Generic simplicity for the solutions of a nonlinear plate equation

Generic simplicity for the solutions of a nonlinear plate equation

A. L. Pereira

^∗

Instituto de Matem´atica e Estat´ıstica da USP

R. do Mat˜ao, 1010 - CEP 05508-900 - S˜ao Paulo - SP BRASIL.

and M. C. Pereira

^†

Escola de Artes, Ciˆencias e Humanidades da USP

Av. Arlindo B´etio, 1000. CEP: 03828-080 - S˜ao Paulo - SP BRASIL

Abstract

In this work we show that the solutions of the Dirichlet problem for a semilinear equation with the Bilaplacian as its linear part are generically simple in the set of C⁴-regular regions.

AMS subject classification: 35J40,35B30.

1 Introduction

Perturbation of the boundary for boundary value problems in PDEs have been inves- tigated by several authors, from various points of view, since the pioneering works of Rayleigh ([11]) and Hadamard ([1]).

In particular, generic properties for solutions of boundary value problems have been considered in [4], [5], [6], [7], [8], [10], [12] and [14].

More recently several works appeared in a related topic, generally known as ‘shape analysis’ or ‘shape optimization’, on which the main issue is to determine conditions for a region to be optimal with respect to some cost functional. Among others, we mention [13] and [15].

Many problems of this kind have also been considered by D. Henry in [2] where a kind of differential calculus with the domain as the independent variable was developed.

This approach allows the utilization of standard analytic tools such as Implicit Function Theorems and the Lyapunov-Schmidt method. In his work, Henry also formulated and

∗Research partially supported by FAPESP-SP-Brazil, grant 2003/11021-7. E-mail address:

alpereir@ime.usp.br

†Research partially supported by CNPq - Conselho Nac. Des. Cient´ıfico e Tecnol´ogico - Brazil.

E-mail address: marcone@usp.br

proved a generalized form of the Transversality Theorem, which is the main tool we use in our arguments.

We consider here the semilinear equation

∆²u(x) +f(x, u(x),∇u(x),∆u(x)) = 0 x∈Ω u(x) =^∂u(x)_∂N = 0 x∈∂Ω

wheref(x, λ, y, µ) is aC⁴ real function inRⁿ×R×Rⁿ×Rwith f(x,0,0,0)≡0 for all x∈Rⁿ.

We show that, for a residual set of regions Ω ⊂ Rⁿ (in a suitable topology), the solutionsuof (1) are all simple, that is , the linearisation

L(u) : ˙u → ∆²u˙+∂f

∂µ(·, u,∇u,∆u)∆ ˙u +∂f

∂y(·, u,∇u,∆u)· ∇u˙+∂f

∂λ(·, u,∇u,∆u) ˙u is an isomorphism.

Our results can be seen as an extension of similar results for reaction-diffusion equa- tions obtained by Saut and Teman ([12]) and Henry ([2]).

This paper is organized as follows: in section 2 we collect some results we need from [2]. In section 3 we prove that the differential operator

L= ∆²+a(x)∆ +b(x)· ∇+c(x) x∈Rⁿ

is, generically, an isomorphism in the set of C⁴-regular regions Ω ⊂Rⁿ. This result is used in section 4 to prove our main result, the generic simplicity of solutions of (1). The most difficult point there is the proof that a certain (pseudo differential) operator is not of finite range. This was proved in a separate work ([9]).

2 Preliminaries

The results in this section were taken from the monograph of Henry [2], where full proofs can be found.

2.1 Differential Calculus of Boundary Perturbation

Given an open bounded, C^m region Ω0 ⊂ Rⁿ, consider the following open subset of C^m(Ω,Rⁿ)

Diff^k(Ω) ={h∈ C^k(Ω,Rⁿ)|his injective and 1/|deth⁰(x)|is bounded in Ω}

We introduce a topology in the collection of all regions {h(Ω) | h ∈ Diff^k(Ω)} by defining (a sub-basis of) the neighborhoods of a given Ω by

{h(Ω0)| kh−i_Ω0k_Ck(Ω0,Rⁿ)< , sufficiently small}.

Whenkh−iΩk_Cm(Ω,Rⁿ)is small,his aC^mimbedding of Ω inRⁿ, aC^mdiffeomorphism to its image h(Ω). Micheletti shows in [4] that this topology is metrizable, and the set of regionsC^m-diffeomorphic to Ω may be considered a separable metric space which we denote byMm(Ω), or simplyMm. We say that a functionF defined in the spaceMm

with values in a Banach space isC^m or analytic ifh7→F(h(Ω)) isC^m or analytic as a map of Banach spaces (hneari_ΩinC^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. More specifically, consider a formal non-linear differential operatoru→v

v(x) =f

x, u(x), Lu(x)

,x∈Rⁿ where

Lu(x) =

u(x), ∂u

∂x₁(x), ..., ∂u

∂x_n(x),∂²u

∂x²₁(x), ∂²u

∂x₁∂x₂(x), ...

,x∈Rⁿ

More precisely, suppose Lu(·) has values in R^p and f(x, λ) is defined for (x, λ) in some open setO⊂Rⁿ×R^p. For subsets Ω⊂Rⁿ defineF_Ωby

F_Ω(u)(x) =f(x, Lu(x)),x∈Ω (1)

for sufficiently smooth functionsuin Ω such that (x, Lu(x))∈O for anyx∈Ω.¯

Leth: Ω −→Rⁿ be C^m imbedding. We define the composition map (orpull-back) h^∗ ofhby

h^∗u(x) = (u◦h)(x) =u(h(x)),x∈Ω

where u is a function defined in h(Ω). Then h^∗ is an isomorphism from C^m(h(Ω)) to C^m(Ω) with inverseh^∗−1= (h⁻¹)^∗. The same is true in other function spaces.

The differential operator

F_h(Ω):DF_h(Ω) ⊂ C^m(h(Ω))−→ C⁰(h(Ω))

given by (1) is called theEulerian form of the formal operatorv7→f(·, Lv(·)), whereas h^∗F_h(Ω)h^∗−1:h^∗D_Fh(Ω) ⊂ C^m(Ω)−→ C⁰(Ω)

is called theLagrangian form of the same operator.

