Continuity of attractors for a reaction-diffusion problem with nonlinear boundary conditions with respect to variations of the domain

Continuity of attractors for a reaction-diffusion problem with nonlinear boundary conditions with respect to variations of the domain

Antˆ onio L. Pereira

^∗

Instituto de Matem´atica e Estat´ıstica - Universidade de S˜ao Paulo (IME-USP) R. do Mat˜ao, 1010 - Cidade Universit´aria CEP 05508-090 - S˜ao Paulo - SP BRAZIL

Marcone C.Pereira

^†

Escola de Artes, Ciˆencias e Humanidades - Universidade de S˜ao Paulo

Av. Arlindo Bettio, 1000 - Ermelino Matarazzo CEP 03828-000 - S˜ao Paulo - SP BRAZIL

January 31, 2013

Abstract

In this work we show, for a class of dissipative semilinear parabolic problems, that the global compact varies continuously with respect to parameters in the equations. Applications to a parabolic problem with nonlinear boundary conditions are also obtained.

AMS subject classification: 35B40, 35K20, 58D25.

Keywords: Parabolic equations; attractors; lower semicontinuity; nonlinear boundary condition;

perturbation of the domain.

1 Introduction

Let Ω be an open bounded region in Rⁿ with smooth boundary, a a positive number, and consider the semilinear parabolic problem with nonlinear Neumann boundary conditions

ut(x, t) = ∆u(x, t)−au(x, t) +f(u(x, t)), x∈Ω, t >0

∂u

∂N(x, t) =g(u(x, t)), x∈∂Ω, t >0. (1.1)

It has been show that, under appropriate growth and dissipative conditions on the nonlinearities f and g, the problem (1.1) has a global attractor (see, for example [5] and [12]).

We are interested here in studying the dependence of the global attractor of (1.1) with respect to perturbations of the domain Ω. In fact, we shall prove the attractor changes continuously with respect toregularvariations the domain.

Results on this direction have been obtained in [13] for the problem with Dirichlet boundary conditions. In [4] the authors prove continuity of the attractor for Neumann homogeneous boundary conditions and some kinds of singular

∗e-mail address: alpereir@ime.usp.br. Research partially supported by FAPESP-SP-Brazil, grant 2003/11021-7

†e-mail address: marcone@usp.br.

perturbations of the boundary. The continuity of the equilibria of (1.1) has been proved in [3], where the authors also allow some kinds of singular perturbations.

One of the difficulties here is that the functional spaces change as we change the region. Our first task is then to find a way to compare the attractors of problem (1.1) in different regions. One possible approach is the one taken by Henry in [8] which we describe very briefly . For a different approach, see [4] or [3].

Given an open boundedC^mregion Ω⊂Rⁿ, consider the following open subset ofC^m(Ω,Rⁿ) Diff^m(Ω) = {h∈C^m(Ω,Rⁿ) ;his injective and

1

|deth⁰(x)| is bounded in Ω}.

We introduce a topology in the collection of regions

{h(Ω);h∈Diff^m(Ω)}

by defining a (sub-basis of) the neighborhoods of a given Ω by

{h(Ω);kh−iΩkC^m(Ω,Rⁿ)< ε, ε >0 sufficiently small}.

Michelleti [11] has shown this topology is metrizable. We denote by Mm(Ω) or simply Mm this (separable) metric space. We say that a function F defined in the space Mm with values in a Banach space is C^m or analytic if h7→F(h(Ω)) isC^mor analytic as a map of Banach spaces (hnear iΩ inC^m(Ω,Rⁿ)). In this sense, we may express problems of perturbation of the boundary of a boundary value problem as problems of differential calculus in Banach spaces.

Ifh: Ω7→Rⁿ is aC^k embedding, we may consider the ‘pull-back’ of h h^∗:C^k(h(Ω))→C^k(Ω) (0≤k≤m)

defined by h^∗(ϕ) =ϕ◦h, which is an isomorphism with inverseh^−1∗. Other function spaces can be used instead of C^k, and we will actually be interested mainly in Sobolev spaces and fractional power spaces. IfF_h(Ω):C^m(h(Ω))→ C⁰(h(Ω)) is a (generally nonlinear) differential operator in Ωh = h(Ω) we can consider h^∗F_h(Ω)h^∗−1, which is a differential operator in the fixed region Ω.

Now it is easily seen that v(·, t) satisfies (1.1) in Ωh if and only ifu(·, t) =h^∗v(·, t) (that is, u(x, t) =v(h(x), t)) satisfies

( u_t(x, t) =h^∗∆_Ωhh^∗−1u(x, t)−au(x, t) +f(u(x, t)), x∈Ω, t >0 h^∗_∂N^∂

Ωh

h^∗−1u(x, t) =g(u(x, t)), x∈∂Ω, t >0, (1.2) where h^∗∆Ω_hh^∗−1andh^∗_∂N^∂

Ωh

h^∗−1are defined by

h^∗∆Ω_hh^∗−1u(x) = ∆Ω_h(u◦h⁻¹)(h(x)) and

h^∗ ∂

∂NΩ_h

h^∗−1= ∂

∂NΩ_h

(u◦h⁻¹)(h(x))

(in appropriate spaces). In particular, ifAh is the global attractor of (1.1) inH¹(Ωh), then ˜Ah={v◦h|v∈ Ah} is the global attractor of (1.2) in H¹(Ω) and conversely. In this way we can consider the problem of continuity of the attractors as h→iΩin a fixed phase space.

We will assume that f and g are continuous and satisfy a suitable ‘dissipative condition’ stated in terms of a associate linear system which we describe precisely in the next section.

Under these conditions, it has been proved in [2] that the attractors of (1.2) are uniformly bounded inL^∞, forh in a neighborhood of the inclusion (see also [12] for similar arguments in fractional power spaces). To accomplish that we may work in aL^p-setting withpbig enough so that no growth condition on eitherf org is necessary. After that, we can perform the standard trick of ‘cutting’ f and g outside a ball containing the attractors so as to have them (and as many of their derivatives as wished), globally Lipschitz without changing the attractors. As shown in [2], the attractor do not change if we then change the phase space to a L²-setting. Therefore we may and will, afterwards, work in fractional spaces associated to our operator defined inL²-like spaces.

This paper is organized as follows: in section 2 we fix some notation, recall some background results and state precisely our hypotheses. In section 3 we prove continuity of attractors in a abstract setting. These results are applied to our context in section 4 to obtain the upper and lower continuity of attractors. To this end we needed to prove some auxiliary results of Calculus in Banach spaces. We leave to the last section a technical part of the proof concerning the continuity of the unstable manifolds of the equilibria.

