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Interaction of a bulk and a surface energy with a geometrical constraint A. Chambolle, M. Solci

R.I. 584

November, 2005

Interaction of a bulk and a surface energy with a geometrical constraint

Antonin Chambolle∗ and Margherita Solci† Abstract This study is an attempt to generalize in dimension higher than two the mathematical results in [8] (Computing the equilibrium conguration of epi-

taxially strained crystalline lms, SIAM J. Appl. Math. 62 (2002), no. 4, 10931121) by E. Bonnetier and the rst author. It is the study of a physical system whose equilibrium is the result of a competition between an elastic energy inside a domain and a surface tension, proportional to the perimeter of the domain. The domain is constrained to remain a subgraph. It is shown in [8] that several phenomenon appear at various scales as a result of this competition. In this paper, we focus on establishing a sound mathematical framework for this problem in higher dimension. We also provide an approximation, based on a phase-eld representation of the domain.

1

Introduction

In this paper, we seek to extend to higher dimension the results of the rst author and Eric Bonnetier in [8]. There, the authors modelize the physical system which consists in a thin lm of atoms deposited on a substrate, made of a dierent crystal. Such systems are common in the engineering of devices such as electronic chips, which are obtained by growing epitaxial lms on at surfaces. In such a situation, the mist between the crystalline lattices of the substrate and the lm induces strains in the lm. To release the elastic energy due to these strains, the atoms of the free surface of the lm may diuse and a reorganization occurs in the lm. The result of this mechanism is a competition between the surface energy of the crystal, and the bulk elastic energy. The former is roughly proportional to the free surface of the crystal, and therefore favors at congurations. The bulk energy, on the contrary, is best released if oscillatory patterns develop.

We refer to [8]

and the former study [9] for a more complete explanation of the phenomenon, and for references on stress driven rearrangement instabilities (SDRI) and epitaxial growth. ∗ CMAP (CNRS UMR 7641), Ecole Polytechnique, 91128 Palaiseau cedex, France. † DAP, Università di Sassari, Palazzo Pou Salit, 07041 Alghero, Italia.

1

Here, we restrict our study to the mathematical model which is proposed in [8] in dimension two.

We extend to higher dimension the relaxation result (implic-

itly contained in Lemma 2.1 and Theorem 2.2 in [8]), and show the correctness of the phase-eld approximation, extending [8, Thm 3.1].

Observe however that

in that paper, the bulk energy is a linearized elasticity energy that involves the symmetrized gradient of the displacement.

It seems that up to now, the theory

of special bounded deformation functions [5, 7] is not well-enough developped to make possible the generalization of our results to that case, so that we only work with

W 1,p -coercive

bulk energies. Alternatively, we could have decided to impose

an additional (articial)

L∞

constraint to the displacements, in which case the

extension to linearized elasticity energies would have been relatively easy (see for instance [13]). Numerical experiments conducted by François Jouve and Eric Bonnetier (at CMAP, Ecole Polytechnique, France, and LMC/Imag, Grenoble, France) show that the phase-eld energy introduced in Section 5, in dimension 3, yield results similar to the 2D plots in [8]. See Figure 1 which shows how an island is formed, as a result of the competition between the surface energy and the strains in the material. Here the stretch (the lattice mist) along the

y -direction,

x-direction

in stronger than in the

explaining the shape of the island. (In this example, the bulk energy is

a linearized elasticity energy.)

Figure 1: Example of an island.

To be precise, we consider in this paper a displacement in a material domain which is the subgraph of an unknown nonnegative function on an open Lipschitz set subgraph

ω ⊂ R

n−1

, the displacement

0

h.

u 0

h is dened

will be dened on the

Ωh := {x = (x , xN ) ∈ ω × (0, +∞) : xN < h(x )}

2

Assuming

of

h.

We will consider

energies of the form:

Z F (u, h) =

W (∇u) dx +

Z p 1 + |∇h|2 dx0

Ωh

u

where

sastises a prescribed boundary condition on the boundary

this paper,

u

ω

ω

will be the

(N − 1)-dimensional

ω × {0}.

In

torus and the boundary condition of

on  ∂ω  will be of periodic type, as in [8] (however, adaption to other situations

will not be dicult as long as

∂ω

is Lipschitz).

The goal of our paper is to show that the relaxed functional of

Z F (u, h) =

F

can be written

W (∇u) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Σ),

Ωh where

Σ,

is now a

the internal discontinuity set of

BV

u,

inside the subgraph

function), will be a vertical rectiable set, so that

Ωh

h

(which

Ωh ∪ Σ

can be

of

viewed as a generalized subgraph. In an article written almost simultaneously by Andrea Braides and the authors of the present paper [10], a similar problem is studied, without the constraint that the domain is the subgraph of a function.

Although this may seem more gen-

eral, showing that recovery sequences can be built, so that

F

is not only a lower

bound, but also an upper bound for the lower semicontinuous envelope of

F,

is

considerably more dicult in our setting, since the sequence which is found must satisfy the constraint, and therefore has to be built in a constructive way (and not using some general existence result). This construction follows the discretization/reinterpolation technique introduced in [12, 13]. On the other hand, the lower bound in this work is almost a straightforward consequence of [10]. Eventually, the last section in this paper deals with the phase-eld approximation of

2

F,

using the same approach as in [8].

Setting of the problem and statement of the result

2.1 Functions of bounded variation We start by recalling some denition and results, useful in this paper, concerning spaces of function of bounded variation; for this topic, we refer essentially to [6]. Let

Ω

be an open subset of

as

RN .

Given

Z u div ψ dx : ψ ∈

sup

u ∈ L1 (Ω),

Cc∞ (Ω; RN ),

its total variation is dened

 |ψ(x)| ≤ 1 ∀x ∈ Ω

.

Ω One may check that it is nite if and only if the distributional derivative is a bounded Radon measure in

Ω.

the total variation of the measure At each

x ∈ Ω,

In this case, the total variation of

Du,

and is classically denoted by

one can dene upper and lower values of

u

u

Du

u

is equal to

|Du|(Ω).

as follows: the upper

value is

  |{y ∈ Ω : u(y) > t}| ∩ Bρ (x) u+ (ξ) = inf t ∈ [−∞, +∞] : lim sup =0 |Bρ (x)| ρ→0 3

of

where

Bρ (x) is the ball of radius ρ, centered at x.

Dening the jump set of

u ∈ BV (Ω), Su

if

(H

is a

u

as

N −1

so that it admits a normal

Su := {x ∈ Ω : u− (x) < u+ (x)},

, N − 1)-rectiable

νu (x)

H

at

N −1

-a.e.

one can show that

set (in the sense of Federer [16]),

x ∈ Su ,

and

Du

Du = ∇u(x) dx + (u+ (x) − u− (x))νu (x) dHN −1 where

−(−u)+ .

The lower value is simply

decomposes as

Su (x) + Dc u

Dc u, the Cantor part, is singular with respect to the Lebesgue measure and

vanishes on any set with nite

Du

Nikodym derivative of

(N − 1)-dimensional Hausdor measure.

with respect to the Lebesgue measure

∇u(x), is a.e. the approximate gradient

of

u at x, see [6].

The Radon-

dx,

Of course, if

denoted by

u ∈ W 1,1 (Ω),

it coincides with the weak gradient. Up to now, we have considered real-valued functions. If valued,

Su

will be the union of the jumps sets of the

d

u : Ω → Rd

is vector-

components of

u.

One

shows, then, that when two of these jumps sets intersect, the corresponding normals coincide

HN −1 -everywhere in the intersection up to a change of sign.

of the derivative

Du is given by (u+ − u− ) ⊗ νu dHN −1

The jump part

Su , where now, u+

are not the upper and lower values (since there is no natural order in the orientation depends on the choice of the direction of the normal

(u− , u+ , ν)

being equivalent to

The space

c

D u = 0, Then, for

SBV (Ω)

that is,

p > 1,

u: Ω → R

we say that a function

N

u ∈ SBV (Ω), ∇u ∈ L (Ω; R )

H

and

N −1

1

u ∈ L (Ω)

We say that a function

u−

) but

(the triple

u

of functions

dx + H

belongs to the space

such that

N −1

Su .

SBVp (Ω)

if

(Su ) < +∞.

is a generalized function of bounded varia-

uT := (−T ) ∨ u ∧ T belongs to BV (Ω) for every T ≥ 0. S u ∈ GBV (Ω), setting Su = T >0 SuT , a truncation argument allows to dene

tion (u If

BV (Ω)

is absolutely continuous with respect to

p

R

d

(u+ , u− , −νu )).

is dened as the subset of

Du

νu

and

∈ GBV (Ω))

the traces

u− (x)

if

and

u+ (x)

for a.e.

x ∈ Su .

c

Dening, for

u ∈ GBV (Ω),

the

c T

Cantor part of the derivative as

|D u| = supT >0 |D u |, we say that a function u in

GBV (Ω)

if

to

belongs to

GSBVp (Ω),

for

GSBV (Ω)

p > 1,

if

|Dc u| = 0, p

N

∇u ∈ L (Ω; R )

The following compactness result for

and moreover and

SBV

H

N −1

u

in

GSBV (Ω)

belongs

(Su ) < +∞.

is proven in [3, 4] (see also [6, Thm.