The Eulerian form is often simpler for computations, while the Lagrangian form is usually more convenient to prove theorems, since it acts in spaces of functions that do not depend onh, facilitating the use of standard tools such as the Implicit Function or the Transversality theorem. However, a new variable,his introduced. We then need to study the differentiability properties of the map

(u, h)7→h^∗F_h(Ω)h^∗−1u (2)

This has been done in [2] where it is shown that, if (y, λ)7→f(y, λ) isC^k or analytic then so is the map above, considered as a map fromDif f^m(Ω)× C^m(Ω) toC⁰(Ω) (other function spaces can be used instead of C^m). To compute the derivative we then need

only compute the Gateaux derivative that is, the t-derivative along a smooth curve t7→(h(t, .), u(t, .))∈Dif f^m(Ω)× C^m(Ω). For this purpose, it is convenient to introduce the differential operator

Dt= ∂

∂t−U(x, t) ∂

∂x, U(x, t) = (∂h

∂x)⁻¹∂h

∂t

which is is called theanti-convective derivative. The results below (theorems 2.3, 2.6) are the main tools we use to compute derivatives.

Theorem 1 Suppose f(t, y, λ) is C¹ in an open set in R×Rⁿ×R^p, L is a constant- coefficient differential operator of order≤mwithLv(y)∈R^p (where defined). For open setsQ⊂Rⁿ andC^m functionsv onQ, letF_Q(t)v be the function

y−→f(t, y, Lv(y)),y∈Q.

where defined.

Supposet−→h(t,·) is a curve of imbeddings of an open setΩ⊂Rⁿ,Ω(t) =h(t,Ω) and for |j| ≤ m,|k| ≤ m+ 1 (t, x) −→∂t∂_x^jh(t, x),∂^k_xh(t, x),∂^k_xu(t, x) are continuous onR×Ωneart= 0, andh(t,·)^∗−1u(t,·)is in the domain ofF_Ω(t). Then, at points ofΩ

Dt(h^∗FΩ(t)(t)h^∗−1)(u) = (h^∗F˙Ω(t)(t)h^∗−1)(u) + (h^∗F_Ω(t)⁰ (t)h^∗−1)(u)·Dtu whereDt is the anti-convective derivative defined above,

F˙Q(t)v(y) = ∂f

∂t(t, y, Lv(y)) and

F_Q⁰(t)v·w(y) =∂f

∂λ(t, y, Lv(y))·Lw(y),y∈Q is the linearisation ofv−→FQ(t)v.

E

^XAMPLE. Letf(x, λ, y, µ) be a smooth function inRⁿ×R×Rⁿ×Rand consider the nonlinear differential operator

FΩ(v)(x) = ∆²v(x) +f(x, v(x),∇v(x),∆v(x))

which does not depend explicitly on t. Suppose also thath(t, x) =x+tV(x) +o(t) in a neighborhood oft= 0 andx∈Ω.Then, since _∂t^∂

FΩ(u)

= 0 andF_Ω⁰(u)·w=L(u)w, we have, by Theorem 1

∂

∂t

h^∗F_h(Ω)h^∗−1(u)

= Dt

h^∗F_h(Ω)h^∗−1(u)

_t=0−h⁻¹_x ht∇

h^∗F_h(Ω)h^∗−1(u) _t=0

= h^∗F_h(Ω)⁰ h^∗−1(u)·D_t(u)

_t=0−h⁻¹_x h_t∇

h^∗F_h(Ω)h^∗−1(u) _t=0

= L(u)∂u

∂t −V · ∇u

−V · ∇

∆²u+f(x, u(x),∇u(x),∆u(x)) where

L(u) = ∆²+∂f

∂µ(·, u,∇u,∆u)∆ +∂f

∂y(·, u,∇u,∆u)· ∇+∂f

∂λ(·, u,∇u,∆u).

2.2 Change of origin

We can always transfer the ‘origin’ or reference region from any Ω ⊂ Rⁿ, to another diffeomorphic region. Indeed, if ˜H : Ω → Ω is a diffeomorphism we define, for any˜ imbeddingh: Ω→Rⁿ, another imbedding ˜h=h◦H˜⁻¹: ˜Ω→Rⁿ.

If ˜x= ˜H(x), ˜u= ˜H⁻¹∗u NΩ˜(˜x) =NH(Ω)˜ ( ˜H(x)) = ^H˜xtNΩ(x)

kH˜_x^tNΩ(x)k then h(Ω) = ˜h( ˜Ω), h^∗F_h(Ω)h^∗−1u(x) = ˜h^∗F˜h( ˜Ω)(˜h^∗)⁻¹u(˜˜ x),

h^∗B_h(Ω)h^∗−1u(x) = ˜h^∗B˜h( ˜Ω)(˜h^∗)⁻¹u(˜˜ x), using the normal

Nh( ˜˜Ω)(˜h(˜x)) = (˜h⁻¹)^t_x˜NΩ˜(˜x) k(˜h⁻¹)^t_˜xNΩ˜(˜x)k

= (h⁻¹)^t_xNΩ(x) k(h⁻¹)^t_xN_Ω(x)k

= N_h(Ω)(h(x)).

This ‘change of origin’ will be frequently used in the sequel, as it allow us to compute derivatives with respect tohath=iΩ, where the formulas are simpler.

2.3 The Transversality Theorem

A basic tool for our results will be the Transversality Theorem in the form below, due to D. Henry [2]. We first recall some definitions.

A map T ∈ L(X, Y) where X and Y are Banach spaces is a semi-Fredholm map if the range of T is closed and at least one (or both, for Fredholm) of dim N(T), codim R(T) is finite; theindexofT is then

index(T) =ind(T) =dimN(T)−codimR(T).

We say that a subsetF of a topological spaceX israreif its closure has empty interior andmeagerif it is contained in a countable union of rare subsets of X. We say thatF isresidual if its complement inX is meager. We also say thatX is aBaire spaceif any residual subset ofX is dense.

Letf be aC^kmap between Banach spaces. We say thatxis aregular pointoff if the derivativef⁰(x) is surjective and its kernel is finite-dimensional. Otherwise,xis called a criticalpoint off. A point is acritical valueif it is the image of some critical point off. Let nowX be a Baire space andI= [0,1]. For any closed orσ-closedF ⊂X and any nonnegative integermwe say that the codimension ofF is greater or equal tom(codim F≥ m) if the subset{φ∈ C(I^m, X) |φ(I^m)∩F is non-empty} is meager in C(I^m, X).

