On some flows in shape optimization

On some flows in shape optimization

Pierre Cardaliaguet^∗and Olivier Ley^†

Abstract. Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a non- local term of Hele-Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain ex- istence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed: we prove that the solutions converge to a generalized Bernoulli exterior free boundary problem.

R´esum´e. On ´etudie des flots g´eom´etriques li´es des probl`emes d’optimisation de forme du type “probl`eme de Bernoulli”. La loi d’´evolution consid´er´ee a la forme d’une somme d’un terme de courbure et d’un terme non-local de type Hele-Shaw.

Notre d´efinition de solution g´en´eralis´ee est fortement inspir´ee de la notion de solu- tions de viscosit´e. Le r´esultat central est un principe d’inclusion pour les ensembles solutions. Nous en d´eduisons l’existence, une unicit´e g´en´erique et des propri´et´es de stabilit´e des solutions. Enfin, nous ´etudions le comportement asymptotique des solutions en montrant qu’elles convergent vers la solution d’un probl`eme `a fronti`ere libre de Bernoulli.

1 Introduction

In recent years several works have been devoted to the study of viscosity solution for moving boundary problems whose evolution law is governed by a nonlocal equation. See in particular [2, 7, 8, 9, 12, 20, 21]. In this paper, we consider subsets Ω(t) ofIR^N (withN ≥2) whose boundary∂Ω(t) evolves with a normal velocity of the type

V_(t,x)^Ω =F(ν_x^Ω(t), H_x^Ω(t)) +λ¯h(x,Ω(t)) (1)

∗Universit´e de Bretagne Occidentale, UFR des Sciences et Techniques, 6 Av. Le Gorgeu, BP 809, 29285 Brest, France; e-mail: <[email protected]>

†Laboratoire de Math´ematiques et Physique Th´eorique. Facult´e des Sci- ences et Techniques, Parc de Grandmont, 37200 Tours, France; e-mail:

<[email protected]>

where λ≥ 0, ν_x^Ω(t) is the outward unit normal to ∂Ω(t) at x, H_x^Ω(t) is the curvature matrix of∂Ω(t) atx(nonpositive for convex sets),F is continuous and elliptic, i.e., nondecreasing with respect to the curvature matrix. The nonlocal term ¯h is of Hele-Shaw type:

¯h(x,Ω(t)) =|Du(x)|² ,

whereu: Ω(t)→IRis the solution to the following p.d.e.





−∆u= 0 in Ω(t)\S, u=g on∂S, u= 0 on∂Ω(t).

(2)

The setS6=∅ is a fixed source with a smooth boundary andg:∂S→IRis positive and smooth. We always assume thatS ⊂⊂Ω(t).

The motivation to study such problems comes from several numerical works using the “level-set approach” in shape optimization [1, 23, 24, 25, 28]. The idea of these papers is to use formally a gradient method for the minimization of an objective functionJ(Ω) where Ω is a subset ofIR^N. The use of the level-set method for building the gradient flow has then the major advantage to allow topological changes. Let us underline that this technique is up to now purely heuristic. One of the goals of this paper is to justify it for some simple shape optimisation problems.

In order to make our purpose more transparent, a brief description of the level-set approach to shape optimization problem is now in order (see also the discussion in [1] for a more detailed presentation concerning more realistic shape optimization problems). Consider the problem of minimizing the capacity of a set under volume constraints:

min

S⊂⊂Ω⊂⊂IR^N{cap(Ω) with vol(Ω) = constant} , (3) where

cap(Ω) = Z

Ω\S|Du(x)|²dx and vol(Ω) = Z

Ω\S

dx

and u is the solution of (2) with Ω instead of Ω(t). For any local diffeo- morphismθ, we can compute the shape derivatives with respect to θ of the capacity and of the volume. By Hadamard formulas we get

cap⁰(Ω)(θ) = Z

∂Ω|Du(σ)|²hθ(σ), ν_σ^Ωidσ and vol⁰(Ω)(θ) = Z

∂Ωhθ(σ), ν_σ^Ωidσ .

Assuming that the optimal shape Ω is smooth, the necessary conditions of optimality states that there is a Lagrange multiplier Λ>0 such that

cap⁰(Ω)(θ) + Λ vol⁰(Ω)(θ) = 0. So it is natural to set

J_λ(Ω) = vol(Ω) +λcap(Ω),

where λ= 1/Λ. If we choose θ(x) = (−1 +λ|Du(x)|²)ν_x^Ω on ∂Ω, then, at least formally, we get

J_λ⁰(Ω)(θ) =− Z

∂Ω

(−1 +λ|Du(σ)|²)²dσ≤0.

Therefore the velocity θ(x) = (−1 +λ|Du(x)|²)ν_x^Ω appears as a descent direction for the optimization problem (3) and for the set Ω. The heuristic method for solving (3) is now clear: fix an initial position Ω₀, consider the evolution (Ω(t))_t≥0 with normal velocity given by (1) and F ≡ −1, and compute the limit of Ω(t) as t → +∞: this limit is the natural candidate minimizer for (3).

It is worth noticing that problem (3) has for necessary condition the classical Bernoulli exterior free boundary problem

Find a setK ⊂⊂IR^N, withS⊂⊂K and |Du(x)|=k for all x∈∂K, (4) where k >0 is a fixed constant and u is the solution of (2). We refer the reader to the survey paper [15] for a complete description of this problem.

If one considers a perimeter constraint instead of a volume constraint:

min

S⊂⊂Ω⊂⊂IR^N{cap(Ω) with per(Ω) = constant} ,

one is naturally lead to consider the evolution equation (1) withF(ν, A) =

1

N−1T r(A) (i.e., the mean curvature). The flow is then formally a descent direction for

J_λ(Ω) = per(Ω) +λcap(Ω).

Let us underline that this problem has for necessary condition the gener- alization of the free boundary problem (4) with curvature dependance (see (53)).

Of course all the above computations are only formal: in general, solu- tions to the evolution equation do not remain smooth, even when starting from smooth initial data. Numerically, this difficulty is overcome by using

the level-set approach, which allows to define the solution after the onset of singularities. The aim of this paper is to define and study generalized solu- tions of the evolution equation, and to investigate the asymptotic behavior of the solution ast→+∞.