4.8]).

Theorem 2.1 (Compactness in SBV ) sup

nZ

n

with

un

(un )n ⊂ SBV (Ω) o |∇un |p dx + HN −1 (Sun ) < +∞,

such that

satisfy

Ω

uniformly bounded in

u ∈ SBVp (Ω)

Let

L∞ (Ω).

u nk → u

Then, there exists a subsequence

a.e. in

Ω, ∇uk * ∇u

in

p

N

L (Ω; R ),

(unk )k

and

and

HN −1 (Su ) ≤ lim inf HN −1 (Suk ) . k→∞

If

un

is bounded only in

holds, with

L1 (Ω),

one shows easily by truncation that the results still

u ∈ GSBVp (Ω). 4

2.2 Subgraphs of nite perimeter In this paper, to simplify,

ω

is the torus

(R/Z)N −1 ;

however, the extension of our

RN −1

does not raise

will be denoted by

(x0 , xN ), x0 =

results to the case of a Lipschitz bounded open subset of any diculties.

A generic point

(x1 , . . . , xN −1 ) ∈ ω , xN ∈ R.

For

x ∈ ω×R

h : ω → R+

measurable, we consider:

Ωh = {x ∈ ω × (−1, +∞) : xN < h(x0 )}

and

0 Ω+ h = {x ∈ ω × (0, +∞) : xN < h(x )} = Ωh ∩ (ω × (0, +∞)) . If

h ∈ BV (ω; R+ ),

ω × (−1, +∞)

(i.e.,

Ωh

the set

has nite perimeter in the sense of Caccioppoli in

|DχΩh |(ω × (−1, +∞)) ≤ |ω| + |Dh|(ω) < +∞,

BV (ω × (−1, +∞))).

At each point

χΩh ∈

ξ ∈ ω one can dene the upper and lower values

h+ (ξ) and h− (ξ) as in the previous section. in

so that

h+ = h−

As before, it is known that

a.e.

ω and the set of points where h− < h+ , called the jump set of h, is denoted by Sh .

Then, if

x = (x0 , xN ) ∈ ω × (−1, +∞), xN < h− (x0 ) ⇒ x ∈ Ω1h

(the set of points

0

0 where Ωh has Lebesgue density 1), xN > h+ (x ) ⇒ x ∈ Ωh (the set of points where 0 1 it has density 0), and ∂∗ Ωh = ω × (−1, +∞) \ (Ωh ∪ Ωh ), the measure-theoretical

HN −1 -a.e.

boundary, is a subset of (and

equal to)

known that the measure-theoretical boundary is

S

ξ∈ω {ξ}

HN −1 -a.e.

× [h− (ξ), h+ (ξ)].

equal to a subset

called the reduced boundary of De Giorgi, that contains only points blow-ups

νΩh (x)

(Ωh −x)/ρ converge as ρ → 0

(hence,

Ωh

has density exactly

x

It is

∂ ∗ Ωh

where the

1 N (in Lloc (R )) to a half-space of outer normal

1/2

at

x).

Let us emphasize the fact that the boundaries paper, be intended as boundaries inside

∂Ωf , ∂∗ Ωh

will always, in this

ω × (−1, +∞), that is, they do not contain

ω × {−1}.

2.3 The relaxation result Let

W : M d×N → [0, +∞),

satisfying a For

p-growth

with

d ≥ 1,

condition. Let

h ∈ C 1 (ω; [0, +∞)),

and

Z F (u, h) =

be a continuous and quasi-convex function

0

u ∈ W 1,p (ω × (−1, 0); Rd ).

d 0 u ∈ W 1,p (Ω+ h ; R ), with u = u in ω × {0}, Z p W (∇u) dx + 1 + |∇h|2 dx0 ;

Ω+ h

ω

clearly, the same denition can be done for

u ∈ L1 (ω × (0, +∞); Rd )

such that the

+ restriction to Ωh satises the previous properties; moreover, we dene +∞ otherwise in L1 (ω × (0, +∞); Rd ) × BV (ω; [0, +∞)). It is clear that equivalently one can write that in

we set:

u ∈ W 1,p (Ωh ; Rd ),

F (u, h) =

with

u = u0

ω × (−1, 0). The main result of this paper is the proof of the following relaxation result for

the functional

F,

here written in the case

remark in Section 2.4).

5

d=1

(for the general case, see the 4th

Theorem 2.2 to the

The lower-semicontinuous envelope of the functional

1

1

L (ω × (0, +∞)) × L (ω)

topology, is the functional

F

with respect

1

F : L (ω × (0, +∞)) ×

1

L (ω) → [0, +∞] dened as:  Z   W (∇u) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Su0 ∩ Ω1h )    Ω+  h  if h ∈ BV (ω; [0, +∞)) and uχΩ+ ∈ GSBV (ω × (0, +∞)) F (u, h) = h        +∞ otherwise, where

Su0 = {(x0 , xN + t) : x ∈ Su , t ≥ 0} . Observe that, denoting sometimes write

Σ = Su0 ∩ Ω1h , Σ is a vertical

Γ = ∂∗ Ωh ∪ Σ,

rectiable set, and we will

the generalized interface.

The proof of Theorem 2.2 will be given by showing a lower and an upper bound, respectively in Section 3 (Prop.

3.1) and in Section 4 (Prop.

4.1); the thesis of

Theorem 2.2 immediately follows from these results.

2.4 Some remarks 1. In [10], a similar result is shown, with mainly two dierences, that both follow from the constraint that the set where

u is dened is a subgraph:

lim inf

in the

0

inequality, we have to keep the track of vertical parts of the boundary (Su ) that might not be in the jump set of In the

u (that is, one might have (Su0 \ Su ) ∩ Ω1h 6= ∅).

lim sup inequality, one needs to build a recovery sequence which remains

a subgraph, leading to a much more complex proof than in [10]. 2. In [8], one also considers the case where the surface tension for the substrate (of boundary

ω × {0}), σS ,

the crystal (of boundary

can be dierent from the surface tension

∂Ωh ∩ (ω × (0, +∞)),

if

h

σC

of

is smooth). In this case,

two dierent phenomena occur, depending on the fact

σS ≤ σC

or

σC < σS .

In the latter case, it is always energetically convenient to cover (or wet) all the surface of the substrate with an innitesimal layer of crystal, so that the global surface tension in the relaxed energy is

σC .

In case

σS

is less than

σC ,

then parts of the substrate might remain uncovered by the crystal, and the surface energy in the relaxed functional will be given by

σC (HN −1 (∂∗ Ωh ∩ (ω × (0, +∞))) + 2HN −1 (Su0 ∩ Ω1h )) + σS HN −1 ({x0 ∈ ω : h(x0 ) = 0}) . We do not prove this result here: we fear it would make the paper harder to read, mostly because of the notation. See also Remark 4.4. 3. Still in [8], the (2D) functional

R

constraint (

ω

h dx = 1).

F

is minimized with an additional volume

It is easy to show that the relaxed functional

not change under this constraint  see Remark 4.2 below.

6

F

does

4. In the sequel, we will assume that

d = 1, u

Adapting the proofs to the vectorial case (and

W

is scalar, hence

W

is convex.

quasiconvex) is straightfor-

ward (and would just make the notation more tedious). 5. In [8] and the problem mentionned in the introduction, it is not

x1

which is 1-periodic in the rst variable.

written with

u ∈ GSBVp (ω × (−1, +∞)):

rst directions (we recall

ω

is the

3

u−

but

Here, to simplify, everything is

that is,

u

is periodic in the

(N − 1)-dimensional

results to extend them to the case where (for instance)

GSBVp (ω × (−1, +∞)), α > 0,

u

(N − 1)

torus). Adapting the

u − α(x1 , 0, . . . , 0) ∈

would not be dicult.

A lower bound for the relaxed envelope of F

In this section we obtain a lower bound for the relaxed functional

F

by proving the

following proposition.

Proposition 3.1 un = u0

in

For every sequence

ω × (−1, 0),

(un , hn ) ∈ W 1,p (Ωhn ) × C 1 (ω; [0, +∞)),

with

such that

sup F (un , hn ) < +∞, n

there exist

Ωh )

h ∈ BV (ω; [0, +∞))

such that

and

u ∈ GSBV (ω × (0, +∞))

(with

χΩhn un → u in L1 (ω × (0, +∞)), hn → h in L1 (ω), Z Z |∇un (x)|p dx , |∇u(x)|p dx ≤ lim inf n→∞

Ω+ h

u=0

out of

and

(1)

Ω+ hn

and

HN −1 (∂∗ Ωh ) + 2HN −1 (Su0 ∩ Ω1h ) ≤ lim inf n→∞

Z p 1 + |∇hn (x0 )|2 dx0

(2)

ω

This Proposition implies immediately the lower bound for the relaxed envelope of

F,

that is the rst part of the proof of Theorem 2.2. Indeed, we obtain in the

proof that the sequence since the function functional

G(u) =

W R

(un )n

converges in fact weakly in the

W 1,p -topology,

is lower semicontinuous and quasi-convex, with growth

Ω+ h

W (∇u) dx

is weakly lower semicontinuous in

W

1,p

p,

and the

; then, in

the same hypotheses, we get the inequality:

Z Ω+ h

W (∇u(x)) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Su0 ∩ Ω1h ) Z Z p ≤ lim inf W (∇un (x)) dx + 1 + |∇hn (x0 )|2 dx0 , n→∞

Ω+ hn

Let us consider a sequence

ω

(un , hn )

such that

sup F (un , hn ) < +∞ ; n≥1

7

(3)

un → u

we show that, up to a subsequence,

1

L (ω),

L1 (ω × (0, +∞))

in

and

hn → h

in

with

F (u, h) ≤ lim inf F (un , hn ).