We saycodim F =kifk is the largest integer satisfyingcodim F ≥m. codim F ≥m.

Theorem 2 Suppose given positive numberskandm; Banach manifoldsX, Y, Zof class C^k; an open setA⊂X×Y ; a C^k map f :A→Z and a pointξ∈Z. Assume for each (x,y)∈f⁻¹(ξ)that:

1. ^∂f_∂x(x, y) :TxX −→TξZ is semi-Fredholm with index< k.

2. (α) Df(x, y) :TxX×TyY −→TξZ is surjective or

(β)dim n_R(Df(x,y))

R(^∂f_∂x(x,y))

o≥m+dim N(^∂f_∂x(x, y)).

3. (x, y)7→y:f⁻¹(ξ)−→Y isσ-proper, that isf⁻¹(ξ)is a countable union of setsMj

such that(x, y)7→y :Mj −→Y is a proper map for each j.[Given (xn, yn)∈Mj

such that{yn} converges inY, there exists a subsequence (or subnet) with limit in Mj.]

We note that 3 holds if f⁻¹(ξ)is Lindel¨of [ every open cover has a countable subcover]

or, more specifically, iff⁻¹(ξ)is a separable metric space, or ifX, Y are separable metric spaces.

Let A_y ={x|(x, y)∈A} and

Y_crit={y |ξ is a critical value off(·, y) :A_y7→Z}.

ThenYcrit is a meager set in Y and, if (x,y)7→ y : f⁻¹(ξ)7→Y is proper, Ycrit is also closed. If ind ^∂f_∂x ≤ −m <0 onf⁻¹(ξ), then(2(α))implies(2(β))and

Ycrit={y |ξ∈f(Ay, y)}

has codimension≥m in Y.[ Note Ycrit is meager iff codim Ycrit≥1 ].

3 Genericity of the isomorphism property for a class of linear differential operators

Leta:Rⁿ→R,b:Rⁿ →Rⁿ and c:Rⁿ →Rbe functions of classC³ and consider the (formal) differential operator

L= ∆²+a(x)∆ +b(x)· ∇+c(x) x∈Rⁿ. We show in this section that the operator

L_Ω:H⁴∩H₀²(Ω) → L²(Ω) (3) u → Lu

is, generically, an isomorphism in the set of open, connected, boundedC³-regular regions ofRⁿ. More precisely, we show that the set

I = {h∈Diff⁴(Ω)| the operatorh^∗L_h(Ω)h^∗−1from H⁴∩H₀²(Ω) intoL²(Ω)

is an isomorphism} (4)

is an open dense set in Diff⁴(Ω). Observe that the operatorh^∗L_h(Ω)h^∗−1 is an isomor- phism if, and only if the operatorL_h(Ω)from H⁴∩H₀²(h(Ω)) toL²(h(Ω)) is an isomor- phism, sinceh^∗andh^∗−1are isomorphisms fromL²(h(Ω)) toL²(Ω) andH⁴∩H₀²(Ω) to H⁴∩H₀²(h(Ω)) respectively. Consider the differentiable map

K:H⁴∩H₀²(Ω)×Diff⁴(Ω) → L²(Ω)

(u, h) → h^∗L_h(Ω)h^∗−1u. (5) Proposition 3 Let a : Rⁿ → R, b : Rⁿ → Rⁿ, c : Rⁿ → R be C² functions, Ω⊂ Rⁿ an open, connected boundedC⁴-regular region, and h∈Diff⁴(Ω). Then zero is a regular value of the map (application)

Kh:H⁴∩H₀²(Ω) −→ L²(Ω)

u −→ h^∗L_h(Ω)h^∗−1u, if and only ifh^∗L_h(Ω)h^∗−1 is an isomorphism.

P

roof. First observe thatKhis a Fredholm operator of index 0 sinceLh(Ω)is Fredholm of index 0 andh^∗, h^∗−1 are isomorphisms. If 0 is a regular value then the linearisation ofKh at 0, which is again Kh, must be surjective. Being of index 0 it is also injective and therefore an isomorphism by the Open Mapping Theorem. Reciprocally, ifKhis an isomorphism, it is surjective at any point.

From 3 and the Implicit Function Theorem it follows thatI is open . We thus only need to show density. For that we may work with more regular regions.

It would be very convenient for our purposes to have the following ‘unique continua- tion’ result.

If uis a solution ofLΩu= 0 with _∂N^∂2u2 = 0 in a open set of∂Ω, thenu≡0.

Such a result is not available, to the best of our knowledge, but the following ‘generic unique continuation result’ will be sufficient for our needs. We will not prove it here since the argument is very similar to the one of 11 below.

Lemma 4 LetΩ⊂Rⁿ be an open, connected, bounded C⁵-regular region withn≥2and J an open nonempty subset of∂Ω. Consider the differentiable map

G:B_M×Diff⁵(Ω)→L²(Ω)×H³²(J) defined by

G(u, λ, h) =

h^∗Lh(Ω)h^∗−1u, h^∗∆h^∗−1u _J

whereBM ={u∈H⁴∩H₀²(Ω)− {0} | kuk ≤M}. Then, the set C_M^J ={h∈Diff⁵(Ω)|(0,0)∈G(BM, h)}

is meager and closed inDiff⁵(Ω).

Theorem 5 Let a : Rⁿ →R, b : Rⁿ → Rⁿ and c : Rⁿ →R be functions of class C³. Then the operator LΩ defined in (3) is generically an isomorphism in the set of open, connected C⁴-regular regions Ω⊂Rⁿ, n ≥2. More precisely, if Ω⊂Rⁿ (n≥2) is an open, connected C⁴-regular region, than the set I defined in (4) is an open dense set in Diff⁴(Ω).

P

roof. By proposition 3, all we need to show is that 0 is a regular value of Kh in a residual subset of Diff⁴(Ω). Since our spaces are separable and Kh is Fredholm of index 0, this would follow from the Transversality Theorem if we could prove that 0 is a regular value ofK. Let us suppose that this is not true, that is, there exists a critical point (u, h)∈ K⁻¹(0). As explained in (2.2 ), we may suppose that h=iΩ. Since we only need to prove density, we may also suppose that Ω isC⁵-regular. Then, there exists v∈L²(Ω) such that

Z

Ω

vDK(u, iΩ)( ˙u,h) = 0 for all ( ˙˙ u,h)˙ ∈H⁴∩H₀²(Ω)× C⁵(Ω,Rⁿ) (6) whereDK(u, i_Ω) fromH⁴∩H₀²(Ω)× C⁵(Ω,Rⁿ) toL²(Ω) is given by

DK(u, iΩ)( ˙u,h) =˙ LΩ( ˙u−h˙ · ∇u).