Our concept of solutions is widely inspired by the definition of viscosity solution for the mean curvature motion, which corresponds to equation (1) with F(ν, A) = _N−1¹ T r(A) and λ = 0. Motivated by the numerical work of Osher and Sethian [22], a weak notion of solution for this motion was introduced in the articles of Chen, Giga and Goto [10] and Evans and Spruck [13]. In this so-called level-set method, the evolution is described as the level set of the solution of an auxiliary pde, the level set equation. This equation is solved in the sense of viscosity solutions (see [11]). This powerful method leads to plenty of results, we refer for instance to the survey book of Giga [16]. Note that the level-set approach in shape optimization is a natural–but up to now formal–generalization of these ideas.

As pointed out in [3, 4, 26], the generalized solutions obtained by the level set approach can also be defined in more geometric and intric ways (see also the related notion of barrier solutions introduced by De Giorgi). We use here a definition introduced in [2], and used repetitively in [7, 8, 9]. In the case of the mean curvature motion, Giga [16] proved this definition is equivalent to the level-set one. Compared with the already quoted studies on viscosity solutions of front propagation problems with nonlocal terms, the main novelty of this paper is the fact that we are able to treat signed velocities which also involve curvature terms. We learnt recently that a similar result (for a Stefan problem) has been obtained by Kim in [21].

Our main result is an inclusion principle, which is the equivalent of the maximum principle for geometric evolutions. It states that viscosity subso- lutions for the flow remain included into viscosity supersolutions, provided the initial positions are. For this we have to generalize Ilmanen interposi- tion Lemma, which was already the key tool of [7, 9]. This Lemma allows to separate disjoint sets by a smooth (that isC^1,1) surface in a clever way. We improve this result in two directions (see Theorem 3.3). At first we show that, when dealing with subsets of IR×IR^N, the smooth separating hyper- surfaces in IR×IR^N can be chosen to be smoothly evolving hypersurfaces of IR^N. Secondly, we build in a carefull way a C² approximation of these evolving hypersurfaces which allows to treat problems with curvature as in (1).

Let us finally explain how this paper is organized. In Section 2, we de- fine the notion of generalized solutions and state the main properties of the velocity law. Section 3 is devoted to the interposition Theorems. In Section

4 we state and prove the inclusion principle for our generalized solutions.

As a consequence, we derive results about existence, uniqueness and stabil- ity of generalized solutions. Finally, Section 5 is devoted to the asymptotic behaviour of the solutions in terms of a generalized Bernoulli exterior free boundary problem.

Acknowledgment. The authors are partially supported by the ACI grant JC 1041 “Mouvements d’interface avec termes non-locaux” from the French Ministry of Research.

2 Definitions and preliminary results

2.1 Definition of the solutions

Let us first fix some notations: throughout the paper | · | denotes the eu- clidean norm (of IR^N orIR^N+1, depending on the context) and B(x, R) the open ball centered atxand of radiusR. IfKis a subset ofIR^N andx∈IR^N, thend_K(x) denotes the usual distance fromx to K: d_K(x) = inf_y∈K|y−x| andd_K is the signed distance to ∂K defined by

d_K(x) =

d_K(x) if x /∈K,

−d_∂K(x) if x∈K. (5)

Finally, in the whole paper, ifK₁ andK₂ are subset of IR^M forN ≥1, then K₁ ⊂⊂K₂

means that K₁ is bounded and, either K₁ ⊂int(K₂) or equivalently K₁∩ IR^N\K₂ =∅.

We intend to study the evolution of compact hypersurfaces Σ(t) =∂Ω(t) ofIR^N, where Ω(t) is an open set, evolving with the following law:

∀t≥0, x∈Σ(t), V_(t,x)^Ω =h_λ(x,Ω(t)) (6) whereV_(t,x)^Ω is the normal velocity of the evolving set, h_λ=h_λ(x,Ω) is given, for any set Ω⊂IR^N with smooth boundary by

h_λ(x,Ω) =F(ν_x^Ω, H_x^Ω) +λ¯h(x,Ω) (7) whereν_x^Ω is the outward unit normal to Ω at x, H_x^Ω the curvature matrix.

Throughout this paper we assume that (ν, A)∈S^N−1×SN 7→F(ν, A)∈IRis

continuous and elliptic, i.e., nondecreasing with respect to the matrix. Here S^N−1 denotes the (N −1)−dimensional unit sphere, and SN the space of N−dimensional symmetric matrices. Typical examples forF areF(ν, A) =

−1 (this corresponds to the flow associated to Bernoulli problem in the introduction) or F(ν, A) = Tr(A) (for the flow arising in the minimization of the capacity under perimeter constraints). As for ¯h, it is a nonlocal evolution term of Hele-Shaw type: the example we consider here is

¯h(x,Ω) =|Du(x)|² , (8)

whereu: Ω→IRis the solution of the following p.d.e.





i) −∆u= 0 in Ω\S, ii) u=g on∂S, iii) u= 0 on∂Ω.

(9) The setS6=∅is a fixed source and we always assume above thatS ⊂⊂Ω(t).

Here and throughout the paper, we suppose that





i) S ⊂IR^N is bounded and equal to the closure of an open set with a C² boundary,

ii) g:∂S→(0,+∞) isC^1,α (for someα ∈(0,1)).

(10)

Let us underline that ¯h(x,Ω) is well defined as soon as Ω has a “smooth”

(say for instanceC^1,α) boundary and that S⊂⊂Ω. In the sequel, we set D={K ⊂IR^N : K is bounded andS ⊂int(K)}, (11) where int(K) denotes the interior ofK.

From now on, we consider the graph

K={(t, x)∈IR⁺×IR^N : x∈Ω(t)}.

of the evolving sets Ω(t). Note that K is a subset of IR⁺×IR^N. The set K is our main unknown. We denote by (t, x) an element of such a set, where t∈IR⁺ denotes the time andx∈IR^N denotes the space. We set

K(t) = {x∈IR^N : (t, x)∈ K}.

The closure of the set K in IR^N+1 is denoted by K. The closure of the complementary ofK is denotedKb:

Kb = (IR⁺×IR^N)\K

and we set

Kb(t) ={x∈IR^N : (t, x)∈K}b .

