, 4.4, 5.5 provide a general framework for applications to evolutionary boundary value problems where a loss of derivatives appears in the estimates of the linearized problem. This might be interesting for further applica- tions which are not accessible to standard methods due to a loss of derivatives, for instance to other evolution equations or to coupled systems involving Navier Stokes system and heat equation where a loss of derivatives appears due to the coupling.

2. Preliminaries

We shall consider Fr´echet spacesE, F, . . .equipped with a fixed sequence k k0≤ k k1≤ k k2≤. . .of seminorms defining the topology. The productE×F is endowed with the seminormsk(x, y)kk = max{kxkk,kykk}. A linear mapT:E→Fis called tame (cf. [13]) if there exist an integerband constantsc_kso thatkT xkk≤c_kkxkk+b

for all k and x. A linear bijection T is called a tame isomorphism if both T and T⁻¹ are tame.

A continuous nonlinear map Φ : (U ⊂E)→F between Fr´echet spaces,U open, is called aC¹-map if the derivative Φ⁰(x)y= lim

t→0 1

t(Φ(x+ty)−Φ(x)) exists for all x∈U, y∈E and is continuous as a map Φ⁰:U ×E →F. Φ is called a C²-map if it is C¹ and the second derivative Φ⁰⁰(x){y1, y2}= lim

t→0 1

t(Φ⁰(x+ty2)y1−Φ⁰(x)y1) exists and is continuous as a map Φ⁰⁰ :U ×E²→F. Similar definitions apply to higher derivatives Φ⁽ⁿ⁾; Φ is called C^∞ if it isCⁿ for all n. Given a function of two variables Φ = Φ(x, y) we can also consider the partial derivatives Φx and Φy

where e.g. Φ_x(x, y)z= lim

t→0 1

t(Φ(x+tz, y)−Φ(x, y)). One-dimensional derivatives Φt, t ∈ R are alternatively considered as a map Φt : U ×R → F or as a map Φt:U→F, respectively. For these notions we refer to [13], I.3.

Let Ω ⊂Rⁿ be bounded and open with C^∞-boundary ∂Ω. In this paper, we restrict ourselves to the case of bounded domains Ω; most results are formulated in a way such that a generalization to uniformly regular domains of class C^∞ in the sense of [9], section 1 or [5], Ch. III, p. 642 is obvious (cf. [36]). For any integer k≥0 the Sobolev space H^k(Ω) is equipped with its natural norms (where

|α|=α₁+. . .+α_n forα∈Nⁿ0) kukk = (X

|α|≤k

Z

Ω

|∂^αu(x)|²dx)^1/2, u∈H^k(Ω). (2.1)

The spaceH^∞(Ω) =T∞

k=0 H^k(Ω) is a Fr´echet space with the norms (k kk)^∞_k=0. On the algebraH^∞(Ω) we can consider sup-norms

kuk^∞_k = sup

|α|≤k

sup

x∈Ω

|∂^αu(x)|, u∈H^∞(Ω) (2.2) since by Sobolev’s imbedding theorem there are constantsck >0 such that

kuk^∞_k ≤ckkukk+b, u∈H^k+b(Ω), b:= [n/2] + 1> n/2. (2.3) The Sobolev space (H^s(∂Ω),k ks) is defined as usual for a real s ≥ 0 using a partition of unity (cf. [54], I. 4.2.). In particular, for an integer k ≥1 the space H^k−1/2(∂Ω) is the class of functionsφwhich are the boundary values of functions u∈H^k(Ω); the spaceH^k−1/2(∂Ω) can be equipped with the equivalent norm

kφk_k−1/2= inf{kukk :u∈H^k(Ω), u=φon∂Ω}, φ∈H^k−1/2(∂Ω). (2.4) The Fr´echet spaceH^∞(∂Ω) =T∞

k=0H^k(∂Ω) is equipped with these norms.

The Fr´echet spaceC^∞(Ω) of allC^∞-functions on Ω such that all partial deriva- tives are uniformly continuous on Ω is equipped with the norms (k k^∞_k )^∞_k=0. The Fr´echet spaceC^∞(∂Ω) of all smooth functions on the manifold∂Ω is endowed with the norm system (k k^∞_k )^∞_k=0 defined as usual using cutoff functions and a partition of unity (cf. [33], 4.14.). It is well known that there exists a linear continuous extension operatorR_Ω:C^∞(∂Ω)→C^∞(Ω) such thatkRΩfkk ≤c_kkfkk for all k and constantsc_k>0; this follows e.g. from [46] using a partition of unity.

A vector valued functionu= (u₁, . . . , u_M) belongs toH^∞(Ω,R^M) if each coor- dinateu_j is in H^∞(Ω); the same applies toH^∞(∂Ω,R^M). We use of the following symbolic calculus introduced in [34]. Let p, q ≥0 be integers, p+q ≥1, and let E1, . . . , Ep, F1, . . . , Fq be linear spaces each equipped with a sequence| |0≤ | |1≤

| |2≤. . . of seminorms. For any integerm, k≥0 andx1 ∈E1, . . . , xp ∈Ep, y1 ∈ F1, . . . , yq ∈Fq we define