(4)

n→∞

To prove the lower inequality, it is sucient to consider sequences

∞

hn ∈ C (ω; [0, +∞))

and

un ∈ W

1,p

(Ω+ hn ), and

u=u

0

on

ω × {0};

(un , hn )

with

however, this

compactness property, as well as inequality (4), will still hold if we just assume that

hn ∈ W 1,1 (ω) and un ∈ SBVp (ω × (−1, +∞)) with un = u0 in ω ×(−1, 0), u(x) = 0 0 ˜ ∗ Ωhn (where A⊂B ˜ means HN −1 (A \ B) = 0). a.e. in {xN > h(x )}, and Su ⊂∂ Let us consider rst the compactness and lower semicontinuity of the jump term, and for this we will use a special notion of convergence for jump set of

SBVp

functions.

3.1 Jump set convegence The following notion of jump set convergence is introduced by Dal Maso, Francfort

p

and Toader [14, Def. 4.1] and [15, Def. 3.1]. It is called  σ -convergence. A variant, which is independent on the exponent

p > 1,

has been introduced more recently by

Giacomini and Ponsiglione, see [18]. In the sequel, we denote respectively equality and inclusion up to a negligible set by the symbols

Denition 3.2 quence

+∞

(Γn )n∈N

Let

Ω

= ˜

and

˜. ⊂

be an open set in

of subsets of

Ωσ

p

HN −1 -

RN ,

and

converges to

p ∈ (1, +∞).

We say that a se-

Γ if and only if supn∈N HN −1 (Γn )
0 with i=1 ci < +∞ P∞ S∞ ∞ that v := ˜ i=1 Svi . i=1 ci vi ∈ SBVp (Ω) ∩ L (Ω) and Sv =

N −1

such

Let us mention the following variant of the proof of Theorem 3.3, still based on

Γ ⊂ Ω, we introduce   Z p ˜ X(Γ) = v ∈ SBVp (Ω; [−1, 1]) : Sv ⊂Γ , |∇v| dx ≤ 1 .

Lemma 3.4: given

Ω 8

Then, if

HN −1 (Γ) < +∞, by Ambrosio's compactness theorem 2.1, X(Γ) is compact

L1loc (Ω)

in

supn H

(which is metrizable).

If

(Γn )n

is a sequence of jumps sets with

L =

N −1

(Γn ) < +∞, then the sets X(Γn ) all belong to   Z N −1 p XL = v ∈ SBVp (Ω; [−1, 1]) : H (Sv ) ≤ L, |∇v| dx ≤ 1 . Ω

L1loc (Ω).

(X(Γnk ))k converges in the 1 Hausdor sense (with the Hausdor distance in Lloc (Ω) induced by a distance in L1loc (Ω)) to a compact K ⊂ XL . We show that K ⊆ X(Γ) for some Γ. ∞ Let (vi )i=1 be a dense sequence in the compact set K . We rst observe that which is also compact in

since

K

is convex, given any

for an appropriate choice of

H

N −1

any let

(Sv ∪ Sv0 ) ≤ L.

θ,

Hence, a subsequence

v, v 0

in

K

w

there exists

(given by

see for instance [17]) such that

In particular, we deduce that

H

N −1

θv + (1 − θ)v 0

Sw =S ˜ v ∪ Sv0 , hence Sk ( i=1 Svi ) ≤ L for

N −1

k ≥ 1, and passing to the limit, that H (Γ) ≤ L < +∞, where we S∞ Γ = i=1 Svi . Using Lemma 3.4, we deduce that there exists v ∈ K

Γ=S ˜ v.

Hence

Γ

v ∈ K

is the limit of an appropriate subsequence

satises axiom (ii) in Denition 3.2.

and a consequence of Ambrosio's compactness axiom (i) in Denition 3.2 is satised. Hence

have with

On the other hand, any

˜ , vi(k) , k ≥ 1, with Svi(k) ⊂Γ ˜ , so that also theorem is that Sv ⊂Γ

Γnk σ p -converges

to

Γ.

We observe that an obvious consequence of Ambrosio's theorem is that if

σ

p

Γn

Γ,

-converges to

HN −1 (Γ) ≤ lim inf HN −1 (Γn ) .

(5)

n→∞

3.2 Proof of the lower inequality Let

Γn = ∂Ωhn = {x ∈ ω × (−1, +∞) : xN = hn (x0 )}

hn .

Up to a subsequence, we know by Theorem 3.3 that

as

n → ∞.

Since

subsequence,

is uniformly bounded in

hn → h

1

L (ω × (0, +∞)) Clearly,

hn

(un ),

We show that

whose jump set is

Γ

Σ

in the three parts

Sv =Γ ˜ ,

vn

˜ n. Svn ⊂Γ

t ≥ 0.

Indeed, let

Ωh

in the

Consider the functions

x 7→ v(x , xN − t)χΩh (x),

Σ0 = Γ ∩ Ω0h .

limits of converging

v ∈ SBVp (ω × (−1, +∞)) v

in

x 7→ vn (x , xN − t)χΩhn (x),

N −1

9

with

t < 1,

These functions will converge

(Sv + teN ) ∩ Ω1h ⊂ Γ, -a.e. in

be such that

SBVp (ω × (−1, +∞))

0

showing that

H

u,

and

x = (x0 , xN ) ∈ Σ, (x0 , xN + t) ∈

ω × (−1, −1 + t).

claim. In particular, we deduce that

vn = χΩhn ,

Ωh .

be a sequence weakly converging to

extended in an appropriate way in to

∂∗ Ωh , Σ = Γ ∩ Ω1h ,

is vertical: that is, for any

for any

and let

converge to

∂∗ Ωh .

will all vanish outside of

Σ ∪ (RN \ Ω1h )

0

Ωhn

is irrelevant in our study, since the functions

subsequences of

with

Equivalently, the sets

possibly extracting another

indeed, if we take in Denition 3.2 the sequence

vn → χΩh

Let us decompose

Σ0

L (ω).

W 1,1 (ω),

Γn σ p converges to some Γ

topology for the characteristic functions.

∂∗ Ωh ⊆ Γ,

we nd that

The part

in

1

be the graph of the function

Σ, νΣ · eN = 0.

which shows our

HN −1 (∂∗ Ωh ) + HN −1 (Σ) ≤ lim inf n→∞ HN −1 (Γn ).

By (5), we have

We claim

that, in addition,

HN −1 (∂∗ Ωh ) + 2HN −1 (Σ) ≤ lim inf HN −1 (Γn ). n→∞

This follows from [10] and the denition of

σ p -convergence.

Indeed, it is a conse-

lim inf -inequality in [10], applied to a sequence (vn )n≥1 ˜ v. converging in SBVp (ω × (−1, +∞)) to a v such that Σ⊂S

quence of the weakly

Let us now conclude.

If

F (un , hn )

× (−1, +∞)).

there exists

∇u

in

out of

(un )

is uniformly bounded in

Then, it is a consequence of Ambrosio's Theorem 2.1 that

u ∈ GSBVp (ω × (−1, +∞))

p

N

L (ω × (−1, +∞); R ), Ωh .

˜ n, Svn ⊂Γ

is uniformly bounded, then by integra-

tion along vertical segments we easily check that

Lploc (ω

with

such that

un (x) → u(x)

∇un *

a.e., and

u

vanishes

By point (i) in Denition 3.2, which is easily generalized to

GSBVp

so that the inequality (1) holds. Clearly,

functions (see [14, Prop. 4.6]), we have that

0 vertical, Su

∩ Ω1h

⊂ Σ.

˜ ∪ ∂∗ Ωh . Su ⊂Σ

In particular since

Σ

is

We deduce (2). Clearly, the inequality (4) follows from (1)

and (2).

4

An upper bound for the relaxed envelope of F

We now get the upper bound for the relaxed envelope of the functional

F

by proving

the following proposition.

Proposition 4.1 and

un ∈ W 1,p (Ωhn )

un χΩ+ → uχΩ+ hn

h

u, h with F (u, h) < +∞, there exist hn ∈ C 1 (ω; [0, +∞))

For any with

un = u0

in

L1 (ω × (0, +∞)),

in

Z lim sup n→∞

Ω+ hn

ω × (−1, 0),

such that

hn → h

in

L1 (ω),

and:

|∇un (x)|p dx =

Z

|∇u(x)|p dx

(6)

Ω+ h

and

lim sup n→∞

Z p 1 + |∇hn (x0 )|2 dx0 ≤ HN −1 (∂∗ Ωh ) + 2HN −1 (Su0 ∩ Ω1h ).