We use here repetitively the terminology and the notations introduced in [8, 9] and [7]:

• A tube K is a subset of IR⁺×IR^N, such that K ∩([0, t]×IR^N) is a compact subset of IR^N+1 for any t≥0.

• A setK ⊂IR⁺×IR^N isleft lower semicontinuous if

∀t >0, ∀x∈ K(t), ift_n→t⁻, then∃x_n∈ K(t_n) such thatx_n→x .

• If s = 1,2 or (1,1), a C^s regular tube Kr is a tube with a nonempty interior and whose boundary has a C^s regularity, and is such that at any point (t, x)∈∂Kr, the outward unit normalν_(t,x)^Kr = (ν_t, ν_x) toKr

at (t, x) satisfies

ν_x6= 0. (12)

• Thenormal velocityV_(t,x)^Kr of aC¹ regular tubeKrat the point (t, x) ∈

∂Kr is defined by

V_(t,x)^Kr =− ν_t

|ν_x|, (13)

whereν_(t,x)^Kr = (νt, νx) is the outward unit normal toKr at (t, x).

• AC¹ regular tube Kr isexternally tangentto a tube K at (t, x)∈ K if K ⊂ Krand (t, x)∈∂Kr .

It isinternally tangentto K at (t, x)∈Kb if Kr ⊂ Kand (t, x)∈∂Kr .

• We say that a sequence ofC^1,1 tubes (Kn) converges to someC^1,1tube Kin the C^1,b senseif (Kn) converges toK and (∂Kn) converges to∂K for the Hausdorff distance, and if there is an open neighborhoodO of

∂K such that, ifd_K (respectivelyd_Kn) is the signed distance (5) to K (respectively to Kn), then (d_Kn) and (Dd_Kn) converge uniformly to d_K and Dd_K on O and kD²d_Knk∞ are uniformly bounded onO.

Remark 2.1

1. The reason to introduce C^1,1 and C² tubes is clear when looking at (6) and (7): ¯h is well defined if Kr is a C^1,1 tube (see Section 2.2) and F is well defined ifKr is a C² tube (to be able to compute the curvature ofKr).

Therefore, (6) is well defined whenKr is a C² regular tube and, following the ideas of viscosity solutions (see [11]),C² regular tubes will play the role of “test-functions” in the following definition.

2. For simplicity, we gave all the above definitions with tubes defined for all time t≥0. But this is not the point, everything in the sequel is local in time, so we use the same definitions with tubesKwhereK(s) is only defined in a neighborhood of some fixed t. Note that smooth tubes will always be defined locally in time.

Definition 2.1 Let K be a tube and K₀∈ D be an initial set.

1. K isa viscosity subsolution to the front propagation problem (in short FPP) (6) if K is left lower semicontinuous and K(t) ∈ D for any t, and if, for any C² regular tube Kr externally tangent to K at some point (t, x), with Kr(t)∈ D and t >0, we have

V_(t,x)^Kr ≤h_λ(x,Kr(t)) where V_(t,x)^Kr is the normal velocity of Kr at (t, x).

We say that Kis a subsolution to the FPP (6) with initial position K₀ ifK is a subsolution and if K(0)⊂K₀.

2. K is a viscosity supersolution to the FPP (6) if Kb is left lower semi- continuous, and K(t) ⊂ D for any t, and if, for any C² regular tube Kr internally tangent to K at some point (t, x), with Kr(t) ∈ D and t >0, we have

V_(t,x)^Kr ≥h_λ(x,Kr(t)).

We say that K is a supersolution to the FPP (6) with initial position K0 if K is a supersolution and if Kb(0)⊂IR^N\K0.

3. Finally, we say that a tube K is a viscosity solution to the front prop- agation problem (with initial position K0) ifK is a sub- and a super- solution to the FPP (with initial position K₀).

Remark 2.2 The operator h_λ defined in (7) is the sum of a local operator F and a nonlocal one ¯h.As in the theory of viscosity solutions, we can local- ize arguments related to the local part of the operator. More precisely, C²

regularity of the boundary of the tube is required to compute the curvature in F, but only C^1,1 regularity is needed to compute the nonlocal part ¯h.

Therefore, the above definition is equivalent if we replace “for anyC² regu- lar tubeKr internally (respectively externally) tangent to K at some point (t, x)...” by “for anyC^1,1regular tubeKrinternally (respectively externally) tangent toK at some point (t, x) such that ∂Kr isC² in a neighborhood of (t, x)...” We will use this equivalent definition in the proof of Theorem 4.1.

2.2 Regularity properties of the velocity ¯h

We complete this part by recalling the regularity properties of the nonlocal term ¯hdefined by (8) and (9). These results were already given in [7], so we omit the proofs. Here we assume that the set S and the function g satisfy assumptions (10).

Because of the maximum principle, the function ¯h is nonnegative and nondecreasing: ifK₁ ∈ D andK₂ ∈ D are closed and with aC^1,1 boundary, ifK₁⊂K₂ and if x∈∂K₁∩K₂, then 0≤¯h(x, K₁)≤¯h(x, K₂).

Furthermore, ¯h is continuous in the following sense: IfK_n and K ∈ D are closed subsets of IR^N with C^1,1 boundary such that Kn converge to K in theC^1,b sense, ifx_n∈∂K_n converge to x∈∂K, then

limn

h(x¯ _n, K_n) = ¯h(x, K).

This is a straightforward application of [17, Theorem 8.33].

Next we give a result describing the behaviour of ¯h for large ball:

Lemma 2.2 For any x0 ∈IR^N, there are constants r0 >0 and α >0 such that

∀r≥r₀, ∀x∈∂B(x₀, r), ¯h(x, B(x₀, r))≤

( αr^2−2N ifN 6= 2, α

r²|log(r)|² ifN = 2.

Moreover, the constants r₀ and α only depend on S and on kgk∞.

The proof is based on standard construction of supersolutions to (9) for Ω =B(0, r), and so we omit it.

Lemma (2.2) states that ¯h is small when Ω is a large ball. On the contrary, the following lemma means that ¯h is large when “Ω is close toS.”