[x1, ..., xp;y1, ..., yq]m,k= sup{|xk₁|m+i₁...|xk_r|m+i_r|y1|m+j₁...|yq|m+j_q} the ’sup’ running over all i1, . . . , ir, j1, . . . , jq ≥ 0 and 1 ≤ k1, . . . , kr ≤ p with 0 ≤ r ≤ k and i1+. . .+ir+j1+. . .+jq ≤ k (for r = 0 the |x|-terms are omitted). For q = 0 we write [x1, . . . , xp]m,k (the |y|-terms are omitted) and for p= 0 we write [;y₁, . . . , y_q]_m,k. For m= 0 we write [. . .]_k = [. . .]_0,k. Observe that [x₁, . . . , x_p;y₁, . . . , y_q]_m,kis a seminorm seperately in each componenty_j while it is completely nonlinear in the x_i-components. The weighted multiseminorms [ ]_m,k are increasing inmand ink. For the purely nonlinear terms (i.e.,q= 0) we have [x₁, . . . , x_p]_m,0= 1 and [x₁, . . . , x_p]_m,k≥1 for allm, k. For properties of the terms [ ]_m,k we refer to [34], 1.7.; we shall often apply rules like [x]_m,k·[x]_m,i≤[x]_m,k+i and [x]m,i+k ≤ max{1,|x|^i+k_m+i}[x]m+i,k ≤ C⁰[x]m+i,k if |x|m+i ≤ C. If Sobolev spacesH^∞(Ω) are involved then the following applies. The expressions [u]m,k and [u;v]m,k are defined by the corresponding Sobolev norms kuki,kvkj. The terms kuk^∞_m,korku, vk^∞_m,k (i.e.,p= 2, q= 0) are defined by sup-normskuk^∞_i ,kvk^∞_j . The expression [u;v]^∞_m,k (i.e., p= q= 1) is defined by means of the sup-norms kuk^∞_i and Sobolev norms kvkj. For a real number t let [t] denote the largest integer j withj≤t.

3. A smoothing property for Fr´echet spaces

In the inverse function theorem 3.4 the Fr´echet spaces are assumed to satisfy smoothing property (S) introduced in [32], 3.4 and property (DN) of Vogt [53]. A Fr´echet space E has property (DN) if there isb such that for any nthere are k_n andc_n>0 such that for allx∈Ewe have

kxk²_n≤cnkxkbkxkk_n (3.1) We say that E has smoothing property (S) if there exist b, p ≥ 0 and constants cn >0 such that for anyθ≥1 and anyx∈Eand for any sequence (An)nsatisfying kxkn ≤An≤An+1 andA²_n≤A_n−1An+1 for allnthere exists an elementSθx∈E (which may depend onxand on the sequence (An)) such that

kS_θxk_n ≤c_nθ^n+p−kA_k, b≤k≤n+p

kx−Sθxkn ≤ckθ^n+p−kAk, k≥n+p. (3.2) Smoothing property (S) generalizes (cf. [32]) the classical smoothing operators (cf. [13], [15], [25]). For a Fr´echet spaceEand T >0 we put

C₀^∞([0, T], E) =n

u∈C^∞([0, T], E) :u^(j)(0) = 0, j= 0,1,2, . . .o

. (3.3)

In caseE is one-dimensional we writeC₀^∞[0, T] instead ofC₀^∞([0, T], E).

Lemma 3.1. Let T1 >0. The spaces C₀^∞[0, T] have property (S) with b=p= 0 wherecn in(3.2) may be chosen uniformly for all0< T ≤T1.

Proof. The spaceD[0,2] of all smooth function with support in [0,2] has property (S) withb=p= 0 (cf. [32], 5.1). The spaceC₀^∞[0,1] is a quotient space ofD[0,2]

by means of restriction and hence a direct summand ofD[0,2] using an extension operator (cf. Seeley [46] or [33], 4.8). Therefore, C₀^∞[0,1] inherits property (S) fromD[0,2] withb=p= 0. To prove uniformity we assume thatT₁= 1. We have

kfk^[0,T]_k =sup^k

j=0

sup

t∈[0,T]

|f^(j)(t)|= sup

t∈[0,T]

|f^(k)(t)|=:|f|^[0,T]_k (3.4) forf ∈C₀^∞[0, T] and 0< T ≤1. Put Γ_T :C₀^∞[0,1]→C₀^∞[0, T],Γ_Tf(x) =f(x/T).

Notice that |Γ_Tf|^[0,T]_k =T^−k|f|^[0,1]_k . If S_θ is induced by property (S) in C₀^∞[0,1]

then Γ_T ◦S_{T θ}◦Γ⁻¹_T gives property (S) forC₀^∞[0, T] with the same constants.

The uniformity part of Lemma 3.1 does not work e.g. forC^∞[0, T]. For a Fr´echet space E and a sequence 0≤α0 ≤α1 ≤ . . . %+∞ we consider the power series space ofE-valued sequencesx= (x_j)^∞_j=1⊂E defined by

Λ^∞_∞(α;E) ={(xj)j ⊂E:kxkk =sup^k

i=0

sup

j

kxjk_k−ie^iαj <∞, k= 0,1, . . .}.

In case dimE= 1 we write Λ^∞_∞(α) instead of Λ^∞_∞(α;E). The corresponding space defined byl²-norms instead of sup-norms is denoted by Λ²_∞(α).

Lemma 3.2. If E has property(S) thenΛ^∞_∞(α;E)has (S)as well.

Proof. Let 06=x∈Λ^∞_∞(α;E) and kxkk ≤Ak ≤Ak+1, A²_k ≤A_k−1Ak+1. We may assume thatt7→logAtis convex and increasing. We have

kx_jk_i≤ inf

i≤k∈N0

e^(i−k)αjA_k=:B^j_i ≤D^j_i+1:= inf

i+1≤t∈R

e^(i+1−t)αjA_t≤A_i+1

for any i, j. It is easy to see thatD^j_i+1≤D^j_i+2 and (D^j_i+1)²≤D_i^jD^j_i+2 for alli, j.