2 Notations and background results

Let Ω be a smooth (C²) region. For eachh∈Diff²(Ω), letBhbe the Laplacian operator with Neumannhomogeneous boundary conditions in h(Ω), that isBh:D(Bh)⊂L²(h(Ω))→L²(h(Ω)), where

D(Bh) =H²(h(Ω))N ={v∈H²(h(Ω))| ∂v

∂N_h(Ω) = 0}

B_hv= ∆_h(Ω)v.

Consider now, the operator Ah =h^∗(Bh−a)h^∗−1 defined in the fixed region Ω. Since h^∗ andh^∗−1 are isomor- phisms (in the appropriate spaces), the operator −Ahis a self-adjoint positive operator in L²(Ω), with

D(−Ah) =H²(Ω)N^∗={u∈H²(Ω)|h^∗ ∂

∂N_h(Ω)h^∗−1u= 0}.

Thus for any λ > 0 the fractional power spaces X_h^α = D((−Ah)^α) generate a scale of Banach spaces which coincide with the complex or real interpolation spaces (see [14]-pp 141-142). We have for α ≥ 0, X_h^α ,→ H^2α and X_h^−α = X_h^α0, where ⁰ means the duality with respect to the bilinear pairing h·,·iL² (see [7]). Furthermore, X_h¹=H²(Ω)_N^∗,X_h^1/2=H¹(Ω) andX_h⁰=L²(Ω).

We may extend the operatorAh to an operatorA_h,−1:L²→X_h⁻¹=H⁻²(Ω) by duality and then we may define, by interpolation, for any 0≤ε≤1, the operatorA_h,−ε:X_h^1−ε→X_h^−ε(see [12] for details).

From now on, we write Ah for A_h,−1/2 to simplify the notation. We also denote by J h the determinant of the Jacobian matrix hx=

∂hi

∂x_j

ⁿ

i,j=1 andv=u◦h⁻¹. Ifu∈D(Bi_Ω), we obtain, integrating by parts

hAhu+au, φi_−1,1 = Z

Ω

h^∗∆h(Ω)h^∗−1u

(x)·φ(x)dx

= Z

Ω

∆h(Ω)(u◦h⁻¹)(h(x))·φ(x)dx

= Z

Ω_h

∆h(Ω)v(y)·φ(h⁻¹(y)) 1

|J h(h⁻¹(y))|dy

= −

Z

Ωh

∇_h(Ω)v(y)· ∇_h(Ω)φ(h⁻¹(y)) 1

|J h(h⁻¹(y))|dy

= −

Z

Ω

∇h(Ω)v(h(x))· ∇h(Ω)φ◦h⁻¹ 1

|J h◦h⁻¹|(h(x))|J h(x)|dx

= −

Z

Ω

h^∗∇_h(Ω)h^∗−1u(x)·h^∗∇_h(Ω)h^∗−1 φ

|J h|(x)|J h(x)|dx. (2.1) Since (2.1) is well defined for anyu∈H¹(Ω), it gives the expression ofAh for anyuin its domain.

Supposev(·, t) is a solution of (1.1) in Ωh. Multiplying byψ∈H¹(h(Ω)) and integrating by parts, we obtain Z

h(Ω)

v_tψ dy = − Z

h(Ω)

∇v· ∇ψ dy+ Z

h(Ω)

(f(v)−a)ψ dy+ Z

∂h(Ω)

ψ g(v)dσ(y) Changing variables back to the fixed region Ω, we obtain

Z

Ω

v_t(h(x))ψ(h(x))|J h(x)|dx = − Z

Ω

∇h(Ω)v(h(x))· ∇h(Ω)ψ(h(x))|J h(x)|dx +

Z

Ω

(f((v◦h)(x))−av◦h(x))ψ(h(x))|J h(x)|dx +

Z

∂Ω

g((v◦h)(x))ψ(h(x))|J_∂Ωh(x)|dσ(x) (2.2) for allψ∈H¹(h(Ω)), whereJ his the determinant of the Jacobian matrixh_x=_∂h

i

∂xj

ⁿ

i,j=1

andJ_∂Ωhthe determinant of the Jacobian matrix of the diffeomorphismh:∂Ω→∂h(Ω).

For eachh∈Diff²(Ω) consider the (nonlinear) operators:

(i) Fh=F(·, h) :H^r(Ω)→H⁻¹(Ω) defined by

hFh(u), φi_−1,1= Z

Ω

f(u)φ dx (2.3)

(ii) Gh=G(·, h) :H^r(Ω)→H⁻¹(Ω) defined by hG_h(u), φi_−1,1=

Z

∂Ω

Γ(g(u)) Γ(φ)

J_∂Ωh J h

dσ(x) (2.4)

where ¹₂ ≤r≤1 and Γ :H^r(Ω)→H^r−12(∂Ω) is the trace map.

We also denoteHh=H(·, h) :=Fh+Gh.

In view of (2.1) and (2.2), it is natural to consider our problem in the abstract form:

˙

u=Ahu+Hh(u), t > t0

u(t0) =u0∈H^r(Ω) (2.5)

with ¹₂ ≤r <1.

In fact, it has been proved in [12] that, with appropriate hypotheses, (2.5) is ‘equivalent’ to problem (1.2). We now state precisely these hypotheses.

(H1)f ∈C¹(R,R) andg∈C²(R,R) are bounded functions with bounded derivative such that, there are constants c0 andd0 with

lim sup

|s|→∞

f(s) s ≤c0

lim sup

|s|→∞

g(s) s ≤d₀ (H2)c₀andd₀ are such that the first eigenvalueµ₁ of the problem

−∆u+ (a−c0)u=µuin Ω

∂u

∂NΩ

=d0uon∂Ω is positive.

Remark 2.1 Since h^∗∆_h(Ω)h^∗−1 is analytic inhthe hypotheses (H2) also holds in h(Ω) if his close enough to the inclusion of i: Ω,→Rⁿ.

Under hypotheses(H1),(H2), it has been shown that problem (1.2) is well posed, its solutions are globally defined and admits a global compact attractor (see [1] or [12]). As shown in [2] these attractors are uniformly bounded inL^∞ forhclose enough to the inclusion in Ω.

3 Continuity of attractors in an abstract setting

We work first in an abstract setting.