For all γ ≥0, we introduce

S_γ ={x∈IR^N , d_S(x)≤γ}. (14) Then, we have

Lemma 2.3 There exist γ₀ >0 and a constant α >0 which depends only on g and S such that, for all γ ∈(0, γ₀),

¯h(x, S_γ)≥ α

γ² ∀x∈∂S_γ.

Proof of Lemma 2.3. Since S has a C² boundary, we can fix γ₀ > 0 small enough such that dS defined by (5) is C² in S2γ0\{dS <−2γ0}. We fixγ ∈(0, γ₀) and setK ={d_S ≤ −γ}. We note that d_K =d_S+γ isC² on S_2γ0\{d_S <−2γ₀}. Moreover d_K =γ on ∂S and d_K= 2γ on∂S_γ. Set

M = max{|∆d²_K(x)| |0≤d_S(x)≤γ}andm= min{g(x)|x∈∂S}. (15) Finally we set Ω =S_γ and, forβ = e^−3M/4m, we define

ϕ(r) =β(e^{−M r/(4γ2)}−1) ∀r ∈IR . We claim that

u(x) =ϕ(d²_K(x)−(2γ)²)

is a subsolution of (9). Indeed, since ϕ(0) = 0, for all x ∈ ∂S_γ, u(x) = 0.

From the definition ofβ, for all x∈∂S, u(x) =ϕ(−3γ²)≤m≤g. Setting rx=d²_K(x)−(2γ)², an easy computation gives

−∆u(x) =−ϕ⁰⁰(r_x)|D(d²_K)|²−ϕ⁰(r_x)∆(d²_K).

ButD(d²_K) = 2d_KDd_K and |Dd_K|= 1. From (15), we get

−∆u(x)≤ −4ϕ⁰⁰(r_x)d²_K+M|ϕ⁰(r_x)|. A computation of the derivatives ofϕgives

−∆u(x)≤ βM²

4γ² e^{−M rx/(4γ2)}

1−d²_K(x) γ²

.

For x ∈ Ω\S, we have d_K(x) ≥ γ and therefore we obtain −∆u(x) ≤ 0.

Finally u is a subsolution with u≥0 in Ω and u= 0 on ∂S_γ. Thus, for all x∈∂S_γ,

¯h(x, Sγ)≥ |Du(x)|²= M²e^−3M/2m² 4γ² .

QED We now recall the main regularity property of the map ¯h:

Lemma 2.4 Let R > 0 be some large constant and γ > 0 be sufficiently small such that S_γ defined by (14) has a C² boundary. There is a constant θ > 1/γ such that, for any compact set K with C^1,1 boundary such that S_γ ⊂int(K) and K⊂B(0, R−γ), for anyv∈IR^N with |v| <1/θ and any x∈∂K, we have

¯h(x+v, K +v)≥(1−θ|v|)²¯h(x, K). (16) For the proof, see [7, Proposition 2.4].

3 Interposition theorems

This part is devoted to interposition theorems in space and in space-time.

Such results are fondamental in the proof of the inclusion principle. They play the same role as Jensen’s maximum principle (see [19]) or Ishii’s lemma (see [11, Theorem 8.3]) in the standard theory of viscosity solutions.

3.1 An interposition theorem in IR^N

Let us start with an interposition result for subsets of IR^N. The following proposition is a direct consequence of Ilmanen interposition lemma [18] and can be found in [7, Proposition 3.7].

Proposition 3.1 (Interposition) Let K₁ andK₂ be two closed subsets of IR^N, with K₁ compact and such that K₁ ⊂⊂K₂. Lety₁ ∈K₁ andy₂ ∈∂K₂ be such that

|y₁−y₂|= min

z1∈K1,z2∈∂K2|z₁−z₂|.

Then there is some open subset Σ₁ of IR^N with a C^1,1 boundary, such that Σ₁ is externally tangent to K₁ aty₁ (i.e., K₁ ⊂Σ₁ andy₁ ∈∂Σ₁) and such that Σ2:= Σ1+y2−y1 is internally tangent to K2 aty2 (i.e., Σ2 ⊂K2 and y₂∈∂Σ₂).

See Figure 1 for an illustration of this proposition. The key point in this result is that the smooth set Σ₂ internally tangent toK₂is just a translation of the smooth set Σ₁ externally tangent to K₁.

The C^1,1 regularity of the sets Σ₁ and Σ₂ turns out to be optimal: one cannot expect Σ₁and Σ₂to beC²in general. Unfortunately theC²regularity will be required in the sequel to be able to deal with curvature terms. In order to overcome this difficulty, one can approximate the sets Σ₁ and Σ₂ in the following way:

K₁ y₁ y₂

Σ1

Σ2

K₂

Figure 1: Illustration of the result of Proposition 3.1.

Theorem 3.2 (Approximation) Let K₁, K₂, y₁, y₂, Σ₁ and Σ₂ be as in Proposition 3.1 and δ >0 be sufficiently small. Then there exists Σ_1,n and Σ_2,n open subsets of IR^N with C^1,1 boundary, converging respectively to Σ₁ and Σ₂ in the C^1,b sense, there exists y_1,n ∈K₁ and y_2,n ∈∂K₂ converging respectively to y1 and y2, and there exists (N −1)×(N −1) matrices X1, X₂ such that

(i) Σ1,n is externally tangent to K1 aty1,n andΣ2,n is internally tangent toK₂ at y_2,n.

(ii) For i = 1 and 2, Σ_i,n is of class C² in a neighbourhood of y_i,n with limn H_y^Σ_i,n^i,n →X_i, and

−1

δI_2(N−1) ≤

X₁ 0 0 −X₂

≤ 1 δ

I_N−1 −I_N−1

−I_N−1 I_N−1

. (17) Remark 3.1

1. Note carefully that the two approximations are not independent because of the inequalities (17), which implies in particular thatX₁≤X₂.

2. By “δ >0 sufficiently small”, we mean δ∈(0,|y₁−y₂|/(2 +|y₁−y₂|)).

The proof of Theorem 3.2 is very similar to the (more difficult) proof of the second part of Theorem 3.3 below, so we omit it.