We note that the proposition completes the proof of Theorem 2.2. if we nd a sequence convergence

(un )n

∇un χΩ+ → ∇uχΩ+ hn

h

in

Lp ; the continuity of W

Ω+ h

lim sup

Indeed,

satisfying the equation (6), we can deduce the strong gives the general result

Z Z p lim sup W (∇un (x)) dx + 1 + |∇hn (x0 )|2 dx0 + n→∞ Ω ω Z hn ≤ W (∇u(x)) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Su0 ∩ Ω1h ), which is the

(7)

ω

inequality for the functional

10

F.

(8)

Remark 4.2 (that is,

In case one adds in the denition of functional

F (u, h) = +∞ if

R ω

h dx 6= V

where

a volume constraint

V > 0 is a xed volume), then it is easy

to show that Proposition 4.1 still holds, with the sequence

(hn )

satisfying the same

(hn ) provided by the R R proposition (without volume constraint), one clearly has rn = h dx/ ω h dx → 1 ω n volume constraint as the limit

h.

F

Indeed, given the sequence

n → ∞, and an appropriate scaling (of the form x 7→ (x0 , xN /rn )) of the functions R R and the domain will provide new sequences (un , hn ) with h dx = ω h dx, and ω n

as

still satisfying (6) and (7).

Proof of the proposition.

Let us consider, now,

u

and

h

such that

First step: approximation of (most of) the graph. approximate a generalized graph

(∂∗ Ωh , Σ),

We show that we can

Σ ⊂ Ω1h ∩ (ω × (0, +∞))

where

0

x ∈ Σ ⇒ (x , xN + t) ∈ Σ for any t 1 Ωh , with the graph of a smooth function f : ω

vertical in the sense that as

F (u, h) < +∞.

0

(x , xN + t) ∈

≥ 0

is

as long

→ R+ ,

with

Ωf ⊂ Ωh \Σ up to a small part, and a good approximation of the total surface energy R p HN −1 (∂∗ Ωh ) + 2HN −1 (Σ) (by the surface of the smooth graph ω 1 + |∇f |2 dx). Let us rst state the following lemma, which will be useful in the sequel:

Lemma 4.3 H

N −1

that

g ∈ BV (ω; R+ )

Let

(∂∗ Ωg \ ∂∗ Ωg ) = 0.

0≤f ≤g

and assume

Then, for any

ε > 0,

ω , kf − gkL1 (ω) ≤ ε

a.e. in

∂∗ Ωg

is essentially closed, that is,

there exists

f ∈ C ∞ (ω; R+ )

such

and

Z p 2 dx − HN −1 (∂ Ω ) ≤ ε . 1 + |∇f | ∗ g ω

We do not give the proof of this lemma, which is obtained by regularizing (at a scale smaller than

δ ∈ (0, 1))

the function

gδ+ = gδ ∨ 0,

where

gδ

is dened by

{x = (x0 , xN ) ∈ ω × (−1, +∞) : xN ≤ gδ (x0 )} = {x ∈ ω × (−1, +∞) : The as

(N − 1)dimensional

δ→0

dist(x, (ω

× (0, +∞)) \ Ωg ) > δ} .

measure of the boundary of this set goes to

HN −1 (∂∗ Ωg )

(along well chosen subsequences) because of the assumption that

∂∗ Ωg

is

closed. Now, let us rst assume that

ε > 0,

there exists

∞

Σ = ∅:

f ∈ C (ω; R+ )

we claim that for any

h ∈ BV (ω; R+ )

such that

kf − hkL1 (ω) + HN −1 (∂∗ Ωh ∩ Ωf ) ≤ ε . and

and

Z p 2 dx − HN −1 (∂ Ω ) ≤ ε . 1 + |∇f (x)| ∗ h

(9)

(10)

ω

We x

ε > 0.

Let us consider a mollifying kernel

the unit ball, and for any

N

ρ ∈ Cc∞ (RN ),

η > 0 let ρη (x) = (1/η) ρ(x/η).

11

For

with support in

n ≥ 1 we consider the

function strongly

wn = ρ1/n ∗ χΩh : ω × R → [0, 1]. It is well known that not only wn → χΩh R 1 in L , but also that |∇wn (x)| dx → |DχΩh |(ω × (−1, +∞)) = ω×(−1,+∞)

HN −1 (∂∗ Ωh )

as

n → +∞.

One has, for every

x ∈ Ω1h ∪ ∂ ∗ Ωh ∪ Ω0h (hence, HN −1 -a.e. x ∈ ω × (−1, +∞)):  1 if x ∈ Ω1h    1 ∗ lim wn (x) = (11) if x ∈ ∂ Ωh 2 n→∞    0 if x ∈ Ω0h

The same properties are true for the sequence of (l.s.c.) functions

(w ˜n )n≥1

de-

ned by

 wn (x) w ˜n (x) = 1

if

x ∈ ω × [0, +∞)

if

x ∈ ω × (−1, 0) .

Indeed, using the coarea formula, one sees that

Z

1

HN −1 (∂{w ˜n > s}) ds

|Dw ˜n |(ω × (−1, +∞)) = 0 1

Z

HN −1 (∂{wn > s}) ds =

≤

Z

0 since

|∇wn (x)| dx , ω×(0,+∞)

HN −1 (∂{wn > s} ∩ (ω × (−1, 0))) ≥ HN −1 ({x0 ∈ ω : wn (x0 , 0) ≤ s}) =

HN −1 (∂{w ˜n > s} ∩ (ω × (−1, 0))), of the rst one. We deduce that but since

w ˜n → χΩh ,

the second set being the projection onto

lim supn→∞ |Dw ˜n |(ω × (−1, +∞)) ≤ H

(∂∗ Ωh ),

limn→∞ |Dw ˜n |(ω × (−1, +∞)) = H

N −1

(∂∗ Ωh ).

it yields

Clearly, (11) is also true for w ˜ , since Ω1h the sequel and just write For a.e.

s ∈ (0, 1),

wn

instead of

⊃ ω × (−1, 0).

lim inf n→∞ H

N −1

limn→∞ |{wn > s}4Ωh | = 0,

s}) = H

N −1

(∂∗ Ωh ).

Let us consider

s),

and using

s ∈ (0, 1), {wn > s}

(∂{wn > s}) = H

a subsequence (possibly depending on

We drop the tilde in

w ˜n .

one also checks that

Fatou's lemma and the co-area formula, that for a.e. open set such that

ω × {0}

N −1

N −1

we may assume

(∂∗ Ωh ).

is an

Thus, up to

limn→∞ H

N −1

(∂{wn >

∗

s ∈ (2/3, 3/4) and an appropriate subsequence

such that this property is true, and we consider the corresponding sequence of sets

{x ∈ ω × (−1, +∞) : wn (x) > s∗ }. We have that HN −1 (∂∗ Ωh ∩ {wn > s∗ }) = R N −1 ∗ (x) dH χ (x), and since by (11), χ{wn >s∗ } (x) → 0 HN −1 a.e. in ∂∗ Ωh {wn >s } ∂∗ Ωh ,

we nd

HN −1 (∂∗ Ωh ∩ {wn > s∗ }) → 0

as

n → ∞.

We x

|{wn > s∗ }4Ωh | + HN −1 (∂∗ Ωh ∩ {wn > s∗ }) ≤

It is clear that

N −1 ε H (∂{wn > s∗ }) − HN −1 (∂∗ Ωh ) ≤ . 2 there exists g : ω → [0, +∞) a BV function such

{xN < g(x0 )}.

By Lemma 4.3 applied to

g,

n

large, such that

ε , 2

that

{wn > s∗ } =

we nd a smooth function

f ≤ g , f ≥ 0,

satisfying both (9) and (10). Now, assume

Σ 6= ∅.

First, possibly replacing

h by h ∧(M −1) = min{h, M −1},

M > 1 large, we may assume without loss of generality that h is bounded by M − 1. 12

Σ0

Let us then dene

0

Ω1h

Σ ∩

= Σ.

Σ

if

K

K ⊆ Σ0

Σδ = Σ ∩ Ωhδ :

with

xN ≤ hδ (x0 ) + δ} ⊆ Σ 0

0 x∈Σ {x }

∪ [xN , M ]

so that

hδ = (h − δ)+ ,

indeed, one will have that

Σ0δ ∩ {hδ (x0 ) ≤

is dened as

Σ,

H

also

as previously.

while it tends to

HN −1 (K 0 ∩ Ω1h )

N −1

0

HN −1 (Σ0 \ K) ≤ ε/10.

0

(Σ \ K ) ≤ ε/10,

Ω1h ) ≤ HN −1 (Σ).