Choosing ˙h= 0 and varying ˙uinH⁴∩H₀²(Ω), we obtain Z

Ω

vL_Ωu˙ = 0 for all ˙u∈H⁴∩H₀²(Ω) andvis therefore a weak, hence strong, solution of

L^∗_Ωv= 0 in Ω (7)

whereL^∗_Ωfrom H⁴∩H₀²(Ω) toL²(Ω) is given by

L^∗_Ωv= ∆²v+a∆v+ (2∇a−b)· ∇v+ (c+ ∆a− divb)v.

By regularity of solutions of strongly elliptic equations,v is also a strong solution, that is, v ∈ H⁴∩H₀²(Ω)∩ C^4,α(Ω) for someα > 0 and satisfies (7) [Note that u∈ H⁵(Ω), since Ω isC⁵-regular.]

Choosing ˙u= 0 and varying ˙hin C⁵(Ω,Rⁿ), we obtain 0 =−

Z

Ω

vLΩ( ˙h· ∇u) = Z

Ω

{( ˙h· ∇u)L^∗_Ωv−vLΩ( ˙h· ∇u)}= Z

∂Ω

h˙ ·N∆v∆u,

for all ˙h ∈ C⁵(Ω,Rⁿ) since ∆u|∂Ω = _∂N^∂2u2

_∂Ω. Thus ∆v∆u≡0 in∂Ω. This is not a contradiction (or at least it is not clear that it is). We show now, however, that it isa contradiction generically; it can only happen in an ‘exceptional’ set of Diff⁴(Ω) . The result then follows by reapplying the argument above outside this exceptional set. To be more precise, consider the map

H :B²_M×Diff⁵(Ω)→L²(Ω)²×L¹(∂Ω) defined by

H(u, v, h) = (K(u, h), h^∗L^∗_h(Ω)h^∗−1v, h^∗∆h^∗−1uh^∗∆h^∗−1v|∂Ω)

whereB_M² ={(u, v)∈H⁴∩H₀²(Ω)²| kuk,kvk ≤M}. We show, using the Transversality Theorem that the set

HM ={h∈Diff⁵(Ω)|(0,0,0)∈H(B²_M, h)}

is meager and closed in Diff⁵(Ω) for all M ∈ N. We may, by ‘changing the origin’ if necessary assume that the ‘generic uniqueness property’ stated in Lemma 4 holds in Ω We then apply the Transversality Theorem again for the mapK, withhrestricted to the complement ofHM, obtaining another subset ˜HM of Diff⁵(Ω). such that 0 is a regular value ofKh for anyh∈H˜M Taking intersection forM ∈N, the desired result follows.

To show thatHM is a meager closed set we apply Henry’s version of the Transversality Theorem for the mapH. Since our spaces are separable and ^∂K_∂u(u, i∂Ω) is Fredholm it remains only to prove that the map (u, v, h)7→h:H⁻¹(0,0,0)→Diff⁵(Ω) is proper and the hypothesis (2β). We first show that (u, v, h)→h:H⁻¹(0,0,0)→Diff⁵(Ω) is proper.

Let{(un, vn, hn)}_n∈N⊂H⁻¹(0,0,0) be a sequence withhn→iΩin Diff⁵(Ω) (the general case is similar). Since{(un, vn)}_n∈N⊂B_M² , we may assume, taking a subsequence that there exists (u, v)∈H₀²(Ω)² such thatun →uandvn →v in H₀²(Ω). We have, for all n∈N

h^∗_n

∆²+a∆ +b· ∇+c

h^∗_n⁻¹u_n = 0 ⇐⇒ h^∗_n∆²h^∗_n⁻¹u_n=−h^∗_n

a∆ +b· ∇+c

h^∗_n⁻¹u_n. Since ∆² is an isomorphism, it follows that

u_n=−h^∗_n(∆²)⁻¹

a∆ +b· ∇+c

h^∗_n⁻¹u_n (8)

By results on section 2.1, the right-hand side of (8) is analytic as an application from H₀²(Ω)×Diff⁵(Ω) to H⁴∩H₀²(Ω). Taking the limit as n → +∞, we obtain that u ∈ H⁴∩H₀²(Ω) and satisfies ∆²u+a∆u+b· ∇u+cu= 0.By Lemma 10 of [10] we have, forh∈Diff⁵(Ω) andv∈H⁴∩H₀²(Ω)

h^∗∆²h^∗−1(v) = ∆²(v) +L^h(v) withkL^huk_L2(Ω)≤(h)kuk_H4(Ω)

and(h)→0 ash→iΩin C⁴(Ω,Rⁿ).

Sinceun →uin H₀²(Ω) andhn→iΩin Diff⁵(Ω) asn→+∞, we obtain

k∆²(un−u) +L^hn(un−u)kL²(Ω) = kh^∗_n∆²h^∗_n⁻¹(un−u)kL²(Ω)

= kh^∗_n∆²h^∗_n⁻¹u+h^∗_n

a∆ +b· ∇+c

h^∗_n⁻¹u_nk_L2(Ω)

→ k∆²u+a∆u+b· ∇u+cuk_L2(Ω)= 0 (9)

asn→+∞.

Since{(un, vn)} ⊂B²_M andhn→iΩin Diff⁵(Ω), we have

kL^hn(un−u)kL²(Ω)≤2M (hn). (10) It follows from (9) and (10) that k∆²(u_n−u)k_L2(Ω)→0 asn→+∞. Since ∆² is an isomorphism fromH⁴∩H₀²(Ω) toL²(Ω), we obtainu_n→uinH⁴∩H₀²(Ω) and, therefore kuk_H4∩H₀²(Ω) ≤M, for alln∈ N. Similarly, we prove that v_n →v in H⁴∩H₀²(Ω) and kvk_H4∩H²₀(Ω) ≤M from which we conclude that the map (u, v, h)→h:H⁻¹(0,0,0)→ Diff⁵(Ω) is proper.