3.2 Interposition by regular tubes

The aim of this part is to extend previous results for subsets Σ₁,Σ₂ ⊂ IR×IR^N which are regular tubes. The point here is to be able to construct tubes satisfying the regularity assumption (12).

For this we introduce some notations. In IR×IR^N we work with the norm (whereσ >0 is fixed)

|(t, x)|σ = 1

σ²t²+|x|^{2 1}₂

.

For any subsetE of IR^N+1, we note the distance to E for this norm d^σ_E(t, x) = inf

(s,y)∈E|(s, y)−(t, x)|σ .

For any two subsets A₁ and A₂ of IR^N+1, we define the minimal distance between A₁ andA₂ by

e(A1, A2) = inf

(t1,x1)∈A1,(t2,x2)∈A2

|(t2, x2)−(t1, x1)|σ . We consider the following transversality condition:

forC₁ ⊂⊂C₂⊂IR×IR^N with C₁ compact andC₂ closed, and for any (¯s₁,y¯₁)∈C₁ and any (¯s₂,y¯₂)∈Cc₂,

if |(¯s₁,y¯₁)−(¯s₂,y¯₂)|σ =e(C₁,Cc₂), then ¯s₁>0,s¯₂>0 and ¯y₁6= ¯y₂. (18)

Theorem 3.3 Let C₁ and C₂ be such that (18) holds. Let us fix (¯s₁,y¯₁)∈ C₁ and (¯s₂,y¯₂)∈Cc₂ with

|(¯s₁,y¯₁)−(¯s₂,y¯₂)|σ =e(C₁,Cc₂).

1. Interposition: There exists aC^1,1 regular tubeΣ₁, defined on an open intervalI (see Remark 2.1.2), such thatΣ1 is externally tangent toC1

at(¯s₁,y¯₁), with s¯₁ ∈I, and Σ₂ := Σ₁+ (¯s₂,y¯₂)−(¯s₁,y¯₁) is internally tangent to C₂ at (¯s₂,y¯₂).

2. Joint approximation byC² tubes: Futhermore, for any δ >0 suf- ficiently small, there existsC^1,1 regular tubesΣ_1,n andΣ_2,n converging respectively toΣ1andΣ2in theC^1,bsense, there exists(¯s1,n,y¯1,n)∈C1

and(¯s_2,n,y¯_2,n)∈Cc₂converging respectively to(¯s₁,y¯₁)and(¯s₂,y¯₂), and there exists (N −1)×(N −1) matrices X1, X2 such that

(i) Σ_1,n is externally tangent to C₁ at (¯s_1,n,y¯_1,n) and Σ_2,n is inter- nally tangent to C₂ at (¯s_2,n,y¯_2,n).

(ii) Fori= 1and2,Σi,nis of classC² in a neighbourhood of(¯si,n,y¯i,n) with lim

n H_y^Σ_¯i,n^i,n(¯si,n)→X_i and

−1

δI_2(N−1) ≤

X₁ 0 0 −X₂

≤ 1 δ

I_N−1 −I_N−1

−I_N−1 I_N−1

. (19) The proof of this theorem is done in Section 3.5.

Remark 3.2

1. Inequality (19) implies thatX₁ ≤X₂. Although we only use this latter inequality in the sequel, inequality (19) allows to treat equations with F depending onx (see for instance [11]). Let us once again point out that the two approximations are not independent because of (19).

2. Thanks to theC^1,bconvergence of Σ_1,nand Σ_2,nto Σ₁and Σ₂respectively, one also has:

limn ν_y^Σ_¯1,n^1,n(¯s1,n)= lim

n ν_y¯^Σ_2,n^2,n(¯s2,n)=ν_y¯^Σ₁^1(¯s1)=ν_y¯^Σ₂^2(¯s2), (20) limn V_(¯^Σ_s^1,n

1,n,¯y1,n) = lim

n V_(¯^Σ_s^2,n

2,n,¯y2,n)=V_(¯^Σ_s¹

1,¯y1)=V_(¯^Σ_s²

2,¯y2). (21) 3. By “δ >0 sufficiently small, we mean: δ ∈(0, e(C₁,Cc₂)/(2 +e(C₁,Cc₂))).

3.3 Existence of the regular interposition tubes

Let us introduce a new notation: if Σ is a tube defined on some open interval I (see Remark 2.1.2), then we set

bd(Σ) :=[

t∈I

∂Σ(t). (22)

The following result is the key point in the proof of the existence of the regular interposition tubes of Theorem 3.3 part (i).

Proposition 3.4 Let C₁,C₂, (¯s₁,y¯₁)∈C₁,(¯s₂,y¯₂)∈C₂ be as in Theorem 3.3. There exist a C^1,1 regular tube Σ defined on some interval I and some (t, x)∈](¯s₁,y¯₁),(¯s₂,y¯₂)[such that

t∈I, x∈∂Σ(t) and e(C₁ , Cc₂) =e(C₁ , bd(Σ)) +e(Σ, Cc₂). (23)

Above, ](¯s₁,y¯₁),(¯s₂,y¯₂)[ denotes the open segment joining (¯s₁,y¯₁) and (¯s₂,y¯₂).

Proof of Proposition 3.4. Let us first fix some notation needed through- out the proof: we set

• ¯e:=e(C₁ , Cc₂),

• E:={(t, x)∈IR⁺×IR^N : (t, x)∈](s1, y1),(s2, y2)[ where

(s₁, y₁)∈C₁ and (s₂, y₂)∈Cc₂ satisfy |(s₁, y₁)−(s₂, y₂)|σ = ¯e},

• A_ρ :={(t, x)∈IR⁺×IR^N : d^σ_C1(t, x)> ρandd^σ_c

C2(t, x)> ρ} forρ∈(0,e/2)¯ , and, ifI is an interval, thenA_ρ(I) =A_ρ∩(I×IR^N),

• (t(s), x(s)) :=s(¯s₁,y¯₁) + (1−s)(¯s₂,y¯₂) for alls∈(0,1) and

(¯t,x) := (t(1/2), x(1/2)),¯ (24)

• I_τ := (¯t−τ,t¯+τ) for all τ >0.