Lemma 3.1 Suppose A is a sectorial operator with k(λ−A)⁻¹k ≤ _|λ−a|^M for all λ in the sector S_a,φ0 ={λ | φ₀ ≤

|arg(λ−a)| ≤ π, λ 6= a}, for some a ∈ R and 0 ≤ φ0 < π/2. Suppose also that B is a linear operator with D(B) ⊃ D(A) and kBx−Axk ≤ εkAxk+Kkxk, for any x ∈ D(A), where K and ε are positive constants with ε≤_4(1+LM)¹ , K≤

√5 20M

√2L−1

L²−1 , for someL >1.

ThenB is also sectorial. More precisely, ifb= _L^L2−1² a−

√ 2L

L²−1|a|,φ= max{φ0,^π₄} andM⁰= 2M√ 5 then k(λ−B)⁻¹k ≤ M⁰

|λ−b|, in the sector Sb,φ={λ| φ≤ |arg(λ−b)| ≤π, λ6=b}.

P

roof. Simple computations show that in the sectorSb,φ, we have

|λ|

|λ−a| ≤L, (3.1)

|λ−a| ≥

√2L−1 L²−1

|a|, (3.2)

|λ−b|

|λ−a| ≤√

5. (3.3)

Therefore,

kA(λ−A)⁻¹k ≤ k(A−λ)(λ−A)⁻¹k+|λ|k(λ−A)⁻¹k

=kIk+|λ|k(λ−A)⁻¹k

≤1 +|λ|_|λ−a|^M

≤1 +LM by (3.1). (3.4)

Thus

k(A−B)(λ−A)⁻¹k ≤εkA(λ−A)⁻¹k+Kk(λ−A)⁻¹k

≤ε(1 +LM) +K_|λ−a|^M ≤¹₂

by (3.2). Therefore, I+ (A−B)(λ−A)⁻¹ is invertible withk[I+ (A−B)(λ−A)⁻¹]⁻¹k ≤2. From this we obtain k(λ−B)⁻¹k = k(λ−A+A−B)⁻¹k

= k

(I+ (A−B)(λ−A)⁻¹)(λ−A)−1

k

= k(λ−A)⁻¹(I+ (A−B)(λ−A)⁻¹)⁻¹k

≤ k(λ−A)⁻¹kk(I+ (A−B)(λ−A)⁻¹)⁻¹k

≤ 2M

|λ−a|

= 2M

|λ−b|

|λ−b|

|λ−a|

≤ 2M√ 5

|λ−b| (3.5)

by (3.3) as claimed.

Remark 3.2 Observe thatb can be made arbitrarily close toaby takingLsufficiently large. In particular, ifa >0 then b >0.

Theorem 3.3 Suppose that Ais as in Lemma 3.1, Λ a topological space and{A_γ}_γ∈Λ is a family of operators in X with A_γ0 =Asatisfying the following conditions:

1. D(Aγ)⊃D(A), for allγ∈Λ;

2. kAγx−Axk ≤ (γ)kAxk+K(γ)kxk for any x ∈ D(A), where K(γ) and (γ) are positive functions with lim_γ→γ0(γ) = 0 andlim_γ→γ0K(γ) = 0.

Then, there exists a neighborhood V of γ0 such that Aγ is sectorial if γ ∈ V and the family of (linear) semigroups e^−tAγ satisfy

ke^−tAγ−e^−tAk ≤C(γ)e^−bt kA e^−tAγ−e^−tA

k ≤C(γ)1 te^−bt kA^α e^−tAγ−e^−tA

k ≤C(γ)1

t^αe^−bt, 0< α <1 fort >0, whereb is as in Lemma 3.1 and C(γ)→0 asγ→γ0.

P

roof. Ifγ is sufficiently close toγ₀ we haveε(γ)≤ _4(1+LM)¹ andK(γ)≤

√5 20M

√2L−1

L²−1 . To simplify the notation we suppress, from now on, the dependence of K andεonγ. By Lemma 3.1,Aγ is sectorial and the estimate

k(λ−A_γ)⁻¹k ≤ M⁰

|λ−b|

holds in the sectorS_b,φ={λ|φ≤ |arg(λ−b)| ≤π, λ6=b}withM⁰ = 2√

5M;M,b andφare independent ofγ.

If Γ is a contour in−Sb,φ with|argλ−b| →π−φas |λ| → ∞ then, for anyxinX e^−Aγtx−e^−Atx= 1

2πi Z

Γ

(λ+Aγ)⁻¹x−(λ+A)⁻¹x e^λtdλ.

We estimate the integrand as follows. Firstly we have, for λ∈S_b,φ k(λ−A_γ)⁻¹−(λ−A)⁻¹k ≤ k(λ−A_γ)⁻¹

I−(λ−A_γ)(λ−A)⁻¹ k

≤ k(λ−A_γ)⁻¹

I−(λ−A+A−A_γ)(λ−A)⁻¹ k

≤ k(λ−A_γ)⁻¹

(A−A_γ)(λ−A)⁻¹ k

≤ k(λ−Aγ)⁻¹kk(A−Aγ)·(λ−A)⁻¹)k. (3.6) Proceeding as in the proof of Lemma 3.1, we obtain

k(A−Aγ)·(λ−A)⁻¹)k ≤ε(1 +LM) +K M

|λ−a|. It follows that

k(λ−Aγ)⁻¹−(λ−A)⁻¹k ≤ M⁰

|λ−b| ε(1 +LM) +K M

|λ−a|

. Therefore,

ke^−Aγt−e^−Atk ≤ 1 2π

Z

Γ

k(λ+Aγ)⁻¹−(λ+A)⁻¹k|e^λt||dλ|

≤ M⁰ 2π

ε(1 +LM) +M K(L²−1) (√

2L−1)|a|

e^−bt

Z

Γ

|e^(λ+b)t|

|λ+b| |dλ|

≤C1(γ)e^−bt Z

Γ

|e^µ|

|µ||dµ|,

where C1(γ)→0 asγ→0, as claimed.

From the resolvent identity, we obtain

kA (λ−Aγ)⁻¹−(λ−A)⁻¹

k ≤ kA(λ−Aγ)⁻¹kk(A−Aγ)·(λ−A)⁻¹k. (3.7) Proceeding as in Lemma 3.1

kA(λ−A_γ)⁻¹k ≤ k(A−A_γ)(λ−A_γ)⁻¹k+kA_γ(λ−A_γ)⁻¹k

≤εkA(λ−Aγ)⁻¹k+_|λ−a|^KM + 1 +LM⁰ (3.8) and

k(A−Aγ)(λ−A)⁻¹k ≤ε(1 +LM) + KM

|λ−a|. (3.9)

From (3.8) and (3.9), we obtain

kA (λ−Aγ)⁻¹−(λ−A)⁻¹ k

≤ 1 1−ε

KM

|λ−a|+ 1 +LM⁰

ε(1 +LM) + KM

|λ−a| =C2(γ), where C₂(γ)→0 asγ→0. Then we have

kA e^−Aγt−e^−At k ≤ 1

2π Z

Γ

kA (λ+Aγ)⁻¹−(λ+A)⁻¹

k|e^λt||dλ|

≤ 1

2πC2(γ)e^−bt Z

Γ

|e^(λ+b)t| |dλ|

≤ 1

2πC₂(γ)e^−bt t

Z

Γ

|e^µ|

|µ||dµ|.