It remains only to prove (2β), which we do by showing that dimnR(DH(u, v, h))

R

∂H

∂u(u, v, h) o

=∞for alll (u, v, h)∈H⁻¹(0,0,0).

Suppose this is not true for some (u, v, h)∈H⁻¹(0,0,0). Assuming, as we may, thath= iΩit follows that there existθ1, ..., θm∈L²(Ω)²×L¹(∂Ω) such that, for all ˙h∈ C⁵(Ω,Rⁿ) there exist ˙u,v˙∈H⁴∩H₀²(Ω) and scalarsc₁, ..., c_m∈Rsuch that

DH(u, v, i_Ω)( ˙u,v,˙ h) =˙

m

X

i=1

c_iθ_i, θ_i = (θ_i¹, θ²_i, θ³_i) (11) Using theorem 1, we obtain

DH(u, v, iΩ)(·) =

DH1(u, v, iΩ)(·), DH2(u, v, iΩ)(·), DH3(u, v, iΩ)(·) where

DH₁(u, v, i_Ω)( ˙u,v,˙ h)˙ = L_Ω( ˙u−h˙ · ∇u) DH2(u, v, iΩ)( ˙u,v,˙ h)˙ = L^∗_Ω( ˙v−h˙ · ∇v) DH3(u, v, iΩ)( ˙u,v,˙ h)˙ = n

∆v∆( ˙u−h˙ · ∇u) + ∆u∆( ˙v−h˙ · ∇v) + ˙h·N ∂

∂N(∆u∆v)o _∂Ω It follows from (11) that

LΩ( ˙u−h˙ · ∇u) =

m

X

i=1

ciθ_i¹ (12)

L^∗_Ω( ˙v−h˙ · ∇v) =

m

X

i=1

ciθ_i² (13)

m

X

i=1

c_iθ_i³ = n

∆v∆( ˙u−h˙ · ∇u) + ∆u∆( ˙v−h˙ · ∇v) + ˙h·N ∂

∂N(∆u∆v)o

_∂Ω. (14)

Let{u1, ..., ul} be a basis for the kernel ofLΩand consider the operators A_L:L²(Ω)→H⁴∩H₀¹(Ω)

CL:H⁵²(∂Ω)→H⁴∩H₀¹(Ω) defined by

w=AL(z) +CL(g)

whereLw−z belongs to a (fixed) complement of R(LΩ) inL²(Ω), ^∂w_∂N =g on ∂Ω and R

Ωwφ = 0 for all φ∈ N(L^∗_Ω). Let also {v₁, ..., v_l} be a basis for the kernel of L^∗_Ω and consider the operators

A_L^∗ :L²(Ω)→H⁴∩H₀¹(Ω) and CL^∗ :H⁵²(∂Ω)→H⁴∩H₀¹(Ω)

similarly defined. We have shown in [9] that these operators are well defined.

From equations (12) and (13), we obtain

˙

u−h˙ · ∇u=

l

X

i=1

ξiui+

m

X

i=1

ciALθ¹_i − CL( ˙h·N∆u) (15)

since _∂N^∂ ( ˙u−h˙ · ∇u)

_∂Ω=−h˙ ·N_∂N^∂2u₂

_∂Ω=−h˙ ·N∆u|∂Ω and

˙

v−h˙ · ∇v=

l

X

i=1

η_iv_i+

m

X

i=1

c_iA_L^∗θ_i¹− C_L^∗( ˙h·N∆v) (16)

since _∂N^∂ ( ˙v−h˙ · ∇v)

_∂Ω=−h˙ ·N_∂N^∂2v2

_∂Ω=−h˙ ·N∆v|∂Ω. Substituting (15) and (16) in (14), we obtain that

nh˙ ·N ∂

∂N(∆u∆v)−h

∆u∆C_L^∗( ˙h·N∆v) + ∆v∆C_L( ˙h·N∆u)io

_∂Ω (17) remains in a finite dimensional space when ˙hvaries inC⁵(Ω,Rⁿ).

The setU ={x∈∂Ω|∆u(x)6= 0} is nonempty since we have assumed that ‘generic unique continuation’ holds in Ω. Therefore, we must have ∆v|U ≡0. If ˙h≡0 in∂Ω−U, then ˙h·N∆v≡0 in∂Ω and, therefore

∆u∆CL^∗( ˙h·N∆v) = ∆u∆CL^∗(0)

belongs to the finite dimensional space [∆u∆v₁, ...,∆u∆v_l] where {v1, ..., v_l} is a basis for the kernel ofN(L^∗_Ω). It follows that

nh˙ ·N ∂

∂N(∆u∆v)−∆v∆CL( ˙h·N∆u)o

_U = h˙ ·N ∂

∂N(∆u∆v) _U

= h˙ ·N∆u ∂

∂N(∆v)

_U (18)

remains in a finite dimensional space, when ˙hvaries inC⁵(Ω,Rⁿ) with ˙h≡0 in∂Ω−U.

Since ∆u(x)6= 0 for any x ∈U, this is only possible ( dim Ω≥2) if ^∂∆v_∂N

_U ≡0. But thenv≡0 in Ω by Theorem 6 below, and we reach a contradiction proving the result.

During the proof of theorem 5 we have used the following uniqueness theorem, which is a direct consequence of Theorem 8.9.1 of [3].

Theorem 6 SupposeΩ⊂Rⁿ is an open connected, bounded, C⁴-regular domain and B is an open ball in Rⁿ such that B∩∂Ω is a (nontrivial)C⁴ hypersurface. Suppose also thatu∈H⁴(Ω) satisfies

|∆²u| ≤C

|∆u|+|∇u|+|u|

a.e. inΩ for some positive constantCand u= ∂u

∂N = ∆u= ∂∆u

∂N = 0in B∩∂Ω.Thenuis iden- tically null.

4 Generic simplicity of solutions

Let f(x, λ, y, µ) be a real function of class C⁴ defined in Rⁿ×R×Rⁿ ×R satisfying f(x,0,0,0)≡0 for all x∈Rⁿ. We prove in this section our main result: generically in the set of connected, boundedC⁴-regular regions Ω⊂Rⁿ,n≥2, the solutionsuof

∆²u+f(·, u,∇u,∆u) = 0 in Ω

u=_∂N^∂u = 0 on∂Ω (19)

are all simple. We choosep > ⁿ₂, so that the continuous imbeddingp,W^4,p∩W₀^2,p(Ω),→ C^2,α(Ω) holds for someα >0.