For later use we note that d^σ_c

C2(t(s), x(s)) =s¯e and d^σ_C1(t(s), x(s)) = (1−s)¯e for all s∈(0,1), (25) because of the definition of (¯s1,y¯1) and (¯s2,y¯2). Moreover, for a point (t, x)∈ IR⁺×IR^N, the equality ¯e−d^σ_c

C2(t, x) =d^σ_C1(t, x) holds if and only if (t, x)∈E.

We now reduce the construction of the tube Σ to the construction of a suitable functionw:

Lemma 3.5 Let I be a nonempty open interval of IR⁺ and w:Aρ(I)→IR be of classC^1,1 (for some ρ∈(0,e/2)) and such that¯

¯ e−d^σ_c

C2(s, y)≤w(s, y)≤d^σ_C1(s, y) ∀(s, y)∈A_ρ(I). (26) We also assume that there is someγ ∈(ρ,e¯−ρ)and some(t, x)∈E∩A_ρ(I) with w(t, x) =γ and such that

Dxw(s, y)6= 0 ∀(s, y)∈Aρ(I) with w(s, y) =γ . (27) Then the setΣ ={(s, y)∈A_ρ(I)|w(s, y)≤γ} satisfies the requirements of Proposition 3.4.

Proof of Lemma 3.5. Let us first check that Σ is a tube of class C^1,1 in the intervall I. Because of assumption (27) it suffices to show that bd(Σ)⊂ A_ρ(I) where bd(Σ) is defined by (22). Using (26) and the fact that γ ∈ (ρ,¯e−ρ), we have, for all (s, y)∈bd(Σ),

d^σ_c

C2(s, y)≥e¯−γ > ρ and d^σ_C1(s, y)≥γ > ρ . Hence (s, y)∈A_ρ(I).

Finally we show that (23) holds. Ifw(s, y) =γ, thend^σ_C1(s, y)≥γ while d^σ_c

C2(s, y)≥e¯−γ. Hence e(C1,bd(Σ))≥γ, e(Cc2,Σ)≥¯e−γ and e(C₁,bd(Σ)) +e(cC₂,Σ)≥e(C₁,Cc₂).

For the reverse inequality let us first recall that ¯e−d^σ_c

C2(t, x) = d^σ_C1(t, x) because (t, x) ∈ E. This implies that γ = d^σ_C1(t, x) ≥ e(C₁,bd(Σ)) and

¯

e−γ =d^σ_c

C2

(t, x)≥e(cC₂,Σ). Hence (23) holds.

QED Next we turn to the construction of a functionwsatisfying the assump- tions of Lemma 3.5. We advice the reader to look at Figure 2 to follow the rest of the proof of Proposition 3.4.

The first step is the following result given in [9]: let us set K1:={(t, x)∈IR⁺×IR^N :d^σ_C1(t, x)≤15¯e/16}. Then, we have

Lemma 3.6 [9, Lemma 5.1] The function ϕ(t, x) =d^σ_∂K1(t, x) is C^1,1 in a bounded open neighborhood O1 of E∩(K₁\∂K₁) and

¯ e−d^σ_c

C2(t, x)≤ 15¯e

16 −ϕ(t, x)≤d^σ_C1(t, x) ∀(t, x)∈IR⁺×IR^N . (28) We now show that ϕ has a nonvanishing spatial gradient in a neigh- borhood of E∩ O1. Let (t, x) ∈ E ∩ O1. Then there exists (s₁, y₁) ∈ C₁, (s₂, y₂)∈Cc₂ such that |(s₁, y₁)−(s₂, y₂)|σ = ¯eand (t, x)∈](s₁, y₁),(s₂, y₂)[.

Therefore, Dϕ(t, x) = ((s2, y2)−(s1, y1)))/¯e. From assumption (18), we know thaty₁6=y₂.ThusD_xϕ(t, x) 6= 0 for all (t, x)∈E∩ O1. By continuity of D_xϕin O1, there is an open set O2 ⊂ O1 which contains E∩A_¯e/8 and such that

η := min

(t,x)∈O2

|D_xϕ(t, x)|>0. (29) We are now going to modify ϕ far away from E. For this we need a technical lemma:

IR^N

IR⁺ Cb₂

C₁ K₁

( ¯s1,y¯1) ( ¯s2,y¯2)

(¯t,x)¯

O₁ O₂ E

A_¯e/8





































| {z }

Iτ

| {z }

I























 A_¯e/4

Figure 2: Illustration of the proof of Proposition 3.4.

Lemma 3.7 Let f, g, h be continuous functions in IR^k such that g≤f ≤h inIR^k. Suppose that K:={h=g}is non empty and compact and that there is some open neighbourhood U of K such that f is C^1,1 in U.

Then, for anyη >˜ 0and for any open subsetU⁰ such thatK ⊂U⁰ ⊂⊂U, there is a functionψ:IR^k→IR such that:

(i) ψ isC_loc^1,1 in IR^k and ψ isC^∞ in IR^k\U⁰, (ii) g≤ψ≤h in IR^k and g < ψ < h in IR^k\U⁰, (iii) |D(ψ−f)| ≤η˜ in U⁰.

Proof of Lemma 3.7. Let U₁ and U₂ be two open subsets of IR^k such thatK ⊂U₂ ⊂⊂U₁ ⊂⊂U⁰ and fix some smooth map θ: IR^k→ [0,1] such that θ = 1 in U2 and θ = 0 in IR^k\U1. Then we consider a smooth map ξ:IR^k→IRsuch thatg < ξ < h inIR^k\U₂,

kξ−fkL^∞(U⁰\U2)≤ η˜ 2kDθk∞

and kDξ−DfkL^∞(U⁰\U2)≤ η˜ 2 . The construction of such a functionξ is possible becauseg < f < h outside ofK and f isC^1,1 in U. Then we set

ψ(x) =θ(x)f(x) + (1−θ(x))ξ(x) ∀x∈IR^k .