The last inequality follows immediately from [7, Theorem 1.4.4].

We now recall some results on spectral projections and exponential bounds (see [7]). Letσ(A) denote the spectrum of A. A set σ⊂σ(A)∪ {∞}is called a spectral set if bothσ and (σ(A)∪ {∞})\σare closed in the extended plane C∪ {∞}.

Supposeσ₁ is a bounded spectral set andσ₂ =σ(A)\σ₁, soσ₂∪ {∞} is another spectral set. LetE₁, E₂ be the projections associated with these spectral sets. If Xj =Ej(X), j = 1,2, then X =X1⊕X2, the Xj are invariant under Aand, ifAj is the restriction of AtoXj, then

A1:X1−→X1 is bounded, σ(A1) =σ1; A₂:D(A₂) =D(A)∩X₂−→X₂ is sectorial, σ(A₂) =σ₂.

If Re(σ2) > β, then ||(λ−A2)⁻¹x|| < _|λ−β|^M2 for all λ in the sector Sβ,φ₂ ={λ | φ2 ≤ |arg(λ−β)| ≤ π, λ 6=β}, 0≤φ2< π/2, and some constantM2and the estimates

||e^−A2t|| ≤Ce^−βt, ||Ae^−A2t|| ≤Ct⁻¹e^−βt hold for t >0.

If Re(σ1)<−α, then

||e^−A1t|| ≤Ce^αt fort≤0.

If A and {Aγ}γ∈Λ are as in Theorem 3.3, the corresponding family of projections satisfy good approximation properties, as shown by the result below.

Lemma 3.4 SupposeAand{Aγ}γ∈Λ are as in Theorem 3.3 andσ(A) =σ₁∪σ₂, whereσ₁ andσ₂∪ {∞}are spectral sets with Re(σ₁)<−α <0 and Re(σ₂)> β >0. Then, if γ is sufficiently small, we also haveσ(A_γ) =σ_1γ∪σ_2γ, Re (σ_1γ)<−α < 0 and Re (σ_2γ)> β >0. If we denote by E₁ (resp. E_1γ), E₂ (resp. E_2γ) the corresponding spectral projections, then

||Ejγ−Ej|| →0 and ||A(Ejγ−Ej)|| →0 as γ→γ0, for j= 1,2.

P

roof. Let Γ be a closed continuous curve in the left half plane enclosingσ₁. It follows from [9, Theorems 4.2.14 and 4.3.16] that σ_1γ is also in the interior of Γ for γ sufficiently close toγ₀, whileσ_2γ is in the exterior of another curve passing throughλ=β enclosingσ₁. Therefore, we have

E1γx= 1 2π

Z

Γ

(λ−Aγ)⁻¹x d λ, (3.10)

where Γ is traversed counterclockwise.

We prove the second claim. The first one is similar but easier.

Since Γ is a compact curve in the resolvent set ofA, there exist constantsRandρsuch that||(λ−A)⁻¹|| ≤Rand

|λ| ≤ρfor anyλ∈Γ. Arguing as in in (3.4), we obtain for anyλ∈Γ

||A(λ−A)⁻¹|| ≤ ||I||+|λ| ||(λ−A)⁻¹|| ≤1 +ρR Therefore

||(A−Aγ)(λ−A)⁻¹|| ≤(γ)||A(λ−A)⁻¹||+K(γ)||(λ−A)⁻¹|| ≤(γ)(1 +ρR) +K(γ)R (3.11) From this, it follows that||(A−Aγ)(λ−A)⁻¹|| ≤ ¹₂|| ifγ is sufficiently close toγ0. Thus, working as is in (3.5)

||(λ−A_γ)⁻¹|| ≤ ||(λ−A)⁻¹|||| I+ (A−A_γ)(λ−A)⁻¹−1

|| ≤2R. (3.12)

From (3.8) and (3.12) we obtain, arguing as in (3.4)

||Aγ(λ−Aγ)⁻¹|| ≤ ||I||+|λ|||(λ−Aγ)⁻¹|| ≤1 + 2ρR. (3.13) Thus, using (3.12) and (3.13)

||A(λ−Aγ)⁻¹|| ≤ ||(A−Aγ)(λ−Aγ)⁻¹||+||Aγ(λ−Aγ)⁻¹||

≤ (γ)||A(λ−A_γ)⁻¹||+K(γ)||(λ−A_γ)⁻¹||+||A_γ(λ−A_γ)⁻¹||

≤ (γ)||A(λ−Aγ)⁻¹||+K(γ)2R+ 1 + 2ρR, from which it follows that

||A(λ−Aγ)⁻¹|| ≤ 1

1−(γ){(2K(γ) + 2ρ)R+ 1} (3.14)

Therefore we obtain, by the resolvent identity, (3.11) and (3.14)

||A (λ−Aγ)⁻¹−(λ−A)⁻¹

|| ≤ ||A(λ−Aγ)⁻¹|| ||(A−Aγ)(λ−A)⁻¹||

≤ 1

1−(γ){(2K(γ) + 2ρ)R+ 1}+(γ)(1 +ρR) +K(γ)R, where C4(γ) = _1−(γ)¹ {(2K(γ) + 2ρ)R+ 1}+(γ)(1 +ρR) +K(γ)R→0, as γ→0.

From (3.10), we then obtain

||A(E1γ−E1)x|| ≤ 1 2π

Z

Γ

||A (λ−Aγ)⁻¹x−(λ−Aγ)⁻¹x d λ||

≤ 1 2π

Z

Γ

C4(γ)d|λ| ||x||

≤

C4(γ) 2π

Z

Γ

|d λ|

||x||

and the result for E_1γ is proved. Since E_2γ =I−E_1γ,E₂=I−E₁ the result also follows easily for them.