We hence may chooseSθxj according to the sequence (D_i+1^j )i such that kSθxjkn≤cnθ^n+p+1−kD^j_k, b+ 1≤k≤n+p+ 1

kxj−Sθxjkn≤ckθ^n+p+1−kD^j_k, k≥n+p+ 1.

(3.5) We defineT_θxforθ≥1 by (T_θx)_j= 0 ife^αj ≥θand (T_θx)_j =S_θx_j ife^αj < θ. For e^αj ≥θwe get fork≥n+p+ 1 and 0≤i≤nthe estimate

kxjkn−ie^iαj ≤e^(n−k)αjAk ≤θ^n−kAk. (3.6) Fore^αj < θwe establish fork≥n+p+ 1 and 0≤i≤nthe estimate

kxj−Sθxjk_n−ie^iαj ≤ckθ^{n−i+p+1−k}e^iαjD_k^j ≤ckθ^n+p+1−kAk. (3.7) Lete^αj < θandb+ 1≤k≤n+p+ 1. In the case 0≤i≤k−b−1 we get

kS_θx_jk_n−ie^iαj ≤c_n−iD_k−i^j θ^n+p+1−ke^iαj ≤c_n−iθ^n+p+1−kA_k (3.8) and fork−b−1≤i≤nwe obtain (where we may assume thatp≥b)

kSθxjkn−ie^iαj ≤cn−iθ^n+p−i−bD^j_b+1e^iαj ≤cn−iθ^n+p+1−kAk (3.9) sinceD^j_b+1e^iαj ≤e^{(i+b+1−k)αj}Ak≤θ^i+b+1−kAk. This gives the result.

Proposition 3.3. Let Ω ⊂ Rⁿ, n ≥ 2, be open and bounded with C^∞-boundary.

Let T1>0and an integerm≥1be fixed. Then the spaces C₀^∞([0, T], H^∞(Ω))and C₀^∞([0, T], H^∞(∂Ω)) equipped with the norms

kukk= supn

ku⁽ⁱ⁾(t)k_k−mi:t∈[0, T],0≤i≤k/mo

(3.10) have properties (S), (DN). In addition, the constants c_n, k_n, b, p in the above defi- nitions of (S), (DN)can be chosen uniformly for all0< T ≤T1.

Proof. Clearly the spaces have (DN); the uniformity statement holds sinceC₀^∞[0, T] is a subspace (by trivial extension) of C^∞[−1 +T, T] ∼= C^∞[0,1] if T ≤1. It is enough to show property (S) for the spaces equipped with the new norm system (k kmk)^∞_k=0 (cf. [31], 4.3). There are tame isomorphisms H^∞(Ω) ∼= Λ^∞_∞(α) for αj = (logj)/n and H^∞(∂Ω)∼= Λ^∞_∞(β) for βj = (logj)/(n−1); this is proved in [33], 4.10, 4.14. We put ˜αj =mαj and obtain a tame isomorphism

C₀^∞([0, T], H^∞(Ω)),(k kmk)^∞_k=0

∼= Λ^∞_∞( ˜α;C₀^∞[0, T]). (3.11) The same argument applies toH^∞(∂Ω). Now 3.1, 3.2 give the assertion.

In section 4 we shall apply the following inverse function theorem of Nash-Moser type which is proved in [34], 4.1 (cf. [13], [15], [25]).

Theorem 3.4. LetE, F be Fr´echet spaces with smoothing property(S)and(DN).

Let U0 ={x∈E :|x|b < η} for some b ≥0, η >0. Let Φ : (U0 ⊂E)→ F be a C²-map with Φ(0) = 0such thatΦ⁰(x) :E→F is bijective for allx∈U0. Assume that there are an integerd≥0 such that

kΦ⁰(x)vk_k ≤c_k[x;v]_d,k kΦ⁰(x)⁻¹yk_k ≤c_k[x;y]_d,k kΦ⁰⁰(x){v, v}k_k≤c_k[x;v, v]_d,k

(3.12)

for all x∈ U0, v ∈ E, y ∈ F and all k = 0,1,2, . . . with constants ck > 0. Then there exist open zero neighbourhoodsV ={y∈F :kyks< δ} ⊂F andU ⊂E such that Φ :U →V is bijective andΦ⁻¹: (V ⊂F)→E is aC²-map. If ΦisCⁿ then Φ⁻¹ is Cⁿ as well,2 ≤n ≤ ∞. Moreover, the numbers s≥0 and δ >0 depend only on the constants in the assumption, i.e., on b, d, η, c_k and on the constants in properties (S), (DN).

4. Linearization of boundary-value problems

Let Ω ⊂Rⁿ be bounded and open with C^∞-boundary. We fix a real number T >0 and integersM ≥1, m≥2. We writeH^∞(Ω) =H^∞(Ω,R^M) andH^∞(∂Ω) = H^∞(∂Ω,R^M). We assume thatmis even and putI(m) ={α∈Nⁿ0 :|α| ≤m}. Let A⊂(R^M)^I(m) be open; then the set

U0={u∈H^∞(Ω) :{∂^αu(x)}_|α|≤m∈A, x∈Ω} (4.1) is open in H^∞(Ω) as well. Let F ∈C^∞([0, T]×Ω×A,R^M), F =F(t, x, u). We considerF : [0, T]×(U₀⊂H^∞(Ω))→H^∞(Ω) defined by