It follows then, from the Implicit Function Theorem, that the set of solutions is discrete inW^4,p∩W₀^2,p(Ω) and, in particular finite iff is bounded.

Remark 7 Since we have assumed f(x,0,0,0)≡0 inRⁿ, the null function u≡0 is a solution of (19) for anyΩ⊂Rⁿ. It follows from theorem 5 thatu≡0is simple forΩin an open dense set ofDiff⁴(Ω). We therefore concentrate in the proof of generic simplicity for the nontrivial solutions.

Proposition 8 LetΩ⊂Rⁿbe an open, connected boundedC⁴-regular region andf(x, λ, y, µ) aC² real function defined in Rⁿ×R×Rⁿ×R. Then, zero is a regular value of the map

Fh:W^2,p∩W₀^1,p(Ω) −→ L^p(Ω)

u −→ h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u), if and only if all solutions of (19) inh(Ω)are simple.

P

roof. The proof is very similar to the one of proposition 3 and will be left to the reader.

Proposition 9 A function u∈W^4,p∩W₀^2,p(Ω) is a solution (resp. a simple solution) of

h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u) = 0 inΩ,

u=_∂N^∂u = 0on ∂Ω (20)

if and only ifv=h^∗−1uis a solution (resp. simple solution) of (19) in h(Ω).

P

roof. Letu∈W^4,p∩W₀^2,p(Ω). Sinceh^∗ andh^∗−1are isomorphisms, we have h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u) = 0

⇐⇒ ∆²h^∗−1u+f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u) = 0,

It is clear thatu= 0 in∂Ω if and only ifv= 0 in∂h(Ω). Writingy=h(x), we obtain

∂v

∂N_h(Ω)(y) = N_h(Ω)(y)· ∇y(u◦h⁻¹)(y)

= N_h(Ω)(y)·(h⁻¹)^t_y(y)∇_xu(x)

= N_h(Ω)(y)·(h⁻¹_x )^t(x)∇xu(x)

= N_h(Ω)(y)·((hx)⁻¹)^t(x)∂u

∂N(x)NΩ(x)

= ∂u

∂N(x) 1

||(h⁻¹x )^tNΩ(x)|| (h⁻¹_x )^t(x)N_Ω(x)·(h⁻¹_x )^t(x)N_Ω(x) where we have used that u = 0 in ∂Ω. Since h⁻¹_x (x) is non-singular it follows that

∂v

∂N_h(Ω)(y) = 0 if and only if _∂N^∂u(x) = 0. Thus u is a solution of (20) if and only if h^∗−1u is a solution of (19) in h(Ω). Finally, since h^∗L(u)h^∗−1 is an isomorphism in W^4,p∩W₀^2,p(Ω) if and only ifL(v) is an isomorphism inW^4,p∩W₀^2,p(h(Ω)) so the result follows.

It follows from (9) and (8) that, in order to show generic simplicity of the solutions of (19) is enough to show that 0 is a regular value ofFh, generically inh∈Diff⁴(Ω). We show, using the Transversality Theorem, that 0 is a regular value of

F_M :B_M×V_M → L^p(Ω)

(u, h) → h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u) (21) where B_M ={u∈W^4,p∩W₀^2,p(Ω)− {0} | kuk ≤ M} and V_M is an open dense set in Diff⁴(Ω), for allM ∈N. Taking the intersection ofV_M forM ∈Nwe obtain the desired residual set.

Remark 10 Applying the Implicit Function Theorem to the mapFM defined in (21) we obtain that the set

F_M ={h∈Diff⁴(Ω)| all solutionsuof (20) withkuk_W4,p∩W₀^2,p(Ω)< M are simple}

is open in Diff⁴(Ω) for all M ∈ N. To prove density, we may work with more regular (for exampleC^∞) regions.

If we try to apply the Transversality Theorem directly to the function F defined in kuk_W4,p∩W₀^2,p(Ω)×Diff⁴(Ω) by (21) we do not obtain a contradiction. What we do obtain is that the possible critical points must satisfy very special properties. The idea is then to show that these properties can only occur in a small (meager and closed) set and then restrict the problem to its complement. In our case the ‘exceptional situation’ is characterized by the existence of a solutionuof (19) and a solutionv of the problem

L^∗(u)v= 0 in Ω v= _∂N^∂v = 0 on∂Ω

satisfying the additional property ∆u∆v ≡0 on ∂Ω. We show in Lemma 12 that this situation is really ‘exceptional’, that is, it can only happen ifhis outside an open dense subset of Diff⁴(Ω) (foruandv restricted to a bounded set).

We will need the following ‘generic unique continuation result’.

Lemma 11 Let Ω⊂Rⁿ n≥2 be an open, connected, bounded,C⁵-regular domain,J a nonempty open subset ofΩandf(x, λ, y, µ)aC²real function defined inRⁿ×R×Rⁿ×R withf(·,0,0,0)≡0. Consider the map

G:AM×Diff⁵(Ω)→L^p(Ω)×W^2−1p,p(J) defined by

G(u, h) =

h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u), h^∗∆h^∗−1u _J

whereAM ={u∈W^4,p∩W₀^2,p(Ω)− {0} | kuk ≤M}, andp > ⁿ₂. Then

C_M^J ={h∈Diff⁵(Ω)|(0,0)∈G(AM, h)}

is a closed meager subset ofDiff⁵(Ω).

P

roof. We apply the Transversality Theorem. Observe thatGis differentiable. In fact it is analytic in has observed in section 2.1, and the differentiability inu follows from the smoothness off and the continuous immersion W^4,p∩W₀^2,p(Ω)⊂ C^2,α(Ω) for some α >0. We compute its differential using Theorem 1 (see example 2.1 of section 2.1).

DG(u, iΩ)( ˙u,h)˙ =

L(u)( ˙u−h˙ · ∇u), n

∆( ˙u−h˙ · ∇u) + ˙h·N∂∆u

∂N o

_J

.

To verify (1) and (2), we proceed as in the proof of theorem 5. We prove that (2β) holds showing that

dimnR(DG(u, h)) R

∂G

∂u(u, h) o

=∞for all (u, h)∈G⁻¹(0,0).