Note first that (i) and (ii) obviously hold. As for (iii), it clearly holds inU₂ since ψ=f inU₂. Moreover, forx∈U⁰\U₂, we have

|D(ψ−f)(x)| ≤ |D(ξ−f)(x)|+|f(x)−ξ(x)||Dθ(x)| ≤ η˜

2 +η˜|Dθ(x)| 2kDθk∞ ≤η .˜

QED Next, we apply Lemma 3.7 withk :=N+1,U :=O1,U⁰ :=O2, ˜η:=η/2 whereη is given by (29),

g(t, x) := ¯e−d^σ_c

C2(t, x)−d^σ_A¯e/8(t, x), h(t, x) :=d^σ_C1(t, x) +d^σ_A¯e/8(t, x) andf(t, x) := 15¯e/16−ϕ(t, x) for (t, x)∈IR⁺×IR^N. We extend f,g andh fort≤0 by setting f(t, x) =f(0, x), g(t, x) =g(0, x) andh(t, x) =h(0, x).

From Lemma 3.6 we haveg ≤f ≤h inIR^N+1. Moreover, from assumption (18), the setK :={g=h}=E∩A_¯e/8is compact and contained in (0,+∞)× IR^N and we know from Lemma 3.6 that f isC^1,1 in the neighbourhoodO1

ofK. Lemma 3.7 states that there is a map ψ:IR^N+1 →IRsuch that











(i) ψis C_loc^1,1 inIR^N+1 andC^∞ inIR^N+1\O2, (ii) ¯e−d^σ_c

C2 ≤ψ≤d^σ_C1 inA_¯e/8, (iii) ¯e−d^σ_c

C2

< ψ < d^σ_C1 inA_e/8¯ \O2, (iv)|D(ψ−f)| ≤η/2 in O2.

(30)

Putting together (29) and (iv) implies that

|D_xψ(t, x)| ≥ |D_xf(t, x)| − |D_x(ψ−f)(t, x)| ≥ η

2 ∀(t, x)∈ O2 . We now choose two open subsets U₁ and U₂ of IR^N and some τ > 0 such thatO2(¯t)⊂⊂U₂⊂⊂U₁ (recall that ¯tis defined by (24)) and

|D_xψ(t, x)| ≥ η

4 ∀(t, x)∈I_τ×U₁. (31) We also fix some smooth function θ : IR^N → [0,1] such that θ = 1 in U₂, θ= 0 in IR^N\U1 and we set

w(t, x) =θ(x)ψ(t, x) + (1−θ(x))ψ(¯t, x) ∀(t, x)∈IR^N+1 .

Note thatwbelongs toC_loc^1,1(IR^N+1)∩ C^∞(IR×(IR^N\U1)). We claim that we can chooseτ >0 sufficiently small such that

( (i) ¯e−d^σ_c

C2 ≤w≤d^σ_C1 inA_¯e/4(I_τ),

(ii)|D_xw| ≥η/8 in I_τ ×U₁. (32)

Let us prove the first assertion. Let (t, x) ∈ A_e/4¯ (I_τ). On the one hand, if (t, x) ∈ A_e/4¯ (I_τ)∩(I_τ ×U₂), then θ(x) = 1 and w(t, x) = ψ(t, x). Since A_¯e/4(I_τ) ⊂ A_e/8¯ , we conclude from (30(ii)). On the other hand, suppose (t, x) ∈ A_¯e/4(I_τ)\(I_τ ×U₂). In particular, (t, x) ∈ O/ 2(¯t) and (¯t, x) ∈ O/ 2. Therefore, from (30(iii)), we have

¯ e−d^σ_c

C2(¯t, x) < ψ(¯t, x) < d^σ_C1(¯t, x).

Using the uniform continuity of the above distance functions in the compact setA_e/4¯ (I_τ),we obtain that, for τ small enough,

¯ e−d^σ_c

C2(t, x) < ψ(¯t, x) < d^σ_C1(t, x) ∀(t, x)∈A_¯e/4(I_τ)\(I_τ ×U₂). Combining with (30(ii)), we conclude also in this case.

For the second assertion, we notice that, for (t, x)∈I_τ ×U₁, we have

|Dx(w−ψ)(t, x)| ≤ |1−θ(x)||Dx(ψ(t, x)−ψ(¯t, x))|+|ψ(t, x)−ψ(¯t, x)||Dθ(x)| with a right-handside smaller than η/8 provided τ is sufficiently small, be- cause ψ ∈ C_loc^1,1. Then, for (t, x) ∈Iτ ×U1, we have, from the choice of U1

andτ in (31),

|D_xw(t, x)| ≥ |D_xψ(t, x)| − |D_x(w−ψ)(t, x)| ≥η/8, which proves the second statement.

We now fixσ ∈(0,e/4) such that¯

(t(s), x(s))∈I_τ× O2(¯t) ∀s∈(1/2−σ,1/2 +σ).

This is possible because (¯t,x) = (t(1/2), x(1/2)) belongs to¯ O2(¯t). From (25), an easy calculation gives w(t(s), x(s)) = (1−s)¯e. Since w is smooth inIR×(IR^N\U₁), Sard Lemma states that we can find a level γ ∈((1/2− σ)¯e,(1/2 +σ)¯e) such that γ is a non critical value of w inIR×(IR^N\U1).

We claim that w and γ satisfy the requirements of Lemma 3.5. Note first that, for s= (¯e−γ)/¯e, the point (t(s), x(s)) belongs to E ∩A_¯e/4(I_τ) and satisfies w(t(s), x(s)) = γ. Moreover, (26) holds from (32(i)). Finally we show that (27) holds. Indeed, ifw(t, x) =γ for some (t, x)∈A_¯e/4(I_τ), then either x ∈ U₁, in which case D_xw(t, x) 6= 0 thanks to (32(ii)), or x /∈ U₁ and Dw(t, x) = (0, Dxw(, tx))6= 0 because γ is a non critical value of win I_τ ×(IR^N\U₁). In each case we have D_xw(t, x) 6= 0. This completes the proof of Proposition 3.4.

QED

3.4 C² regularization of a C^1,1 tangent surface near contact points

The aim of this section is to show the following fact: if a C^1,1 surface Σ is externally tangent to a set K at a point y, then it is possible to find a C^1,1 surface ˜Σ which is close to Σ (in theC^1,b sense) and is still externally tangent to K at a point ˜y close to y. Moreover, ˜Σ is more regular than Σ, namely is C² in a neighborhood of ˜y. In particular, we can use ˜Σ as a test set to estimate the curvature (see Remark 2.2).