F(t, u)(x) =F(t, x,{∂^αu(x)}_|α|≤m), u∈U₀, t∈[0, T], x∈Ω. (4.2) It is proved in [36], section 2 (cf. [15]) thatF is a nonlinearC^∞-map between Fr´echet spaces where F⁰ : ([0, T]×U0)×(R×H^∞(Ω)) → H^∞(Ω) is given by F⁰(t, u)(s, v) =Ft(t, u)s+Fu(t, u)v where

Fu(t, u)v= X

|α|≤m

F∂^αu(t,·,{∂^βu(·)}_|β|≤m)∂^αv. (4.3) IfAis bounded then [36], 2.3, 2.4 give withb= [n/2] + 1 the estimates

kF⁰(t, u)(s, v)kk≤ck[(t, u); (s, v)]m+b,k

kF⁰⁰(t, u){(s, v),(s, v)}kk ≤ck[(t, u); (s, v),(s, v)]m+b,k

(4.4) for allt∈[0, T], u∈U0, s∈R, v∈H^∞(Ω) whereck>0 are constants.

We define nonlinear boundary operators Bj and put B = (B1, . . . ,Bm/2). For that we fix integers mj ≥0 and choose open sets Aj ⊂(R^M)^I(mj) and mappings Bj ∈C^∞([0, T]×∂Ω×Aj,R^M), j= 1, . . . , m/2. Then the sets

Uj={u∈H^∞(Ω) :{∂^βu(x)}_|β|≤mj ∈Aj, x∈∂Ω} (4.5) are open inH^∞(Ω). We defineBj: [0, T]×(Uj⊂H^∞(Ω))→H^∞(∂Ω) by

Bj(t, u)(x) =Bj(t, x,{∂^βu(x)}_|β|≤mj), u∈Uj, t∈[0, T], x∈∂Ω. (4.6) The arguments used in [36], section 2 or [15] show that B_j is aC^∞-map between Fr´echet spaces where B_j⁰ : ([0, T]×Uj)×(R×H^∞(Ω)) → H^∞(∂Ω) is given by B⁰_j(t, u)(s, v) = (Bj)t(t, u)s+ (Bj)u(t, u)vwhere

(Bj)u(t, u)v= X

|α|≤mj

(Bj)∂^αu(t,·,{∂^βu(·)})∂^αv. (4.7) For a bounded setA_j the proof of [36], 2.3, 2.4 yields

kB⁰_j(t, u)(s, v)k_k−1

2 ≤ck[(t, u); (s, v)]m_j+b,k

kB⁰⁰_j(t, u){(s, v),(s, v)}k_k−1

2 ≤ck[(t, u); (s, v),(s, v)]m_j+b,k

(4.8)

for allt∈[0, T], u∈Uj, s∈R, v∈H^∞(Ω) with some ck >0 andb as above. Our goal is to solve the nonlinear initial boundary-value problem

u_t=F(t, u) in Ω, t∈[0, T₀] B(t, u) =h(t) on∂Ω, t∈[0, T0]

u(0) =φ .

(4.9)

More precisely, for a given initial valueφ∈U :=Tm/2

j=0 U_j ⊂H^∞(Ω) and a given boundary value h ∈ C^∞([0, T], H^∞(∂Ω)^m/2) we are looking for a solution u of problem (4.9) for some suitable small T0 > 0; by a solution we mean a function u∈C^∞([0, T₀], H^∞(Ω)) such thatu(t)∈U for allt∈[0, T₀] and (4.9) is satisfied.

There are some natural necessary constraints on the given data h, φ. In order such that (4.9) can admit a smooth solution the data h, φ have to satisfy the following necessary compatibility conditions which are obtained by computingh^j(0) as a differential operator acting onφon the boundary of Ω by means of (4.9). For instance, we geth(0) =B(0, φ) =: Γ₀(φ) and

h⁰(0) =Bt(0, φ) +Bu(0, φ)F(0, φ) =: Γ1(φ). (4.10) In a similar way we obtain from (4.9) the necessary compatibility conditions

h^(j)(0) =∂_t^jB(t, u(t))_|t=0=: Γ_j(φ) on∂Ω, j= 0,1,2, . . . (4.11) where differential operators Γ_j acting onφon∂Ω are obtained by first computing

∂_t^jB(t, u(t)), then replacing all derivatives ∂_tⁱu by terms involving u using u_t = F(t, u(t)) and finally evaluating att = 0 usingu(0) =φ. Analogously, the values u^j(0) are a priori determined in Ω by (4.9). For instance, we getu(0) =φ=: Ψ0(φ) andu⁰(0) =F(0, φ) =: Ψ1(φ) and

u⁰⁰(0) =F_t(0, φ) +F_u(0, φ)F(0, φ) =: Ψ₂(φ). (4.12) Using the first and the third equation in (4.9) we see that solutionsusatisfy

u^(j)(0) =∂_t^j−1F(t, u(t))_|t=0=: Ψj(φ) in Ω, j= 1,2,3, . . . (4.13) with differential operators Ψ_j acting on φ in Ω where Ψ_j are defined using u_t = F(t, u) andu(0) =φ. We note that (4.13) are by no means compatibility conditions like (4.11). However, the a priori knowledge of u^(j)(0) can be used to transform problem (4.9) such that solutions v of the transformed problem satisfy v^j(0) = 0 for all j. This simplifies the compatibility conditions (4.11). We shall apply the following version of a theorem of E. Borel’s [8].

Lemma 4.1. LetE be a Fr´echet space. Let(a_j)^∞_j=0⊂E be an arbitrary sequence.

Then there isψ∈C^∞([0,1], E) such thatψ^(j)(0) =aj for allj.

The proof of this lemma follows the standard proof in (cf. [14], 1.2.6 or [30], 1.3).