By (11), we have that

0 HN −1 -a.e. n → ∞,

as

h

HN −1 (Σ0δ ∩ (ω × [0, M ])) ≤ (M/δ)HN −1 (Σ) < +∞.

be a compact set such that

0

HN −1 (Σ0 ∩ (ω ×

with

Let us build the sequence of l.s.c. functions

(2/3, 3/4),

and recall that by assumption,

possibly replacing (in a preliminary step)

small, and

Now, let

S

We may also assume without loss of generality that

[0, M ])) < +∞, δ>0

Σ0 =

by

outside.

and

n

We can hence choose

Observe that,

is compact.

(wn )n≥1 ,

and nd a level

χ{wn >s∗ }

converges to

In particular,

and this limit saties

K

0

1

s∗ ∈ Ω1h ,

in

HN −1 (K 0 ∩ {wn > s∗ }) →

HN −1 (Σ) − ε/10 ≤ HN −1 (K 0 ∩

such that

|{wn > s∗ }4Ωh | + HN −1 (∂∗ Ωh ∩ {wn ≥ s∗ }) ≤

ε , 4

N −1 ε H (∂{wn > s∗ }) − HN −1 (∂∗ Ωh ) ≤ , 4 and

Observe now

N −1 0 ε H (K ∩ {wn > s∗ }) − HN −1 (Σ) ≤ . 8 0 that since the set K is compact, then its Minkowski

0

|{dist(·, K ) < s}|/(2s)

H

converges to

N −1

0

(K )

as

s → 0

(see [16]).

content

Since by

the coarea formula,

|{dist(·, K 0 ) < s}| 1 = 2s 2s

s

Z

HN −1 (∂{dist(·, K 0 ) > t}) dt ,

0

we can deduce (for instance with arguments similar as in Section 3.2) that there exists a sequence

(sk )k≥1

such that

measures. In particular, if that

H

N −1

{0})) = 0

0

k

(K ∩ ∂{wn > s }) = 0

 otherwise

H

∂{dist(·, K 0 ) > sk } * 2HN −1

K0 ∗

s

is large enough, and provided we have chosen

∗

N −1

HN −1

0

(Σ )

(almost any choice suits, since

H

N −1

as

such

0

(K ∩ (ω ×

would be innite), we have

N −1 ε H (∂{dist(·, K 0 ) > sk } ∩ {wn > s∗ }) − 2HN −1 (Σ) ≤ , 2 while

|{dist(·, K 0 ) ≤ sk }| ≤ ε/4

For such values of

and

HN −1 (∂{wn > s∗ } ∩ {dist(·, K 0 ) ≤ sk }) ≤ ε/8.

k , the open set {dist(·, K 0 ) > sk }∩{wn > s∗ }∩(ω × (−1, +∞))

(with piecewise Lipschitz boundary, if of a nonnegative

BV

function

g

with

sk

was properly chosen) is the subgraph

kg − hkL1 (ω) ≤ ε/2, H

HN −1 ((∂∗ Ωh ∪ Σ) ∩ Ωg ) ≤ and

N −1

Ωg

(∂Ωg \ ∂∗ Ωg ) = 0,

ε 2

∂Ωg = (∂{dist(·, K 0 ) > sk } ∩ {wn > s∗ }) ∪ (∂{wn > s∗ } ∩ {dist(·, K 0 ) > sk }),

so that

N −1 3ε H (∂Ωg ) − (HN −1 (∂∗ Ωh ) + 2HN −1 (Σ)) ≤ 4

13

f ≤ g , f ≥ 0,

Then, invoking again Lemma 4.3, we nd a smooth function

with

kf − hkL1 (ω) ≤ ε,

and

HN −1 ((∂∗ Ωh ∪ Σ) ∩ Ωf ) ≤ ε

(12)

Z p 2 dx − (HN −1 (∂ Ω ) + 2HN −1 (Σ)) ≤ ε . 1 + |∇f (x)| ∗ h

(13)

ω

Remark 4.4

We have, in addition,

lim HN −1 ({x0 ∈ ω : fε (x0 ) = 0}) = HN −1 ({x0 ∈ ω : h(x0 ) = 0}) ,

ε→0

(fε denoting the

k > 1

exists with

h(x ) ≥ 1/k

that

H H

and

so that

lim supε→0 H

duce that

N −1

N −1

fε → h

0

x 6∈ K H

N −1

N −1

uniformly in

({fε = 0}) ≤ H

({h = 0} ∩ {fε = 0}) → H

lim inf ε→0 H

N −1

N −1

H

N −1

({h = 0}).

N −1

σs

energy of

hence

there

and

K ⊂ ω

ε

is small

{fε = 0} ⊂

({h = 0}) + 2η .

We de-

On the other hand,

({h = 0} ∩ {fε > 0}) → 0

({h = 0}),

hence

H

N −1

so

({h = 0}) ≤

({fε = 0}).

A consequence is that in case (as in [8]) the substrate tension

η > 0,

Then, if

fε (x ) ≥ 1/(2k),

will yield

we see that

ω \ K.

0

({fε = 0}) ≤ H

(∂∗ Ωh ∩ Ωfε ) → 0, N −1

Indeed, for

HN −1 ({h < 1/k}) ≤ HN −1 ({h = 0}) + η

such that

0

K ∪ {h < 1/k}

ε > 0).

obtained for a particular

HN −1 (K) ≤ η

enough,

since

f

such that

less than the supercial tension

(∂∗ Ωh , Σ)

σc

{xN ≤ 0} has a supercial

of the crystal, that is, the surface

is

σs HN −1 ({h = 0}) + σc (HN −1 (∂∗ Ωh ∩ (ω × (0, +∞))) + 2HN −1 (Σ)) , then

f

can fulll the additional requirement

Z N −1 ({f = 0}) + σc σs H

p 1 + |∇f |2 dx

{f >0}

− σs H

N −1


({h = 0}) + σc (HN −1 (∂∗ Ωh ∩ (ω × (0, +∞))) + 2HN −1 (Σ)) ≤ ε

If on the other hand mating

(h, Σ)

with

σ c < σs ,

this is not optimal (in terms of relaxation: approxi-

(h + δ, Σ + δeN ), δ

small, will reduce the energy).

Second step: approximation of both the graph and displacement. now show that if out of

Ωh ,

and

u ∈ GSBVp (ω × (−1, +∞)) u = u0

Ω1h ∩ (ω × (0, +∞))

∇un → ∇u

un

is given, with

(where

vertical, then there exists

un ∈ W 1,p (Ωhn ), un = u0 (extending both

ω × (−1, 0)

on

and

strongly in

in

Su ⊆ ∂∗ Ωh ∪ Σ, u = 0

u0 ∈ W 1,p (ω × (−1, 0))), Σ ⊂

(un , hn )n≥1 ,

with

hn ∈ C ∞ (ω; R+ ),

ω × (−1, 0), such that as n → ∞, hn → h in L1 (ω) and

∇un

with zero out of

Ωhn ), un → u

in

L1 (ω × (−1, +∞)),

Lp (ω × (−1, +∞); RN ),

Z p lim 1 + |∇hn (x)|2 dx = HN −1 (∂∗ Ωh ) + 2HN −1 (Σ) .

n→∞

We

ω 14

ε > 0.

Let us x

kf − hkL1 (ω) ≤ ε,

First, by the previous step, there exists

and such that both (12) and (13) hold.

function that is equal to Possibly choosing

f

f ∈ C ∞ (ω)

We denote by

u in Ωf , to 0 in (ω × (0, +∞))\Ωf , and to u

in

v

the

ω ×(−1, 0).

h, we may assume, also, that kv−ukL1 (ω×(−1,+∞)) ≤ ε.

closer to

Eventually, we also extend

0

with

v

(by symmetry) slightly below

ω × {−1},

to the set

ω × (−1 − δ, −1), 0 < δ < 1. Let us dene, for

ξ ∈ RN ,

the anisotropic potential

N X

Wp (ξ) :=

|ξi |p .

i=1 Clearly,

v ∈ GSBVp (ω × (−1 − δ, +∞)),

Z

and one has, if

δ

is small enough,

Z Wp (∇v(x)) dx =

Wp (∇v(x)) dx

Ωδf

ω×(−1−δ,+∞)

Z ≤

Wp (∇u(x)) dx + ε,

(14)

ω×(−1,+∞) where

Ωδf = {x ∈ ω × (−1 − δ, +∞) : xN < f (x0 )}.

Sv ⊂ ∂Ωf ∪ ((∂∗ Ωh ∪ Σ) ∩ Ωf ), For

vky,η

n≥1

let

η = 1/n

be a discretization step. Given

= (v(yη + kη)), (k1 , . . . , kN −1 ) ∈ (Z/nZ) ω × (−1 − δ, +∞)

η

,

y ∈ (0, 1)N ,

v y,η .

We let, for

and

(so that

v.

i = 1, . . . , N ,

∪ Σ) ∩ [yη + kη, yη + (k + ei )η] = ∅, + kη)

satises

we denote by

kN ≥ −(1 + δ)/η − y

are considered) a discretization of

Let us also dene a discrete jump of

i,y,η above, lk = 0 if (∂∗ Ωh i,y,η have that l = χS i (yη

v

its surface energy is estimated by (12) and (13).