Theorem 3.5 With the same hypotheses and notations of Lemma 3.4 the family of (linear) semigroupse^−tAγ satisfy ke^−tAγE_2γx−e^−tAE₂xk ≤C₂(γ)e^−βt||x||, kA e^−tAγE_2γx−e^−tA

E₂xk ≤C₂(γ)1

te^−βt||x||

kA^α e^−tAγE_2γx−e^−tAE₂x

k ≤C₂(γ)1

t^αe^−βt||x||, 0< α <1 fort >0, whereC₂(γ)→0 asγ→γ₀.

ke^−tAγE1γx−e^−tAE1xk ≤C1(γ)e^αt||x||

fort <0, whereC1(γ)→0 asγ→γ0.

P

roof. DenoteA_j =AE_j, A_jγ =A_γE_jγ j = 1,2. We prove the estimates fore^−tAγE_2γ =e^−tA2γE_2γ. Since the operators A_1γ are bounded, the corresponding estimates are simpler.

We first show the familyA2γ satisfies an estimate analogous to the one in the hypothesis of Theorem 3.3. We have, for any, for any xin D(A)

||A2γx−A2x|| ≤ ||AγE2γx−AE2γx||+||AE2γx−AE2x||

≤ (γ)||AE2γx||+K(γ)||E2γx||+||AE2γx−AE₂x||

≤ (1 +(γ))||AE2γx−AE2x||+(γ)||AE2x||+K(γ)||E2γx||

≤ {(1 +(γ))||AE_2γ−AE₂||+K(γ)||E_2γ||} ||x||+(γ)||AE₂x||

≤ C2(γ)||x||+(γ)||AE2x|| (3.15)

with C₂(γ)→0 asγ→γ₀by Lemma 3.4.

We now recall thatA2γ is sectorial and, ifγis close enough toγ0an estimate of the form k(λ−A2γ)⁻¹k ≤ M2

|λ−β|

holds in the sector S_β,φ={λ| φ≤ |arg(λ−β)| ≤π, λ6=β}, while a similar estimate holds forA_γ in the sectorS_b,φ (withb≤β).

If Γ is a contour in−Sb,φ with|argλ−b| →π−φas |λ| → ∞ then, for anyxinX e^−AγtE2γx−e^−AtE2x= 1

2πi Z

Γ

(λ+Aγ)⁻¹E2γx−(λ+A)⁻¹E2x e^λtdλ.

Now, ifλis in the resolvent set ofAγ then (λ+Aγ)⁻¹E2γ = (λ+A2γ)⁻¹. Therefore e^−AγtE2γx−e^−AtE2x= 1

2πi Z

Γ

(λ+A2γ)⁻¹x−(λ+A2)⁻¹x e^λtdλ.

But, since the resolvent set of A_2γ contains the sectorS_β,φ, by Cauchy’s Theorem, the contour Γ can now be shifted to a contour ¯Γ in−S_β,φ with|argλ−β| →π−φas|λ| → ∞. Thus, for anyxinx

e^−AγtE_2γx−e^−AtE₂x = 1 2πi

Z

Γ¯

(λ+A_2γ)⁻¹x−(λ+A₂)⁻¹x e^λtdλ

= e^−A2γtx−e^−A2tx.

Using (3.15) we can see that the familyA2γ satisfies the hypotheses of Theorem 3.3, from which the result follows.

Lemma 3.6 SupposeY is a Banach space, Λ is an open set inY, {−Aλ}_λ∈Λ is a family of operators in a Banach spaceX satisfying the conditions of Theorem 3.3 atλ=λ0,U is an open set inR⁺×X^α,0≤α <1andf :U×Λ→X is H¨older continuous int, continuous inλatλ0 uniformly for(t, x)in bounded subsets ofU,kf(t, x, λ)−f(t, y, λ)k ≤ Lkx−ykα,||f(t, x, λ0)|| ≤R for(t, x),(t, y)inU andλ∈Λ. Suppose the solutionx(t, x0, λ) of the problem

dx

dt =A_λx+f(t, x, λ), t > t₀ x(t₀) =x₀

(3.16) exists for x₀ in bounded subsets ofX^α,λin a neighborhood of λ₀ andt₀≤t≤T.

Then the function λ 7→ x(t, x₀, λ) ∈ X^α is continuous at λ₀ uniformly for x₀ in bounded subsets of X^α and t0≤t≤T .

P

roof. Letb be the exponential rate of decay of the semigroup generated by Aλ, λ in the neighborhood ofλ0, given by Theorem 3.3. We write xλ(t) and x(t) for the solutions of (3.16) with parameter valuesλ and λ0. If x0

belongs to a bounded setB ofX^αwe have, by the variation of constants formula kx_λ(t)k_α ≤ ke^Aλ(t−t0)x₀k_α+

Z t t0

ke^Aλ(t−s)f(s, x_λ(s), λ)k_αds

≤ e^−b(t−t0)kx₀k_α+R Z t

t₀

e^−b(t−s)(t−s)ds

so xλ(t) remains in a bounded set forx0∈B andt−t0 bounded. By hypothesis there exists then a a function θ(λ) such thatθ(λ)→0 asλ→λ0andkf(s, x(s), λ)−f(s, x(s), λ0)k ≤θ(λ), forx0∈B. Using again the the variation of constants formula, we obtain

kxλ(t)−x(t)kα≤ k[e^Aλ(t−t0)−e^A(t−t0)]x0kα+ Z t

t0

ke^Aλ(t−s)[f(s, xλ(s), λ)

−f(s, x(s), λ)]k_αds+ Z t

t₀

ke^Aλ(t−s)[f(s, x(s), λ)−f(s, x(s), λ₀)]k_αds +

Z t t₀

k[e^Aλ(t−s)−e^A(t−s)]f(s,(x(s), λ)kαds

≤C(λ)e^−b(t−t0)kx0kα+LM Z t

t₀

(t−s)^−αe^−b(t−s)kxλ(s)−x(s)kαds +θ(λ)M

Z t t₀

(t−s)^−αe^−b(t−s)ds+C(λ)R Z t

t₀

(t−s)^−αe^−b(t−s)ds (C(λ) +θ(λ))N+LM

Z t t0

(t−s)^−αkxλ(s)−x(s)kαds whereM is suchke^Aλ(t−t0)k ≤M e^−b(t−t0)andN =kx₀k_α+ (M +R)Rt

t0(t−s)^−αe^−b(t−s)ds.

It follows from Gronwall’s inequality (see [7]), thatkxλ(t)−x(t)kα≤(C(λ) +θ(λ))NM, where M=M(L, M, N, α, T)<∞. This proves the claim.