We chooseψ∈C^∞([0, T], H^∞(Ω)) such thatψ(t)∈U fort∈[0, T] and

ψ^(j)(0) = Ψj(φ), j= 0,1,2, . . . (4.14) We putv=u−ψand get from (4.9) the transformed problem

vt=F(t, v+ψ)−ψ⁰(t) in Ω, t∈[0, T0] B(t, v+ψ)− B(t, ψ) =h(t)− B(t, ψ) on∂Ω, t∈[0, T0]

v(0) = 0.

(4.15)

Remark 4.2. (i) If u solves (4.9) then v = u−ψ solves (4.15). On the other hand, ifv solves (4.15) thenu=v+ψsolves (4.9). (ii) Solutionsuof (4.9) satisfy u^(j)(0) =ψ^(j)(0) for allj. On the other hand, solutionsv of (4.15) automatically satisfy v^(j)(0) = 0 for all j. (iii) For γ(t) = B(t, ψ(t)) we have γ^(j)(0) = Γj(φ) for all j. (iv) If h satisfys h^(j)(0) = Γj(φ) for all j then the right hand side

˜h(t) =h(t)− B(t, ψ(t)) in (4.15) satisfies ˜h^(j)(0) = 0 for all j. (v) The left hand side ˜B(t, v) = B(t, v+ψ(t))− B(t, ψ(t)) considered in (4.15) as an operator in v satisfies (∂_t^jB)(0,˜ 0) = 0 for allj. Note that

(∂_tB)(t, v) =˜ B_t(t, v+ψ) +B_u(t, v+ψ)ψ⁰− B_t(t, ψ)− B_u(t, ψ)ψ⁰.

(vi) In the case of linear boundary conditions we have ˜B(t, v) =B(t, v). (vii) The right hand side ˜F(t, v) =F(t, v+ψ(t))−ψ⁰(t) in (4.15) considered as a nonlinear differential operator in v satisfies (∂_t^jF˜)(0,0) = 0 for all j. This follows since F(0,˜ 0) =F(0, φ)−ψ⁰(0) = 0 and

(∂_t^jF)(0,˜ 0) =∂_t^j{F(t, ψ(t))}t=0−ψ^(j+1)(0) = Ψj+1(φ)−ψ^(j+1)(0) = 0.

Using the above notation we hence may consider the normalized problem ut= ˜F(t, u) in Ω, t∈[0, T0]

B(t, u) = ˜˜ h(t) on∂Ω, t∈[0, T0] u(0) = 0,

(4.16)

where we may assume the normalized conditions

(∂_t^jF)(0,˜ 0) = 0, j = 0,1,2, . . . (∂_t^jB)(0,˜ 0) = 0, j = 0,1,2, . . .

˜h^(j)(0) = 0, j= 0,1,2, . . .

(4.17)

where ˜h^(j)(0) = 0 are the natural compatibility conditions for (4.16) if we assume the first two conditions in (4.17). Since solutionsuof (4.16), (4.17) satisfyu^(j)(0) = 0 for all j we have to look for solutions u in the space C₀^∞([0, T0], H^∞(Ω)). We formulate problem (4.16) by a mapping. We fixT >0 and put J = [0, T]. We get an open setW ⊂C₀^∞(J, H^∞(Ω)) by

W =

u∈C₀^∞(J, H^∞(Ω)) :u(t)∈U, t∈J (4.18) where we may assume that 0∈W. We define the nonlinear map

Φ : (W ⊂C₀^∞(J, H^∞(Ω)))→C₀^∞(J, H^∞(Ω))×C₀^∞(J, H^∞(∂Ω)) (4.19) by

Φ(u) = (∂tu−F(t, u) + ˜˜ F(t,0),B(t, u)˜ −B(t,˜ 0)). (4.20) We note that Φ is well defined since ∂_t^j{ut−F(t, u(t)) + ˜˜ F(t,0)}(0) = 0 and

∂_t^j{B(t, u(t))˜ −B(t,˜ 0)}(0) = 0 for allj and every u∈ W in view of (4.17). The map Φ is aC²-map satisfying Φ(0,0) = 0 where the first derivative is

Φ⁰(u)v= (∂_tv−F˜u(t, u)v,B˜u(t, u)v) (4.21) For a fixedT1>0 the first and third estimate in (3.12) hold with uniform constants for 0< T ≤T1where the norms are defined by (3.10). This follows from the proof of [36], 4.3 using (4.4), (4.8). We consider the equation

Φ(u) = ( ˜F(t,0),˜h(t)−B(t,˜ 0)). (4.22)

The inverse function theorem 3.4 requires the smallness condition

kF(t,˜ 0)k_s+kB(t,˜ 0)k_s+kh(t)k˜ _s< δ. (4.23) By (4.17) condition (4.23) holds if T > 0 is chosen sufficiently small. We here shall have to observe that s, δ in Theorem 3.4 can be chosen uniformly for all 0< T ≤T1. We consider the nonlinear problem (4.9) for some given initial value φ∈U andh∈C^∞([0, T1], H^∞(∂Ω)). We assume that the compatibility conditions (4.11) hold.