N −1

only point in

The jump set of

1

and

y, k

as

otherwise. We

i where the set Sη is given by

Sηi = (∂∗ Ωh ∪ Σ) + [−ηei , 0] where

A, B

(e1 , . . . , eN )

is

RN

is the canonical basis of

and as usual the sum of two sets

A + B = {a + b : a ∈ A, b ∈ B}.

The discrete energy of

Dηy =

N X

(vky,η , (lki,y,η )N i=1 )k

p

Dηi,y

with

Dηi,y = (η)N

X

i=1

(1 − lki,y,η )

k

where the sum is taken on all

k

Let us compute the average

y,η |vk+e − vky,η | lki,y,η i + α , (η)p η

such that the segment

δ inside open set Ωf . The parameter

of variable

is dened by

α>0

R y∈(0,1)N

[yη + kη, yη + (k + ei )η]

lies

will be xed later on.

Dyη .

For each i, one has (using the change

(y, k) 7→ x = (y + k)η

Z (0,1)N

Dηi,y

Z = i Oη

(1 − χSηi )(x)

p χSηi (x) |v(x + ηei ) − v(x)| + α dx (η)p η

where the domain of integration is

Oηi

  0 = x ∈ ω × (−1 − δ, +∞) : xN < min f (x + tηei ) 0≤t≤1

OηN = {x ∈ ω × (−1 − δ, +∞) : xN < f (x0 ) − η} 15

if

i ≤ N − 1,

and

Now, using the slicing technique of Gobbino [19], used in a similar setting in [12, 13] (see also [2]), we nd that this integral is less than

p Z ∂v dx + α (x) ∂xi

Z Ωδf

|ei · νv (x)| dHN −1 (x) .

Sv ∩Ωf

HN −1 (Sv ∩ Ωf ) ≤ ε, we deduce Z √ ≤ Wp (∇v(x)) dx + α N ε .

Since by construction, using (12),

Z (0,1)N

Dηi,y

On the other hand, if for any

y and η > 0 (small) we dene the interpolate of (vky,η )k

as

v

y,η

X

(x) =

(15)

Ωδf



vky,η ∆

k∈(Z/nZ)N −1 ×Z where

∆(x) =

N Y

 x − (k + y) , x ∈ ω × R , η

(1 − |xi |)+ ,

(16)

i=1

(ηl )l≥1

then it is classical [2, 11] that there exists a sequence

1

d

L (ω × (−1, +∞); R )

as

l→∞

for a.e.

N

y ∈ (0, 1)

subsequence, we deduce from (15) that there exist

lim Dηyl ≤

and

kv y,ηl − vkL1 → 0

as

In the sequel, we x Consider now a cube If

∂∗ Ωh ∪ Σ

k + {0, 1}N

y ∈ (0, 1)N

y

to this value and drop the corresponding superscript. such that

1 2N −1

N X

i,ηl

happens is bounded by this case, as long as

∂∗ Ωh ∪ Σ 0

C ⊂

∂∗ Ωh ∪ Σ

to the energy

(since at least one l ˆ k

is

1).

Dηl

cubes.

i

and

kˆ ∈

i

2

Ck

N −1

to the energy

N −1

c/(ηl )

Ck

Wp (∇v (x)) dx. Ck ,

crosses one of the edges of

is at least

a.e. in

C

0

Ck ⊂ Ωδf

then the con-

α(ηl /2)N −1 = αHN −1 (∂Ck )/(N 2N )

, hence their total volume by

v=0

since

cubes. By inequality (30) in

must cross an edge of every other cube

Ωδf , or unless

Dηl ,

ηl

R

By (17), the total number of cubes

We call a jump cube a cube

∂∗ Ωh ∪ Σ

for any

(ηl )p

is shared by

Lemma A.1, this sum is larger or equal to On the other hand, if

= 0

p ηl − vkˆηl vk+e ˆ

X

N i=1 k∈k+{0,1} ˆ ˆi =ki k

ˆ l , (y + kˆ + ei )ηl ] [(y + k)η

Ck

Ck ⊂ Ωδf .

The sum

can be interpreted as the contribution of the cube

tribution of

(17)

l → ∞.

(ηl )N

each edge

such that both

ω×(−1−δ,+∞)

Ck = (y + k)ηl + (0, ηl )N

kˆi = ki .

in

. Then, possibly extracting a

i,ηl does not cross any edge of Ck , then l ˆ k

with

v y,η → v

√ Wp (∇v) dx + α N ε ,

Z

l→∞

such that

Ck

cηl

such that this

Notice that, in

0

C = Ck0 ,kN +m , m ≥ 1,

(which may happen if such that either

C 0 ⊂ Ωf \ Ω1h ).

Ck ⊂ Ωf \ Ω1h ,

or

δ crosses an edge of Ck ; the other cubes lying in Ωf are called regular

Let

J

be the union of all jump cubes, and

16

R

be the union of all regular

Cf = R ∪ J

cubes (so that

x∈J

implies

is the union of all cubes

0

(x , xN + t) ∈ J

discussion shows that of the cubes of

J

H

κ=1+

Ωδf ).

0

t ≥ 0 as long as (x , xN + t) ∈ Cf .

Then,

The above

N

(∂J ∩ ∂R) is controlled by (N 2 /α)× the contribution R η energy D l , while Wp (∇v ηl (x)) dx is controlled by the R

contributions of the cubes of Let now

contained in

N −1

to the

√

for any

Ck

R

to the same energy.

N maxξ∈ω |∇f (ξ)|,

this constant is such that

Cf + κηl eN ⊃ Ωf as soon as

l

is large enough (so that

clearly holds as soon as

xN > −1

l

(large enough), the function

sup{xN < f (x0 ) : (x0 , xN − κηl ) ∈ R}, vl (x)

xN − κηl > −1 − δ + ηl

which

ηl ≤ δ/(1 + κ)).

We now dene, for any

dene

yields

and for any

fl ∈ BV (ω)

fl (x0 ) =

by

x ∈ ω × (−1, +∞),

we also

by

 v ηl (x0 , xN − κηl ) vl (x) = 0 By construction, the boundary of

Ω fl

(in

if

− 1 < xN < fl (x0 )

otherwise.

ω × (−1, +∞))

is a piecewise smooth

compact set made of two parts: one part is contained in the (smooth) graph of

∂Ωf ,

and the rest,

∂Ωfl ∩ Ωf ,

(∂J ∩ ∂R) + κηl eN ,

is a subset of

union of facets of hypercubes. On the other hand,

Z Wp (∇vl (x)) dx + Ω fl We x

α = N 2N .

set of measure

O(ηl ) (the union of the cubes of J

We can now x

vl → v

as

which is a nite

(Ωfl ),

with

α HN −1 (∂Ωfl ∩ Ωf ) ≤ Dηl . N 2N

We now make the observation that

of the cube). Therefore,

f ).

vl ∈ W

1,p

l → ∞,

in

f,

(18)

vl = v ηl (·−κηl eN ) except on a

such that

∂∗ Ωh ∪Σ crosses an edge

1

L (ω × (−1, +∞)) (and,

as well,

fl →

l large enough so that kfl − f kL1 (ω) + kvl − vkL1 (ω×(−1,+∞)) < ε,

and

Z

Wp (∇vl (x)) dx + HN −1 (∂Ωfl ) ≤ Dηl + HN −1 (∂Ωf )

Ω fl

Z ≤

√ Wp (∇u(x)) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Σ) + (3 + 2N N N )ε

ω×(−1,+∞) where we have used (13), (14), (17) and (18). Observe eventually that if enough, we also have (since

lim inf l→∞ H

N −1

(∂Ωfl ) ≥ H

N −1

(∂Ωf )

l

is large

and using (13))

HN −1 (∂Ωfl ) ≥ HN −1 (∂∗ Ωh ) + 2HN −1 (Σ)) − 2ε . Using now Lemma 4.3, we can nd a smooth close enough to one has

0

fl , in such a way that if v =

f 0 ∈ C ∞ (ω; RN )

vl in Ω0f and

kf 0 − f kL1 (ω) + kv 0 − vkL1 (ω×(−1,+∞)) < 2ε,

17

with

f 0 ≤ fl ,

0 in (ω × (−1, +∞)) \ Ω0f ,

hence both

kf 0 − hkL1 (ω) < 3ε

kv 0 − ukL1 (ω×(−1,+∞)) < 3ε,

and

Z

and

Wp (∇v 0 )) dx + HN −1 (∂Ωf 0 )

Ωf 0

Z

Wp (∇u(x)) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Σ) + βε,

≤

ω×(−1,+∞)

√

where

β = 4 + 2N N N

is a constant, and, as well,

HN −1 (∂Ωf 0 ) ≥ HN −1 (∂∗ Ωh ) + 2HN −1 (Σ)) − 3ε . Performing this construction for sequences

(fn )n≥1 , (un )n≥1 ,

un → u

L1 (ω × (−1, +∞)),

in

fn ∈ C (ω), un ∈ W 1,p (Ωfn ), fn → h

with

Wp (∇un (x))) dx +

n→∞

yields the existence of two

∞

Z lim sup

ε = 1/n, n ≥ 1,

in

L1 (ω),

Z p 1 + |∇fn (x)|2 dx

Ω fn

ω

Z

Wp (∇u(x)) dx + HN −1 (∂∗ Ωh ) + 2HN −1 (Σ)

≤

(19)

ω×(−1,+∞) and

HN −1 (∂∗ Ωh ) + 2HN −1 (Σ)) ≤ lim inf

Z p

n→∞

The function its gradient is theorem for

un ,

extended with

∇un

GSBV

in

Ω fn

and

0

0

out of

Ω fn ,

1 + |∇fn (x)|2 dx .