One can prove the same result under slightly different hypotheses.

Lemma 3.7 SupposeY is a metric space,Λis an open set inY,{−Aλ}_λ∈Λis a family of operators in a Banach space X satisfying the conditions of Theorem 3.3 atλ=λ0,U is an open set inR⁺×X^α,0≤α <1 and f :U×Λ→X is H¨older continuous in t. Suppose also that, for any bounded subset D⊂U,f is continuous inλatλ0 uniformly for (t, x)inD, and there is a constantL=L(D), such thatkf(t, x, λ)−f(t, y, λ)k ≤Lkx−ykα for(t, x),(t, y)inD and λ∈Λ. Suppose further that the solutionsx(t, x0, λ)of the problem

dx

dt =A_λx+f(t, x, λ), t > t₀ x(t0) =x0

exist and remain in a bounded subset of X^α whenx₀ varies in a bounded subset ofX^α,λin a neighborhood of λ₀ and t0≤t≤T.

Then the function λ 7→ x(t, x0, λ) ∈ X^α is continuous at λ0 uniformly for x0 in bounded subsets of X^α and t0≤t≤T .

P

roof. The proof is similar to Lemma 3.6. We now don’t need now to prove boundedness of the solutions which is part of the hypotheses, but we have to show that the image byf remain in a bounded set using the Lipschitz property.

Lemma 3.8 Suppose, in addition to the hypotheses of Lemma 3.6, that the derivative ^∂f_∂x(t, x, λ)exists, is continuous and bounded for 0≤t≤T, λin a neighborhood of λ0. Then, the map λ7→ ^∂x(t,x_∂x^0,λ)

0 is continuous at λ0 uniformly forx₀ in a bounded subset ofX^αandt₀≤t≤T.

P

roof. The proof is similar to the one of Lemma 3.16, once we observe that the derivative v_λ(t) =^∂x(t,x_∂x^0,λ)

0 ·∆x₀ is the solution of the initial (linear) value problem

dy

dt =Aλy+fx(t, x(t), λ)y, t > t0

y(t₀) = ∆x₀. (See Lemma 3.2 of [13], for details).

We recall now that the family of subsetsAλof a metric space (X, d) is said to beupper-semicontinuousatλ=λ0

if δ(Aλ,Aλ₀)→ 0 asλ→ λ0, where δ(A, B) = sup_x∈Ad(x, B) = sup_x∈Ainfy∈Bd(x, y) and lower-semicontinuousif δ(Aλ₀,Aλ)→0 asλ→λ0.

Theorem 3.9 SupposeY is a metric space,Λ is an open set in Y,{−Aλ}_λ∈Λ is a family of operators in a Banach space X satisfying the conditions of Theorem 3.3 at λ = λ0 with b > 0, U is an open set in X^α, 0 ≤ α < 1 and f :U×Λ→X satisfying the conditions of Lemma 3.7. LetT(t, λ)(x0)be the nonlinear semigroup inX^α given by the solutions of the problem

dx

dt =Aλx+f(x, λ), t > t0

x(t0) =x0.

(3.17) Suppose also that, for each λ∈ Λ there exists a global compact attractor A_λ forT(t, λ)(x), the union∪_λ∈ΛA_λ is a bounded set in X andf maps this union into a bounded set of X. Then the family A_λ is upper semicontinuous.

P

roof. Ifkf(x)k ≤N for anyx∈ ∪λ∈ΛAλ then, taking an initial conditionx0∈ Aλ

kT(t, λ)x₀k ≤ ke^Aλ(t−t0)x₀kα+ Z t

t₀

ke^Aλ(t−s)f(x_λ(s), λ)kαds

≤ (t−t0)^−αe^−b(t−t0)kx0k+N Z t

t₀

(t−t0)^−αe^−b(t−s)(t−s)ds.

Taking t−t0 = 1, and observing that T(1, λ)(Aλ) = Aλ, we obtain that ∪_λ∈ΛAλ is also in a bounded set B of X^α. Given ε > 0, there exists then a t0, such that δ(T(t0, λ0)B,Aλ₀) ≤ ε/2. Now, if λ is close enough to λ0, we have from Lemma 3.7, δ(T(t0, λ)(B), T(t0, λ0)(B)) ≤ ε/2 and, therefore δ(T(t0, λ)B,Aλ₀) ≤ ε. Since Aλ=T(t0, λ)(Aλ)⊂T(t0, λ)(B) it follows thatδ(Aλ,Aλ₀)≤ε. This proves the result.

Lower semicontinuity can also be proved under some additional hypotheses.

Theorem 3.10 Suppose, in addition to the hypotheses of Theorem 3.9, that the system generated by (3.17) is gradient for any λand its equilibria are all hyperbolic and continuous in λ. Then the family Aλ is also lower semicontinuous.

P

roof. The result follows from results in [6] (see also [4]), once continuity of the unstable manifolds of the equilibria is proved. We leave the proof of this last fact to the section 5.

4 Continuity of attractors for the Neumann problem with nonlinear boundary conditions

We now verify that the hypotheses of Theorems 3.9 and 3.10 hold for problem (2.5). We first consider the linear problem.

Lemma 4.1 The family of operators{−A_h}_h∈Diff2(Ω) defined by (2.1) satisfies the hypotheses of the Lemma 3.3.

P

roof. The hypotheses that −AiΩ is a sectorial operator and D(−Ah) ⊂ D(−AiΩ) for all h ∈ Diff²(Ω) are immediate. We only need to show that there exist positive functions (h) andK(h) such that

k(A_h−A_iΩ)uk_H−1(Ω)≤(h)kA_iΩuk_H−1(Ω)+K()kuk_H−1(Ω)

for allu∈D(−Ai_Ω), where lim_h→iΩK(h) = 0 and lim_h→iΩ(h) = 0. More precisely, we show

| h(Ah−Ai_Ω)u, ψi_−1,1| ≤(h)kuk_H1(Ω)kψk_H1(Ω)

for allu∈H¹(Ω) and for allψ∈H¹(Ω) with lim_h→iΩ(h) = 0.

Firstly we have for allh

| h(Ah−Ai_Ω)u, ψi_−1,1| ≤ Z

Ω

h^∗∇_h(Ω)h^∗−1− ∇Ω

u·h^∗∇_h(Ω)h^∗−1 ψ

|J h|

|J h|dx +

Z

Ω

∇_Ωu·

h^∗∇_h(Ω)h^∗−1− ∇_Ω ψ

|J h|

|J h|dx +

Z

Ω

ψ∇Ωu· ∇Ω

1

|J h|

|J h|dx. (4.1)

We need to express the coefficients ofh^∗∇h(Ω)h^∗−1in terms ofh. To this end, we write h(x) =h(x₁, x₂, . . . , x_n) = (h₁(x), h₂(x). . . , h_n(x)) = (y₁, y₂, . . . , y_n) =y.