Theorem 4.3. Let T1 > 0, φ ∈ H^∞(Ω) and h ∈ C^∞([0, T1]), H^∞(∂Ω)) satisfy (4.11). Assume that there are b≥0 andck >0 and an open neighbourhood U of φ inH^∞(Ω) so that for any 0< T ≤T1 andu∈ W ={w∈C^∞([0, T], H^∞(Ω)) : w(t)∈U, t∈[0, T]} the linear problem

zt(t) =Fu(t, u(t))z(t) +f(t) in Ω, t∈[0, T] Bu(t, u(t))z(t) =g(t) on∂Ω, t∈[0, T]

z(0) = 0

(4.24)

admits for anyf ∈C₀^∞([0, T], H^∞(Ω)) andg ∈C₀^∞([0, T], H^∞(∂Ω)^m/2) a unique solution z∈C₀^∞([0, T], H^∞(Ω))satisfying the estimates

kzkk≤ck[u; (f, g)]b,k, k= 0,1,2, . . . (4.25) Then(4.9)has a unique solutionu∈C^∞([0, T0], H^∞(Ω)) for someT0>0.

Proof. We chooseψ∈C^∞([0, T1], H^∞(Ω)) satisfying (4.14) such thatψ(t)∈U for allt∈[0, T1]. By remark 4.2 (i) it is enough to solve problem (4.15) forv=u−ψ.

For that we define Φ by (4.22) where Φ(0,0) = 0 and ˜F,B,˜ ˜h are defined as in Remark 4.2 satisfying (4.17). We have to solve equation (4.22). By our assumption on the linear problem (4.24) the operator Φ⁰(u) is bijective for all uin some zero neighbourhood in C₀^∞([0, T], H^∞(Ω)),0 < T ≤ T1. The inequalities (4.25) yield the second estimate in (3.12) while the first and third estimate in (3.12) hold as observed above. The assumptions of Theorem 3.4 on the spaces are satisfied by Proposition 3.3. Hence there exist numbers s ≥0 and δ > 0 as in Theorem 3.4 which can be chosen uniformly for all 0 < T ≤ T₁. We can choose T₀ > 0 so small such that the smallness condition (4.23) holds in [0, T₀]. Theorem 3.4 gives a solutionv ∈C₀^∞([0, T₀], H^∞(Ω)) of problem (4.15) and thus a solution u=v+ψ of problem (4.9). The uniqueness can be shown using Theorem 3.4 and a standard argument as in [36], Theorem 4.4. This gives the result.

Problem (4.24) can be reduced to a problem with homogeneous boundary condi- tions provided that the boundary conditions can be solved. LetT1>0 and choose U, W as in Theorem 4.3. We assume there existb≥0 andck>0 such that for any u∈W and 0< T ≤T1 there exists a map

R_u:C₀^∞([0, T], H^∞(∂Ω)^m2)→C₀^∞([0, T], H^∞(Ω)), B_u(·, u)Ru= Id (4.26) which satisfies for allg∈C₀^∞([0, T], H^∞(∂Ω)^m/2) the estimates

kRugkk≤ck[u;g]b,k, k= 0,1,2, . . . (4.27) In section 5 we show that suchR_u exist for normal boundary conditions.

Theorem 4.4. Let T1 >0, φ∈H^∞(Ω), h ∈C^∞([0, T1]), H^∞(∂Ω)) satisfy (4.11).

Assume that there are b≥0 andck >0 and open setsU, W as in Theorem 4.3so that for any 0 < T ≤T1 and u∈ W there exist Ru satisfying (4.26), (4.27) such that for anyf₁∈C₀^∞([0, T], H^∞(Ω))the problem

wt(t) =Fu(t, u(t))w(t) +f1(t) inΩ, t∈[0, T] Bu(t, u(t))w(t) = 0 on∂Ω, t∈[0, T]

w(0) = 0

(4.28)

admits a unique solutionw∈C₀^∞([0, T], H^∞(Ω))satisfying the estimates

kwkk≤ck[u;f1]b,k, k= 0,1,2, . . . (4.29) Then(4.9)has a unique solutionu∈C^∞([0, T0], H^∞(Ω)) for someT0>0.

Proof. Let f ∈ C₀^∞([0, T], H^∞(Ω)) andg ∈ C₀^∞([0, T], H^∞(∂Ω)^m/2). We choose v = R_ug satisfying (4.27). We put f₁(t) = f(t)−v_t(t) +Fu(t, u(t))v(t). Then f₁∈C₀^∞([0, T], H^∞(Ω)). By assumption we find a solutionw∈C₀^∞([0, T], H^∞(Ω)) of (4.28) satisfying (4.29). Thenz=v+wis a solution of (4.24) satisfying (4.25).

The solution is unique by means of the unique solvability of (4.28). Hence Theorem

4.3 gives the result.

5. Normal boundary conditions

In this section we are concerned with normal boundary conditions. Let Ω⊂Rⁿ be a bounded open set with C^∞-boundary. Let {Bj}^p_j=1 be a set of differential operatorsB_j=B_j(x, ∂) of orderm_j given by

B_j=B_j(x, ∂) = X

|β|≤mj

b_j,β(x)∂^β, j = 1, . . . , p (5.1) with bj,β ∈C^∞(∂Ω). There is a linear extension operator S :C^∞(∂Ω)→C^∞(Ω) satisfyingkSfkk≤ckkfkkfor allk, fwith constantsck >0; this follows from Seeley [46] using a partition of unity (cf. [33]). We hence may assume thatb_j,β ∈C^∞(Ω).