Let us give the exact assumptions:

Let K be a subset of IR^k for k ≥ 1 and Σ be an open set with a C^1,1 boundary ∂Σ, which is externally tangent to K at some point y∈∂K. Let x /∈K be such that y is the unique projection ofxonto K and p:=Dd_K(x) is the outward normal to Σ aty. Suppose that there is a sequence of pointsxn→x, wheredK is twice differentiable with first and second derivative denoted respectively p_nandX_n, and finally assume thatp_nconverges topwhile X_n converges to someX.

(33)

Note that, by usual properties of the distance function at differentiability points, the projection ofx_nontoKis unique and converges to y. We denote byy_nthis projection.

Proposition 3.8 Under Assumption (33) we can find a sequence of open sets Σ_n with C^1,1 boundary such that

(i) Σ_n is externally tangent to K aty_n,

(ii) Σ_n has a C² boundary in a neighbourhood of y_n, with normal p_n and curvature equal to the restriction to (p_n)^⊥ of −(X_n−_n¹I_k),

(iii) Σ_n converges to Σ in the C^1,b sense.

Before starting the proof of the proposition, we need two lemmas. The first one builds, from the derivatives of the distance function at a pointa,a map φwhich has a local maximum onK at the point b, projection ofa ontoK:

Lemma 3.9 Suppose that a /∈ K and that d_K is twice differentiable at a.

Let b be the projection of a onto K. Then, for any α > 0, the (smooth) function

φ(z) =hDd_K(a), z−bi+1

2h(D²d_K(a)−αI_k)(z−b), z−bi has a strict local maximum at b on K.

Proof of Lemma 3.9. For anyz∈K, we have φ(z) =d_K(z+a−b)− α

2|z−b|²−d_K(a)− |z−b|²(z−b)

becaused_K is has a second order Taylor expansion at a. But, sincez ∈K, we haved_K(z+a−b)≤ |(z+a−b)−z|=|a−b|=d_K(a). Therefore

φ(z)≤ −α

2|z−b|²− |z−b|²(z−b)

which is negative as soon asz∈K is sufficiently close tob andz6=b.

QED From now on, we fix a smooth function θ : IR⁺ → [0,1] such that θ is nonincreasing,θ= 1 on [0,1/2] andθ= 0 on [1,+∞).

We will use several times below the following interpolation. The proof relies on straightforward computations, so we skip it.

Lemma 3.10 Let φ and ψ some C^1,1 functions in some open set O. Let

¯

y∈ O be such that φ(¯y) =ψ(¯y), and let us set, for any ρ >0, ξρ(z) =φ(z)θρ(z) +ψ(z)(1−θρ(z)) where θρ(z) =θ

|z−y¯|² ρ²

. Then, for any ρ >0 such that B(¯y, ρ)⊂⊂ O, we have

kξ_ρ−ψk∞≤C(ηρ+ (M₁+M₂)ρ²), kDξ_ρ−Dψk∞≤C(η+ (M₁+M₂)ρ) and

kD²ξρk∞≤ C

ρ(η+ (M1+M2)ρ)

for some constant C =C(k)>0, where we have set η=|Dφ(¯y)−Dψ(¯y)|, M₁ =kD²φk∞ and M₂=kD²ψk∞.

Remark 3.3

1. The normsk · k∞ are of course taken onB(¯y, ρ), sinceξ_ρ=ψoutside.

2. The key point is thatξρ coincides withφin a small neighbourhood of ¯y, but is not too far fromψ in the full setO providedρ and η are small.

We are now ready to prove the proposition.

Proof of Proposition 3.8. From Lemma 3.9 the function φ_n defined by φ_n(z) =hDd_K(x_n), z−y_ni+1

2h(D²d_K(x_n)− 1

nI_k)(z−y_n), z−y_ni has a strict local maximum aty_non K. Sincey_nis the unique projection of xnonto K, the mapψn(z) =dK(xn)− |z−xn|has a global strict maximum onK at y_n. Hence

ζ_n(z) =φ_n(z)θ_n(z) +ψ_n(z)(1−θ_n(z)) whereθ_n(z) =θ

|z−yn|² ρ²_n

, has a global strict maximum atynonK, provided we chooseρn>0, ρn→ 0 and n large enough. Since φ_n is globally smooth with uniformly bounded second order derivative, and since ψ_n is smooth with uniformly bounded second order derivative ouside of the ball B(x, dK(x)/2), and since finally ψ_n(y_n) = φ_n(y_n) = 0 and Dψ_n(y_n) = Dψ_n(y_n) = p_n, Lemma 3.10 states thatζ_nandDζ_nuniformly converge to the functionz→d_K(x)− |z−x|and its derivative respectively, in the setIR^k\B(x, d_K(x)/2), and thatkD²ζ_nk∞

is uniformly bounded inIR^k\B(x, d_K(x)/2).

Let us now denote byd_Σ the signed distance to Σ (see (5) for a defini- tion). Since∂Σ is aC^1,1 manifold, we can find some open neighbourhood O of∂Σ such that d_Σ isC^1,1 inO, withkD²d_Σk∞bounded inO. Forz∈IR^k, we define

dⁿ(z) =d_Σ(z)−β_n|y−z|² whereβ_n>0, β_n→0.

Notice that dⁿ isC^1,1 inO, thatdⁿ and its derivative converge locally uni- formly to d_Σ whereas kD²dⁿk∞ is bounded in O. The advantage of in- troducing dⁿ is that {dⁿ ≤ 0} is still externally tangent to K at y with

∂K∩∂{dⁿ≤0}={y}(instead, Σ can touch K at many points). We claim that, if we chooseβ_n= 2|y−y_n|^1/3, then, at least for n large,

dⁿ(z)≤dⁿ(yn) for any z∈K\B(yn, βn/2). (34) Indeed, forz∈K\B(y_n, β_n/2),

dⁿ(z)−dⁿ(y_n) ≤ −β_n|y−z|²−d_Σ(y_n) +β_n|y−y_n|²

≤ −β_n(β_n/2− |y−y_n|)²+|y−y_n|+β_n|y−y_n|²

≤ −|y−y_n|(1−4|y−y_n|²³), which is nonpositive for largen since y_n→y.