Then

h^∗ ∂

∂y_ih^∗−1(u)

(x) = ∂

∂y_i(u◦h⁻¹)(h(x))

=

n

X

j=1

∂u

∂xj

(h⁻¹(y))∂h⁻¹_j (y)

∂yi

(y)

=

n

X

j=1

∂h

∂x −1

j,i(x)∂u

∂x_j(x)

=

n

X

j=1

bi,j(x) ∂u

∂x_j(x),

where bij(x) is thei, jentry in the inverse-transpose of the Jacobian matrixhx= [^∂h_∂xⁱ

j]ⁿ_i,j=1.

Therefore

h^∗∇_h(Ω)h^∗−1− ∇Ω

u(x) =







 Pn

j=1(b1j(x)−δ1j)_∂x^∂u

j(x) ...

Pn

j=1(b_nj(x)−δ_nj)_∂x^∂u

j(x)









with δij = 1 ifi=j and 0 otherwise.

Now, we can estimate the first integral of (4.1) as follows.

R

Ω

h^∗∇_h(Ω)h^∗−1− ∇Ω

u·h^∗∇_h(Ω)h^∗−1

ψ

|J h|

|J h|dx

≤R

Ω







 Pn

j=1(b1j(x)−δ1j)_∂x^∂u

j(x) ...

Pn

j=1(bnj(x)−δnj)_∂x^∂u

j(x)









·h^∗∇h(Ω)h^∗−1 _ψ

|J h|

|J h|dx

≤max_x∈Ω¯{|J h|}R

Ω

nPn i=1

hPn j=1bij ∂

∂x_j

_ψ

|J h|

i2o¹₂n Pn

i=1

hPn j=1

bij−δij

∂u

∂x_j

i2o¹₂ dx

≤max_x∈Ω¯{|J h|}n R

Ω

Pn i=1

hPn j=1bij ∂

∂x_j

_ψ

|J h|

i2

dxo¹₂n R

Ω

Pn i=1

hPn j=1

bij−δij

∂u

∂x_j

i2

dxo¹₂

≤C0(h) R

Ω∇Ωψ· ∇Ωψ dx¹₂ R

Ω∇Ωu· ∇Ωu dx¹₂ +C₀⁰(h)

R

Ωψ²dx¹₂ R

Ω∇Ωu· ∇Ωu dx¹₂

where C₀(h) and C₀⁰(h) are functions depending on b_ij and _∂x^∂

j

1

|J h|

with C₀(h) and C₀⁰(h)→ 0 when h→ i_Ω in C²(Ω,Rⁿ).

Analogously, we estimate the other integrals R

Ω

∇_Ωu·

h^∗∇_h(Ω)h^∗−1− ∇_Ω

ψ

|J h|

|J h|dx

≤max_x∈Ω¯{|J h|}n R

Ω

Pn i=1

h∂u

∂x_j

i2

dxo¹₂n R

Ω

Pn i=1

hPn j=1

b_ij−δ_ij

∂

∂x_j

ψ

|J h|

i2

dxo¹₂

≤C₁(h) R

Ω∇_Ωψ· ∇_Ωψ dx¹₂ R

Ω∇_Ωu· ∇_Ωu dx¹₂ +C₁⁰(h)

R

Ωψ²dx¹₂ R

Ω∇_Ωu· ∇_Ωu dx¹₂ and

R

Ω|ψ∇Ωu· ∇Ω

1

|J h|

| |J h|dx

≤max_x∈Ω¯{|J h|}R

Ωψh Pn

j=1

∂u

∂xj

²i¹₂h Pn

j=1

∂

∂xj

1

|J h|

²i¹₂ dx

≤ C2(h)R

Ωψ²dx¹₂R

Ω|∇Ωu|²dx¹₂

with C1(h),C₁⁰(h) andC2(h) go to 0 whenh→iΩin C²(Ω,Rⁿ). From this, the result follows readily.

We now need to obtain regularity properties for the map H_h =F_h+G_h defined in (2.3) and (2.4). To this end, we first recall some definitions.

Definition 4.2 SupposeX, Y are Banach spaces,O⊂X open,f :O→Y andp∈O;

1. f isGateaux differentiable (or G-differentiable) atpwith G-derivativeδf(p,·) :X →Y if, for each q∈X

t→0∈limR

f(p+tq)−f(p)−tδf(p, q)

t = 0.

2. f is Hadamard differentiable (or H-differentiable) atp with H-derivative f⁰(p) :X → Y if f⁰(p) is a bounded linear operator and, for eachq∈X

lim

(t,k)→(0,0)∈R×X

f(p+tq+tk)−f(p)−tf⁰(p)q

t = 0.

3. f isFr´echet differentiable (or F-differentiable)atpwith F-derivativef⁰(p) :X →Y if f⁰(p)is a bounded linear operator and

q→0∈Xlim

||f(p+q)−f(p)−f⁰(p)q||

||q|| = 0.

It is clear that F-differentiability ⇒ H-differentiability ⇒ G-differentiability. On the other hand, we have the following partial converse

Lemma 4.3 Supposef, X, Y andO are as in Definition 4.2,X₁⊂X is a Banach space and the inclusioni:X₁,→X is compact. Then, iff is Hadamard differentiable atp∈X₁, the compositionf˜=f◦iis Fr´echet differentiable at p.

P

roof. LetT =f⁰(p) denote the H-derivative off at p. It is clear that ˜T =T◦i:X1→Y is a bounded linear operator. We now observe that ˜f is F-differentiable atpif and only if

limt→0

f˜(p+tq)−f(p)˜ −tT q˜

t = 0,

uniformly for||q||= 1 (where||q||denotes the norm inX₁). Suppose this is not the case and let (t_n, q_n) be a sequence in R×X₁ such thatt_n→0,||q_n||= 1 and for someε >0 and alln

||f˜(p+tnqn)−f˜(p)−tT q˜ n

tn

||> ε. (4.2)

By compacity, there exists a subsequence ofq_n which we still denote byq_n, converging inX toq∈X. Therefore

||f˜(p+tnqn)−f˜(p)−tnT q˜ n

t_n ||

=||f(p+tnq+tn(qn−q))−f(p)−tnT q−tnT(qn−q)

t_n ||

≤ ||f(p+tnq+tn(qn−q))−f(p)−tnT q tn

||+||T(qn−q)||

Sincef is H-differentiable atp,

f(p+t_nq+t_n(q_n−q))−f(p)−t_nT q tn

→0 as t_n→0 and q_n →q.