The set{Bj}^p_j=1 is called normal (cf. [24], [44], [54]) ifm_j 6=m_iforj6=iand if for anyx∈∂Ω we have B_j^P(x, ν)6= 0, j= 1, . . . , pwhereν =ν(x) denotes the inward normal vector to∂Ω atxand B_j^P denotes the principal part of Bj. A normal set {Bj}^p_j=1 is called a Dirichlet system ifmj =j−1, j= 1, . . . , p. We can consider the Dirichlet boundary conditions u7→ ^∂_∂ν^j−1j−1^u|∂Ω, j = 1, . . . , p, which give for any k≥pa trace operator

T_k^p:H^k(Ω)→

p

Y

i=1

H^k−i+1/2(∂Ω), T_k^pu=n∂^j−1u

∂ν^j−1

|∂Ω

op j=1

. (5.2) The trace operators T_k^p are surjective admitting a continuous linear right inverse Z_k^p which depends on k (cf. [24], [54]). To construct a tame linear right inverse for the induced trace operatorT^p :H^∞(Ω)→ H^∞(∂Ω)^p we apply tame splitting theory in Fr´echet spaces developed by Vogt (cf. [52]).

Let (F_k)_k,(G_k)_k be families of Hilbert spaces with injective linear continuous imbeddings F_k+1 ,→ F_k, G_k+1 ,→ G_k for all k. LetT_k : F_k → G_k be surjective

continuous linear maps such that (Tk)_|Fk+1=Tk+1 for allk. LetEk =N(Tk)⊂Fk

denote the kernel ofTk; we haveEk+1,→Ek and

0 −→ E_k ,→ F_k −→^Tk G_k −→ 0 (5.3)

are exact sequences of Hilbert spaces. We equip the Fr´echet spacesE=T

kEk, F = T

kF_k, G =T

kG_k with the induced norms. We then have a mappingT :F →G defined byT x=T_kx, x∈F where N(T) =E. The following splitting theorem is a simplified version of [40], 6.1, 6.2.

Lemma 5.1. Let E_k, F_k, G_k, T_k andE, F, G, T be as above where(5.3)is an exact sequence of Hilbert spaces for every k. Assume that there are tame isomorphisms E∼= Λ²_∞(α)andG∼= Λ²_∞(β)for someα, β. Then

0 −→ E ,→ F −→^T G −→ 0 (5.4)

is an exact sequence of Fr´echet spaces which splits tamely, i.e., there is a tame linear mapZ:G→F such thatT◦Z = IdG.

Lemma 5.2. Let p≥1. The trace operator T^p : H^∞(Ω) → H^∞(∂Ω)^p admits a tame linear right inverseZ^p:H^∞(∂Ω)^p→H^∞(Ω),T^p◦Z^p= Id.

Proof. The trace operatorT_k^p induces fork≥pan exact sequences 0 −→ N(T_k^p) ,→ H^k(Ω) ^T

p

−→k

p

Y

i=1

H^k−i+1/2(∂Ω) −→ 0 (5.5) of Hilbert spaces. We show using Lemma 5.1 that the sequence

0 −→ N(T^p) ,→ H^∞(Ω) ^T

p

−→ H^∞(∂Ω)^p −→ 0 (5.6) of Fr´echet spaces splits tamely. Note that there are tame isomorphismsH^∞(Ω)∼= Λ²_∞(α) and H^∞(∂Ω)^p ∼= Λ²_∞(β) (cf. the proof of Proposition 3.3). Let ∆ denote the Laplacian. We consider ∆^p as an unbounded operator in L²(Ω) under null Dirichlet boundary conditions, the domain given byDp=N(T_2p^p) ={u∈H^2p(Ω) : T_2p^pu= 0}. It is well known that the spectrum of ∆^p is discrete (cf. [9], Theorem 17, [2], Theorem 2.1). We thus can choose λsuch that ∆^p−λis an isomorphism Dp →L²(Ω). Therefore, ∆^p−λ:N(T^p) →H^∞(Ω) is an isomorphism (cf. [54]) which is a tame isomorphism by means of classical elliptic a priori estimates (cf. [3], Theorem 15.2). HenceN(T^p)∼=H^∞(Ω)∼= Λ²_∞(α) tamely isomorphic. By Lemma 5.1 the sequence (5.6) splits tamely. This gives the result.

For a differential operatorP =P

|α|≤maα(x)∂^α withaα∈C^∞ we put kPk_i=P

|α|≤mka_αk^∞_i , i= 0,1, . . . (5.7) ForP as above andQ=P

|β|≤nbβ(x)∂^β we get kP Qki≤Ci

i

X

j=0

kPk_i−jkQkm+j (5.8)

with constantsC_i >0. For smooth nonvanishing functions f we get with C_i >0 depending only oni, m, nand onk1/fk^∞₀ the estimates

k1/fk^∞_i ≤C_i[f]_i, kP/fki≤C_i[f;P]_i (5.9)

for alli. Here the expressions [f]i and [f;P]i are defined by the normskfk^∞_j and (5.7). To prove a generalization of Lemma 5.2 to normal boundary conditions we first consider the case of a half space. We consider the cubes

Σ ={x∈Rⁿ :|xi|<1 (i= 1, . . . , n), xn>0}. (5.10) σ={x∈Rⁿ:|xi|<1 (i= 1, . . . , n), xn = 0}. (5.11) The following lemma is well known and is due to [7] (see [24, 43, 44, 51, 54]); we prove additional estimates which are important for our purposes. In the following lemma we consider smooth function on Σ.

Lemma 5.3. Let{Bj}^p_j=1 and{B_j⁰}^p_j=1be two Dirichlet systems onσ. Then there exist smooth differential operators Λkj,1 ≤j ≤k ≤p, of order k−j containing only tangential derivatives ∂1, . . . , ∂_n−1 such that

B_k⁰ =

k

X

j=1

ΛkjBj, k= 1, . . . , p, on σ (5.12) whereΛ_kk is a function which vanishes nowhere onσ. In addition, we have

kΛkjki≤C[Bj, . . . , Bk;B⁰_k]i+k−j, 1≤j≤k, i= 0,1, . . . (5.13) with some constant C >0 depending only on i, n, k and on k1/σkk^∞₀ whereσk is the nonvanishing coefficient of the term∂_n^k−1 inBk.