(20)

ω is in

GSBV (ω × (−1, +∞)),

and

outside. Invoking now Ambrosio's compactness

functions, we nd that

∇un * ∇u

in

Lp (ω × (−1, +∞); RN ),

so that

Z

Z Wp (∇u(x)) dx ≤ lim inf n→∞

ω×(−1,+∞)

Wp (∇un (x)) dx , ω×(−1,+∞)

which, combined with (19) and (20), yields that

Z

Z

lim

Wp (∇un (x)) dx =

n→∞

lim

n→∞

∇u

in

(−1, 0)).

(21)

ω×(−1,+∞)

Z p 1 + |∇fn (x)|2 dx = HN −1 (∂∗ Ωh ) + 2HN −1 (Σ)) .

(22)

ω

In particular, we deduce from (21) (since to

Wp (∇u(x)) dx ,

ω×(−1,+∞)

p

N

L (ω × (−1, +∞); R ). Modifying

un

1 < p < +∞)

We also nd that

in order to ensure that

that

un → u

un ≡ u0

in

0

∇un

goes strongly

strongly in

ω × (−1, 0)

W 1,p (ω ×

is now not

dicult. A simple way is as follows: we choose a continuous extension operator from

W 1,p (ω ×(−1, 0)) to W 1,p (ω × (−1, +∞)), and dene, for all n, a function wn extension of

(un |ω×(−1,0) − u ).

The sequence in

0

un

Clearly,

wn → 0

strongly in

W

1,p

is then modied in the following way: we replace

Ωfn , letting it keep the value 0 outside.

as before, but, additionally,

un = u0

This new

a.e. in

18

un

as the

(ω × (−1, +∞)). un

with

un − wn

satises the same properties

ω × (−1, 1).

This shows the thesis.

5

An approximation result

We introduce in this section, as in [8], a phase-eld approximation of the functional

F.

Ωh \ Σ

The idea is to represent the subgraph

v

by a eld

that will be an

approximation of the characteristic function of this set, at a scale of order

ε.

Then,

numerically, the minimization of our new functional will provide an approximation of

(u, h)

F.

minimizing

Our approximated functional is the following:

Z

(ηε + v 2 (x))W (∇u(x)) dx

Fε (u, v) = ω×(0,+∞)

+ cV if

Z

ε 2

1 |∇v(x)| dx + ε ω×(0,+∞)

u ∈ W 1,p (ω × (0, +∞))

with

v = 1

on

ω × {0}

2

with

u, v ∈ L (ω × (0, +∞)),

two-wells potentials with

c−1 V =

R1p 2V (t) dt. 0

ε → 0.

u

The function

still denoted by bounded:

u

0

a.e. in

and

except if

ηε

(23)

v ∈ H 1 (ω × (−1, +∞)),

ω × (0, +∞).

Otherwise, for all

V

is a

t ∈ {0, 1}, V (0) = V (1) = 0,

and

Here the potential

is any function of

ε

with

is assumed to be the trace of a function in

ηε /(εp−1 ) → 0 W

1,p

as

(ω×(−1, 0)),

, and for technical reasons we also have to assume that it is

∞

0

ω × {0},

V (v(x)) dx ω×(0,+∞)

Fε (u, v) = +∞.

we let

V (t) > 0

The parameter

0

on

∂N v ≤ 0

and

1

other

u = u0

!

Z

u ∈ L (ω × (−1, 0)).

The following results generalizes in arbitrary

dimension Theorem 3.1 in [8]. However, its proof also owes a lot to [10, Sec. 5.2], where a similar approximation is studied.

Theorem 5.1 0.

Let

(εj )j≥1

be a decreasing sequence of positive numbers, going to

Then

(uj , vj ),

(i) For any

there exist

{v = 1},

u, v

if

lim supj→∞ Fεj (uj , vj ) < +∞,

such that

and there exists

vj → v

then up to a subsequence

1

L (ω × (0, +∞)), uj (x) → u(x)

in

h ∈ BV (ω; R+ )

such that

{v = 1} = Ωh ,

a.e. in

and

F (u, h) ≤ lim inf Fεj (uj , vj ) .

(24)

j→∞

(ii) For any

h ∈ BV (ω; R+ )

ω × (−1, 0) uj → u

and

and

u(x) = 0

vj → χΩh

in

and

u ∈ GSBVp (ω × (−1, +∞))

a.e. in

0

{xN > h(x )},

1

L (ω × (0, +∞)),

there exists

with

u = u0

(uj , vj )

in

such that

and

lim sup Fεj (uj , vj ) ≤ F (u, h) .

(25)

j→∞

This is almost a

(uj , vj ) in

Ωh ,

is a minimizer of

where

Remark 5.2 set

Ωh

Γconvergence

(u, h)

Fεj ,

result. We deduce in particular that if for all

then, up to a subsequence,

minimize the relaxed functional

vj → χΩh

and

uj → u

j,

a.e.

F.

The thesis of the theorem is still valid if (as in [8, Thm 3.1]) the

must satisfy a volume constraint

approximation by a constraint on

vj :

R ω

|Ωh | = V > 0

vj (x) dx = V ).

is easy, see Remark 4.2 above.

19

(which is imposed in the

The adaption of the proofs

Proof of Theorem 5.1.

Fεj (uj , vj )

Since

We rst show the rst point.

is nite,

vj

must be nondecreasing in

v˜j (x) = 0 ∨ ((vj (x) − δj xN ) ∧ 1),

by

Fεj (uj , vj ) = Fεj (uj , v˜j ) + O(1/j), Assume rst that

(0, 1), let hsj

s ∈ 0

x ∈

vj

: ω → R+

|∇0 hsj (x0 )| =

Z

x0 ∈ ω .

xN .

δj

v˜j

is strictly decreasing.

C (ω),

vj

is small enough one can ensure that

v˜j

is smooth in

{0 < v˜j < 1}. 0

v˜j (x

be the function such that

1

be as in (i).

Now, if we replace

if

is smooth, so that

ω , then clearly, hsj is in

for any

and

(uj , vj )

Let

, hsj (x0 ))

= s

For any for any

with

|∇0 v˜j (x0 , hsj (x0 ))| 1 0 ≤ |∇ v˜j (x0 , hsj (x0 ))| s 0 0 |∂N v˜j (x , hj (x ))| δj

Now, we deduce that

|∇0 v˜j (x0 , hsj (x0 ))|2 0 dx ˜j (x0 , hsj (x0 ))| ω |∂N v q  Z 1 + |∇0 hsj (x0 )|2 1  dx0 |∇0 v˜j (x0 , hsj (x0 ))|2  δj ω |∇˜ vj (x0 , hsj (x0 ))| Z |∇0 v˜j (x)|2 1 dHN −1 (x) . = δj ∂{˜vj >s} |∇˜ vj (x)| 1 δj

|∇0 hsj (x0 )|2 dx0 ≤

ω

Z

Using the coarea formula, we nd that

Z 0

1

Z

|∇0 hsj (x0 )|2

dx

0



1 δj

ds ≤

ω

Z

|∇0 v˜j (x)|2 dx < +∞ .

{1>˜ vj >0}

By approximation, we easily deduce that this remains true when

1

H (ω × (0, +∞)):

s ∈ (0, 1),

we get that for a.e. level

s represented as the subgraph of a function hj this is true for all

∈ H (ω).

the set

is just in

{˜ vj > s}

can be

We may also assume that

j ≥ 1.

Now, we notice that (using

εj 2

1

vj

Z

a2 + b2 ≥ 2ab

1 |∇˜ vj (x)| dx + εj ω×(0,+∞) Z ≥ 2

and the co-area formula)

Z V (˜ vj (x)) dx ω×(0,+∞)

q 2V (˜ vj (x))|∇˜ vj (x)| dx

ω×(0,+∞)

Z ≥

1

p 2V (s)

Z q  0 s 0 0 2 1 + |∇ hj (x )| dx

(26)

ω

0 and in particular, using Fatou's lemma, we see that

Z 0

1

  Z q p 2V (s) lim inf 1 + |∇0 hsj (x0 )|2 dx0 j→∞

≤ lim inf j→∞

ω

εj 2

Z

1 |∇˜ vj (x)| dx + εj ω×(0,+∞) 2

s ∈ (0, 1), hsj ∈ H 1 (ω) q lim inf j→∞ 1 + |∇0 hsj |2 is nite.

In particular, for a.e.

20

!