Since T(qn−q) also goes to 0, we reach a contradiction with (4.2).

Lemma 4.4 Supposeg is aC² bounded function with bounded derivatives. Then the map G:H¹²(Ω)×Diff²(Ω)→ H⁻¹(Ω) defined in (2.4) is Hadamard differentiable with respect to u, for any (u, h) ∈ H¹²(Ω)×Diff²(Ω) with H- derivative given by

∂G

∂u(u, h) ˙u, ψ

−1,1

= Z

∂Ω

Γ(g⁰(u) ˙u) Γ(ψ)

J_∂Ωh J h

dσ(x) (4.3)

for any u˙ ∈H¹(Ω).

P

roof. For any (u, h)∈H¹²(Ω)×Diff²(Ω) andψ∈H¹(Ω), letT :H¹²(Ω)→ B(H¹²(Ω), H⁻¹(Ω)) be the map defined by (4.3). To simplify the notation, let θh=θ(h),θ: Diff²(Ω)→C⁰(Ω,R) be given byθ(h) =

J∂Ωh J h

.Then, we have

| hG(u+th+tk)−G(u)−T(u)(th), ψi_−1,1|

≤ Z

∂Ω

|Γ

g(u+t(h+k))−g(u)−g⁰(u)(th)

Γ(θh)Γ(ψ)|dσ(x)

≤ Z

∂Ω

|Γ

g(u+t(h+k))−g(u)−g⁰(u)(th)

Γ(θ_h)|²dσ(x)

^1/2Z

∂Ω

|Γ(ψ)|²dσ(x) ^1/2

≤K Z

∂Ω

|Γ

g(u+t(h+k))−g(u)−g⁰(u)(th)

Γ(θ_h)|²dσ(x) ^1/2

||ψ||_H1(Ω)

where Kis the constant of the embedding H¹(Ω),→L²(∂Ω).

Now

Z

∂Ω

|Γ

g(u+t(h+k))−g(u)−g⁰(u)(th)

Γ(θh)|²dσ(x) 1/2

≤ Z

∂Ω

|Γ

g(u+t(h+k))−g(u+th) +g(u+th)−g(u)−g⁰(u)(th)

Γ(θh)|²dσ(x) 1/2

≤ Z

∂Ω

|Γ

g(u+t(h+k))−g(u+th)

Γ(θ_h)|²dσ(x) 1/2

+ Z

∂Ω

|Γ

g(u+th)−g(u)−g⁰(u)(th)

Γ(θ_h)|²dσ(x) 1/2

. (4.4)

For the first term in (4.4), we have 1

|t|

Z

∂Ω

|Γ

g(u+t(h+k))−g(u+th)

Γ(θh)|²dσ(x) 1/2

≤ Z

∂Ω

|Γ

g⁰(u+t^∗(h+k))(k)

Γ(θh)|²dσ(x) 1/2

≤2||g⁰||∞kθhk∞

Z

∂Ω

|Γ(k)|²dσ(x) 1/2

≤2K0||g⁰||∞kθhk∞||k||

H¹2(Ω) (4.5)

where 0<|t^∗|<|t|andK0 is the constant of the embeddingH¹²(Ω),→L²(∂Ω).

For the second term

1

|t|

Z

∂Ω

|Γ

g(u+th)−g(u)−g⁰(u)(th)

Γ(θh)|²dσ(x) ^1/2

≤ Z

∂Ω

|Γ

(g⁰(u+t^∗h)−g⁰(u))(h)

Γ(θh)|²dσ(x) ^1/2

. (4.6)

The integrand in (4.6) is bounded by the integrable function 4||g⁰||²_∞kθhk²Γ(h)². Furthermore

|Γ

(g⁰(u+t^∗h)−g⁰(u))(h)

Γ(θh)|²(x)≤ |Γ

(g⁰⁰(u+t^∗∗)t^∗)h²Γ(θh)

|²(x)→0 as t→0, for anyx∈∂Ω, since 0<|t^∗∗|<|t^∗|<|t|.

Therefore

1

|t|

Z

∂Ω

|Γ

g(u+th)−g(u)−g⁰(u)(th)

Γ(θ_h)|²dσ(x) ^1/2

→0 (4.7)

as t→0 by Lebesgue’s Dominated Convergence Theorem.

From (4.4), (4.5) and (4.7), it follows that

|1

t hG(u+th+tk)−G(u)−T(u)(th), ψi_−1,1| →0 as (t, k)→(0,0) as claimed.

We will need the following result. Its simple proof is omitted .

Lemma 4.5 SupposeX, Y are Banach spaces and Tn :X →Y is a sequence of linear operators converging strongly to the linear operator T : X → Y. Suppose also that X1 ⊂X is a Banach space and the inclusion i :X1 ,→X is compact and let T˜n=Tn◦iand T˜ =T◦i. Then T˜n →T˜ uniformly forxin a bounded subset ofX1 (that is, in the or norm of B(X1, Y), the space of linear bounded operators fromX1 toY).

Lemma 4.6 Suppose g is a C² bounded function with bounded derivatives and ¹₂ < r ≤ 1. Then the map G : H^r(Ω)×Diff²(Ω) → H⁻¹(Ω) defined in (2.4) is locally Lipschitz continuous in h, uniformly for u in H^r(Ω) and continuously (Fr´echet) differentiable with respect tou, with derivative given by (4.3).

P

roof. Choose an open setV of Diff²(Ω). Letθ: Diff²(Ω)→C⁰(Ω,R) be as in the proof of Lemma 4.4 and fix h0 ∈Diff²(Ω). There is a positive constant c and a neighborhood V of h0 in Diff²(Ω) such that J h(x)> c for all x∈Ω. It follows that|θ(h)(x)−θ(h1)(x)| ≤M||h−h1||_C2, for any h, h1 in V and for some constantM depending only onV. Therefore, we have foru∈H^r(Ω) and h, h1∈V

| hG(u, h)−G(u, h₁), ψi_−1,1| ≤ Z

∂Ω

Γ(ψ) Γ(g(u)) Γ

θ(h)−θ(h₁) dσ(x)

≤ M||g||_∞||h−h1||_C2kΓ(ψ)k_L2(∂Ω)

≤ K M||g||∞||h−h1||_C²kψkH¹(Ω)