Suppose, by contradiction this is not true for some (u, h)∈G⁻¹(0,0). By ‘changing the origin’ we may suppose thath=iΩ. Then, there exist θ1, ..., θm∈L^p(Ω)×W^2−1p,p(J) for all ˙h∈ C⁵(Ω,Rⁿ) there exists ˙u∈W^4,p∩W₀^2,p(Ω) and scalarsc₁, ..., c_m∈Rsuch that

DG(u, iΩ)( ˙u,h) =˙

m

X

i=1

ciθi,

that is,

L(u)( ˙u−h˙ · ∇u) =

m

X

i=1

ciθ¹_i (22) n

∆( ˙u−h˙ · ∇u) + ˙h·N∂∆u

∂N o

_J =

m

X

i=1

c_iθ²_i (23) where

L(u) = ∆²+∂f

∂µ(·, u,∇u,∆u)∆ + ∂f

∂y(·, u,∇u,∆u)· ∇+∂f

∂λ(·, u,∇u,∆u). (24) Let {u1, ..., ul} be a basis for the kernel of L0(u) =L(u)

W^4,p∩W₀^2,p(Ω)

and consider the operators

AL(u):L^p(Ω)→W^4,p∩W₀^1,p(Ω) C_L(u):W^3−1p,p(∂Ω)→W^4,p∩W₀^1,p(Ω) defined by

w=AL(u)(z) +CL(u)(g)

ifL(u)w−z belongs to a fixed complement ofR(L0(u)) inL^p(Ω), _∂N^∂w =g on∂Ω and R

Ωwφ= 0 for allφ∈ N(L^∗₀(u)). (We proved these operators are well defined in [9]).

Choosing ˙h∈ C⁵(Ω,Rⁿ) such that ˙h≡0 on∂Ω−J, we obtain from (22), that

˙

u−h˙ · ∇u=

l

X

i=1

ξiui+

m

X

i=1

ciA_L(u)(θ_i¹) (25) since ˙u−h˙ · ∇u∈W^4,p∩W₀^2,p(Ω).

Substituting (25) in (23), we obtain that ˙h·N∂∆u

∂N

_J remains in a finite dimen- sional subspace when ˙h varies in C⁵(Ω,Rⁿ). Since dim Ω ≥ 2 this is possible only if

∂∆u

∂N ≡0 onJ sousatisfies







∆²u+f(·, u,∇u,∆u) = 0 in Ω u=_∂N^∂u = 0 on∂Ω

∆u= ^∂∆u_∂N = 0 onJ.

(26)

We claim thatu satisfies the hypotheses of Cauchy’s Uniqueness Theorem 6. Indeed, sinceu∈W^4,p(Ω)∩ C^2,α(Ω) for someα >0 (p > ⁿ₂) and is a solution of the uniformly elliptic equation ∆²u+f(·, u,∇u,∆u) = 0 in Ω thenu∈W^4,p(Ω)∩C^4,α(Ω). Furthermore, u= _∂N^∂u = ∆u= ^∂∆u_∂N = 0 onJ ⊂∂Ω and

|∆²u| ≤ |f(·, u,∇u,∆u)|

≤ |f(·, u,∇u,∆u)−f(·,0,0,0)|

≤ max

Ω¯

{|Df(·, u,∇u,∆u)|}

|u|+|∇u|+|∆u|

, We conclude thatu≡0, which gives the searched for contradiction.

Lemma 12 Let Ω⊂Rⁿ n≥2 be an open, connected, bounded, C⁵-regular domain and f(x, λ, y, µ)aC³ real function defined inRⁿ×R×Rⁿ×Rwithf(·,0,0,0)≡0. Consider the map

Q:AM,p×AM,q×DM →L^p(Ω)×L^q(Ω)×L¹(∂Ω) defined by

Q(u, v, h) =

h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u), h^∗L^∗(h^∗−1u)h^∗−1v, h^∗∆h^∗−1uh^∗∆h^∗−1v

_∂Ω

whereAM,p={u∈W^4,p∩W₀^2,p(Ω)− {0} | kuk ≤M}andp⁻¹+q⁻¹= 1with p > ⁿ₂, AM,q ={u∈W^4,q∩W₀^2,q(Ω)− {0} | kuk ≤ M}, DM = Diff⁵(Ω)− C_M^∂Ω, C_M^∂Ω given by Lemma 11 and

L^∗(w) = ∆²+∂f

∂µ(·, w,∇w,∆w)∆

+h 2∇∂f

∂µ(·, w,∇w,∆w)

−∂f

∂y(·, w,∇w,∆w)i

· ∇

+∆h∂f

∂µ(·, w,∇w,∆w)i

− div∂f

∂y(·, w,∇w,∆w) +∂f

∂λ(·, w,∇w,∆w).

Then

E_M ={h∈D_M |(0,0,0)∈Q(A_M)×A_M,q, h)}

is a meager closed subset ofDiff⁵(Ω).

(Observe thatL^∗(w) is the formal adjoint ofL(w) defined by (24).)

P

roof. We again apply the Transversality Theorem. The differentiability of Q is easy to establish, and its derivative can be computed using theorem 1 (see example 2.1)

DQ(u, v, i_Ω)( ˙u,v,˙ h)˙ =

L(u)( ˙u−h˙ · ∇u), L^∗(u)( ˙v−h˙ · ∇v) +∂L^∗

∂w(u)·v

( ˙u−h˙ · ∇u), n

∆( ˙u−h˙ · ∇u)∆v+ ∆u∆( ˙v−h˙ · ∇v) + ˙h·N ∂

∂N(∆u∆v)o _∂Ω

where ^∂L_∂w^∗(u)·v is the second order differential operator given by ∂L^∗

∂w (u)·v

z = ∂²f

∂λ∂µv+ ∂²f

∂y∂µ · ∇v+∂²f

∂µ²∆v

∆z +h

2∇ ∂²f

∂λ∂µv+ ∂²f

∂y∂µ· ∇v+∂²f

∂µ²∆v

− ∂²f

∂λ∂yv+∂²f

∂y²∇v+ ∂²f

∂µ∂y∆vi

· ∇z

h∂²f

∂λ²v+ ∂²f

∂λ∂y· ∇v+ ∂²f

∂λ∂µ∆v +∆ ∂²f

∂λ∂µv+ ∂²f

∂y∂µ · ∇v+∂²f

∂µ²∆v

−div ∂²f

∂λ∂yv+∂²f

∂y²∇v+ ∂²f

∂y∂µ∆vi z.