Proof. Let first B_k⁰ = ∂^k−1_n . We assume that Bk =Pk

j=1Γkj∂_n^j−1 where Γkj has orderk−j and Γkk is a function not vanishing onσ. We have

∂_n^k−1= Γ⁻¹_kkB_k−Γ⁻¹_kk

k−1

X

j=1

Γ_kj∂_n^j−1=

k

X

j=1

Λ_kjB_j (5.14) where Λ_kk= Γ⁻¹_kk and Λ_kj=−Γ⁻¹_kk Pk−1

l=j Γ_klΛ_lj, j < k. Forj < kwe get kΛ_kjk_i≤C

k−1

X

l=j i

X

m=0

[B_k]_i−mkΓ_klΛ_ljk_m≤C

k−1

X

l=j

[B_k; Λ_lj]_i+k−l (5.15) from (5.8), (5.9) andkΛkkki≤C[Bk]i. By induction we see that

kΛkjki≤C[B_j, . . . , B_k]_i+k−j (5.16) for allk. In the general case we may write

B_l⁰ =

l

X

k=1

Ψ_lk∂_n^k−1=

l

X

k=1 k

X

j=1

Ψ_lkΛ_kjB_j =

l

X

j=1

Φ_ljB_j, l= 1, . . . , p (5.17) where Ψlk are tangential operators of orderl−kand Ψlldoes not vanish onσ; here Λkj as above and Φlj=Pl

k=jΨlkΛkj. From (5.8), (5.16) we get kΦljki≤C

l

X

k=j i

X

m=0

kΨlkk_i−mkΛkjk_l−k+m≤C[Bj, . . . , Bl;B_l⁰]_i+l−j (5.18)

which proves the result.

The assertion of Lemma 5.3 is invariant w.r.t. normal coordinate transformations (cf. [54]). In Theorem 5.4 we follow [54], Theorem 14.1.

Theorem 5.4. Let{Bj}^p_j=1be a smooth normal system. Then there exists a linear map R : H^∞(∂Ω)^p → H^∞(Ω) such that BjRg = gj for j = 1, . . . , p and any g={gj}^p_j=1∈H^∞(∂Ω)^p. There is b≥0 such that

kRgkk≤C

p

X

j=1

[Bj, . . . , Bp;gj]b,k, k= 0,1,2, . . . (5.19) for allg∈H^∞(∂Ω)^p whereC depends only on k, p, n,Ω and on

sup{|B_j^P(x, ν(x))|+|B_j^P(x, ν(x))|⁻¹:x∈∂Ω, j= 1, . . . , p}. (5.20) Proof. We may assume that{B_j}^p_j=1 is a Dirichlet system. We choose forx∈∂Ω an open neighbourhood Ux in Rⁿ and a normal diffeomorphism Ux ↔ {x∈Rⁿ :

|x_i|<1, i= 1, . . . , n} whereU_x∩Ω↔Σ, U_x∩∂Ω↔σ, U_x∩Ω^c↔ −Σ. We cover

∂Ω by finitely many open setsU_i=U_xi and choose a subordinate partition of unity αi. We writeDj=∂^j−1/∂ν^j−1. By Lemma 5.3 we get on Ui∩∂Ω forj= 1, . . . , p a representation

B_j =

j

X

l=1

Λⁱ_jlD_l, D_j =

j

X

l=1

Φⁱ_jlB_l,

j

X

l=m

Λⁱ_jlΦⁱ_lm =δ_jm (5.21) where Λⁱ_jl,Φⁱ_jl are tangential differential operators of order j−l and Λⁱ_jj,Φⁱ_jj do not vanish onUi∩∂Ω. Note that (5.16) holds for Φjl. We choose smooth functions βi such thatβi= 1 on an open neighbourhoodVi of suppαi and suppβi ⊂Ui. Let Z^p be the extension operator from Lemma 5.2. We define

Rg=X

i

β_iZ^p{

j

X

l=1

Φⁱ_jl(α_ig_l)}^p_j=1. (5.22) Letvi =βiZ^p(wⁱ_j) andw_jⁱ =P

Φⁱ_jl(αigl) as in (5.22). We claim thatDjvi=w_jⁱ on Ui∩∂Ω. This holds onVi∩∂Ω by Lemma 5.2. On (Ui\Vi)∩∂Ω all derivatives of order≤p−1 ofZ^p(w_jⁱ) vanish; the normal derivatives arewⁱ_j = 0 and the tangential derivatives vanish sinceZ^p(w_jⁱ) = 0 on (U_i\V_i)∩∂Ω. ThusD_jv_i= 0 =w_jⁱ on this set. We obtain

BjRg=X

i

Bj(vi) =X

i j

X

l=1

Λⁱ_jlwⁱ_l=X

i j

X

m=1 j

X

l=m

Λⁱ_jlΦⁱ_lm(αigm) =gj. By Lemma 5.2 we havekZ^p{gj}kk≤P

kgjkk+a for somea≥0. We get kRgk_k ≤CX

i p

X

j=1 j

X

l=1 k+a

X

m=0

kΦⁱ_jlk_mkα_ig_lk_{k+j+a−l−m}

≤C⁰X

i p

X

j=1 j

X

l=1 k+a

X

m=0

[B_l, . . . , B_j]_m+j−lkg_lk_{k+j+a−l−m}

≤C⁰⁰

p

X

l=1

[Bl, . . . , Bp;gl]_p+a−1,k

which gives the result whereb=p+a−1.