(We have writtenf instead off(·, u,∇u,∆u) to simplify the notation).

The hypotheses (1) and (3) of the Transversality Theorem can be verified as in the proof of Theorem 5. We prove (2β) by showing that

dimn R(DQ(u, v, h)) R

∂Q

∂(u,v)(u, v, h) o

=∞

for all (u, v, h) ∈ Q⁻¹(0,0,0). Suppose this is not true for (u, v, h) ∈ Q⁻¹(0,0,0).

‘Changing the origin’, we may assume that h = iΩ. Then, there exist θ1, ..., θm ∈ L^p(Ω)×L^q(Ω)×L¹(∂Ω) such that for all ˙h∈ C⁵(Ω,Rⁿ) there exists ˙u∈W^4,p∩W₀^2,p(Ω),

˙

v∈W^4,q∩W₀^2,q(Ω) and scalarsc1, ..., cm∈Rsuch that DQ(u, v, iΩ)( ˙u,v,˙ h) =˙

m

X

i=1

ciθi, that is,

L(u)( ˙u−h˙ · ∇u) =

m

X

i=1

c_iθ¹_i (27)

L^∗(u)( ˙v−h˙ · ∇v) +∂L^∗

∂w(u)·v

( ˙u−h˙ · ∇u) =

m

X

i=1

ciθ²_i (28)

and

m

X

i=1

ciθ_i³ = n

∆( ˙u−h˙ · ∇u)∆v+ ∆u∆( ˙v−h˙ · ∇v) + ˙h·N ∂

∂N(∆u∆v)o

_∂Ω. (29) Let{u1, ..., ul} be a basis for the kernel of L0(u) =L(u)

_W4,p

∩W₀^2,p(Ω), {v1, ..., vl} a basis for the kernel ofL^∗₀(u) and consider the operators

A_L(u):L^p(Ω)→W^4,p∩W₀^1,p(Ω) C_L(u):W^3−1p,p(∂Ω)→W^4,p∩W₀^1,p(Ω) defined by

w=A_L(u)(z) +C_L(u)(g)

whereL(u)w−z belongs to a fixed complement of R(L0(u)) in L^p(Ω), _∂N^∂w =g on∂Ω, R

Ωwφ= 0 for allφ∈ N(L^∗₀(u)) and

AL^∗(u):L^q(Ω)→W^4,q∩W₀^1,q(Ω) C_L^∗_(u):W^3−1q,q(∂Ω)→W^4,q∩W₀^1,q(Ω) defined by

t=AL^∗(u)(z) +CL^∗(u)(g)

whereL^∗(u)t−z belongs to a fixed complement of R(L^∗₀(u)) in L^q(Ω), _∂N^∂t =g on∂Ω andR

Ωtϕ= 0 for allϕ∈ N(L₀(u)). (We proved these operators are well defined in [9]).

From (27) and (28) it follows that

˙

u−h˙ · ∇u=

l

X

i=1

ξiui+

m

X

i=1

ciA_L(u)(θ¹_i)− C_L(u)( ˙h·N∆u) (30)

˙

v−h˙ · ∇v =

s

X

i=1

ηivi+

m

X

i=1

ciA_L∗(u)(θ²_i)− C_L∗(u)( ˙h·N∆v)

−A_L^∗_(u)∂L^∗

∂w(u)·v

( ˙u−h˙ · ∇u)

. (31)

Substituting in (29), we obtain that nh˙ ·N ∂

∂N(∆u∆v)−∆v∆

C_L(u)( ˙h·N∆u) +∆u∆h

A_L^∗_(u)∂L^∗

∂w (u)·v

C_L(u)( ˙h·N∆u)

− C_L^∗_(u)( ˙h·N∆v)io _∂Ω

remains in a finite dimensional space when ˙hvaries inC⁵(Ω,Rⁿ), that is, the operator Υ( ˙h) = n

h˙ ·N ∂

∂N(∆u∆v)−∆v∆

CL(u)( ˙h·N∆u)

(32) +∆u∆h

A_L∗(u)

∂L^∗

∂w (u)·v

C_L(u)( ˙h·N∆u)

− C_L∗(u)( ˙h·N∆v)io _∂Ω defined inC⁵(Ω,Rⁿ) is of finite range

We proved in [9] that, if dim Ω≥2, a necessary condition for Υ to be of finite range

is ∂

∂N(∆u∆v)≡0 on∂Ω. (33)

Thus the functionsu,v must satisfy

∆²u−f(·, u,∇u,∆u) = 0 in Ω

u=_∂N^∂u = 0 on∂Ω (34)

L^∗(u)v= 0 in Ω

v= _∂N^∂v = 0 on∂Ω (35)

and also

∆u∆v|∂Ω= ∂

∂N(∆u∆v)

_∂Ω= 0. (36) Let U = {x ∈ ∂Ω | ∆u(x) 6= 0}. Observe that U is a nonempty, since i_Ω ∈ D^∂Ω_M (given by Lemma 11).

By equation (36), we have ∆v|U =∂∆v

∂N U

≡0.Thereforev∈W^4,q∩W₀^2,q(Ω) satisfies the hypotheses of theorem 6. Thusv≡0 in Ω and we reach a contradiction, proving the result.

Theorem 13 Generically in the set of open, connected, bounded C⁴-regular regions of Rⁿ n≥2 the solutions of (19) are all simple.

P

roof.

Consider the differentiable map

F :B_M×U_M →L^p(Ω) defined by

F(u, h) =h^∗∆²h^∗−1u+h^∗f(·, h^∗−1u,∇h^∗−1u,∆h^∗−1u)

whereBM ={u∈W^4,p∩W₀^2,p(Ω)− {0} | kuk ≤M}, p > ⁿ₂, UM =DM −EM, DM

is the complement of the meager closed set given by Lemma 11 andEM is the meager closed set given by Lemma 12. Observe that UM is an open dense subset of Diff⁴(Ω).

We show, using the Transversality Theorem, that the set

{h∈UM | u→F(u, h) has 0 as a regular value}