Z

for all

V (˜ vj (x)) dx ω×(0,+∞)

j ≥ 1

and in addition,

(εj ))

and

limn→∞ sn = 0,

and

By a diagonal argument, we can nd a subsequence (still denoted by a decreasing sequence such that for each

Z q

lim

j→∞

(sn )n≥1

of real numbers in

1 + |∇0 hsj n (x0 )|2 dx0 = lim inf

Z q

j→∞

ω

with

0

sn

xN > h (x )

function is independent on

n

n, hsj n

and

1 + |∇0 hsj n (x0 )|2 dx0 < +∞ .

ω

converges in

V (˜ vj (x)) → 0

and since it is then clear (since

x

with

n,

We can also assume that for each

for a.e.

(0, 1)

ω × (0, +∞))

a.e. in

v˜j (x) → 1

L1 (ω) to some function hsn , x

for a.e.

v˜j (x) → 0

xN < h (x0 ), sn

with

and will be denoted simply by

that

this

h.

sn 0 n For any n ≥ 1, let us denote by uj the function given by uj (x) if xN < hj (x ) n and by 0 otherwise: let us show that (uj )j≥1 is compact in GSBV (ω × (−1, +∞)). s n 1,p One has uj ∈ W ({x : −1 < xN < hj n (x0 )}), hence unj ∈ GSBV (ω × (−1, +∞)) sn 0 0 0 with Sun ⊆ {(x , hj (x )) : x ∈ ω}. In particular, j

H

N −1

is uniformly bounded (in

(

j ).

un j

Z q ) ≤ 1 + |∇0 hsj n (x0 )|2 dx0 ω

On the other hand,

Fεj (uj , v˜j ) ≥ (ηεj +

s2n )

Z

W (∇unj (x)) dx

ω×(0,+∞) showing that

∇unj

is uniformly bounded in

x0 ∈ ω ,

Now, for any

if we denote by

appropriately extended to a function in

1),

one sees that for any

|ˆ unj (x)|

xN

Z ≤

x

xN
0

M ∧hsj n (x0 )

W

1,p

u ˆnj

the function

unj − u0

(where

u0

is

(ω × (−1, +∞)) that vanishes for xN ≥

hsj n (x0 ),

0 so that for any

Lp (ω × (−1, +∞); RN ).

≤

1−1/p xN

Z

xN

|∂N u ˆnj (x0 , s)|p

1/p ds ,

0

x0 ∈ ω ,

and a.e.

M 2−1/p ≤ 21−1/p

|ˆ unj (x0 , s)| ds

0

Z

hsj n (x0 )

!1/p |∂N u ˆnj (x0 , s)|p

ds

.

0

We get

kˆ unj kL1 (ω×(−1,M )) ≤ C(M )k∂N u ˆnj kLp (ω×(−1,+∞)) . Therefore,

unj = u ˆnj + u0

is uniformly bounded in

L1loc (ω × (−1, +∞)).

sio's compactness theorem we deduce that there exists such that

unj (x) → un (x)

a.e. in

ω × (−1, +∞),

By Ambro-

un ∈ GSBVp (ω × (−1, +∞))

up to a subsequence.

By a diagonal argument, we can extract a subsequence (still denoted by such that as

εj → 0,

for each

n≥

0, unj (x)

n

→ u (x)

n0 construction we have that if n ≥ n, then uj (x) = n0 n from this we deduce that u (x) = u (x) a.e. in {xN 0

one checks easily that both functions vanish a.e. in

u

n

, which is simply denoted by

u

almost everywhere. Now, by

unj (x)

a.e. in

{xN < hnj (x0 )}:

< h(x0 )},

and since moreover

0

one deduces that

{xN > h(x )}

in the sequel, is independent on

21

(εj )j≥1 )

n.

We have shown the rst assertion of point (i) of the Theorem: indeed, if we let

v = χΩh , one sees that v˜j (x) → v(x) a.e., and by construction also vj (x) → v(x) a.e. ω × (0, +∞).

in

with

u=u

0

in

uj (x) → u(x)

Moreover,

ω×(−1, 0).

above the graph of

a.e. in

{x ∈ ω × (−1, +∞) : xN < h(x)},

u is in GSBVp (ω × (−1, +∞)) and vanishes

The function

h.

Let us now show (24). We follow a similar proof in [10]. We have

Z (ηεj +

v˜j2 (x))W (∇uj (x)) dx

Z ≥

ω×(0,+∞)

v ˜j (x)

 Z 2 0

ω×(0,+∞) 1

Z ≥

 s ds W (∇uj (x)) dx !

Z 2s

W (∇uj (x)) dx

ds .

{˜ vj (x)>s}

0 This inequality, together with (26), yields

Fεj (uj , v˜j ) ≥ Z 1 Z 2s

Z q p 1 + |∇0 hsj (x0 )|2 dx0 W (∇uj (x)) dx + cV 2V (s)

{˜ vj (x)>s}

0

! ds .

ω

By Fatou's lemma, we deduce that

1

Z

lim inf 0

Z q p 1 + |∇0 hsj (x0 )|2 dx0 W (∇uj (x)) dx + cV 2V (s)

Z 2s

j→∞

{˜ vj (x)>s}

j→∞

lim inf 2s

W (∇uj (x)) dx + cV {˜ vj (x)>s}

Z q p 2V (s) 1 + |∇0 hsj (x0 )|2 dx0 < +∞ . ω

Let us choose such a consider a subsequence

s,

with additionnally

(jk )k≥1

W (∇ujk (x)) dx + cV {˜ vjk (x)>s}

and let us

Z q p 2V (s) 1 + |∇0 hsjk (x0 )|2 dx0

{˜ vj (x)>s}

ω

As above, let us introduce the sequence of functions

usjk (x) = ujk (x)

easily check that

j ≥ 1,

Z q p W (∇uj (x)) dx + cV 2V (s) 1 + |∇0 hsj (x0 )|2 dx0 .

= lim inf 2s

such that

for all

ω

Z j→∞

hsj ∈ H 1 (ω)

such that

Z lim 2s

k→∞

(27)

s ∈ (0, 1),

Z j→∞

ds

ω

≤ lim inf Fεj (uj , v˜j ) < +∞ . Therefore for a.e.

!

if

xN < hsjk (x0 )

ujk (x) → u(x)

a.e. in

and

0

usjk ∈ GSBVp (ω × (−1, +∞))

otherwise. By compactness, we

ω × (−1, +∞),

while

P1), we deduce

hsjk → h

in

L1 (ω).

By

the lower semicontinuity property (

Z 2s Ω+ h

W (∇u) + cV

p 2V (s)(HN −1 (∂∗ Ωh ) + 2HN −1 (Su0 ∩ Ω1h ))

Z ≤ lim 2s k→∞

{˜ vjk (x)>s}

Z q p 1 + |∇0 hsjk (x0 )|2 dx0 . W (∇ujk (x)) dx + cV 2V (s) ω

22

Integrating then (27) on

Fεj (uj , vj ) + o(1),

(0, 1)

and recalling that by construction,

Fεj (uj , v˜j ) =

we deduce (24).

Let us now show point (ii) of Theorem 5.1. The proof follows the lines in [8], where the same inequality is shown in the 2D case, and we will only sketch it.

h ∈ BV (ω; R+ )

Let

(−1, 0)

and

u(x) = 0

there exists

1

n

h →h

in

hn 1

in

L (ω)

and let

0

{xN > h(x )},

a.e. in

C (ω; R+ )

and

u ∈ GSBVp (ω × (−1, +∞)),

n

u →u

and

n

u ∈W

with

1,p

F (u, h) < +∞.

(Ωh ; R),

ω × (0, +∞),

a.e.in

with

with

n

u = u0

in

ω×

By Theorem 2.2,

u = u0

in

ω × (−1, 0),

with

lim sup F (un , hn ) = F (u, h) . n→∞ By construction (since we have assumed

∞

n

u ∈L in

u0 ∈ L∞ (ω × (−1, 0))),

L1 (ω

(ω ×(0, +∞)). Now, we construct sequences (unj )j and × (0, +∞)) vjn → χωhn in L1 (ω × (0, +∞)) such that

one also has that

(vjn )j with

unj → un

lim sup Fεj (unj , vjn ) ≤ F (un , hn ).

(28)

j→∞

Let us condider the sequence of functionals

Hε (v) = the

ε 2

Z

|∇v(x)|2 dx +

ω×(0,+∞)

1 ε

Z V (v(x)) dx; ω×(0,+∞)

Γ-convergence result of Modica and Mortola for such functionals (see [1]) allows

us to nd, for each

χΩhn

n,

a sequence

(vjn )j

converging to the characteristic function

such that

lim sup Hεj (vjn )

Z =

j→∞

1

p

2V (s) ds HN −1 (Sχωhn ∩ ω × (0, +∞))

0 −1 N −1 cV H (∂Ωhn ).

=

We recall that the explicit construction of the recovery sequence obtained in the following way: one considers

γj

(vjn )j

can be

solution of the Euler's equation of

the functional with appropriate boundary conditions, namely:

  −γj00 + V 0 (γj ) = 0  1   γj (0) = 1, γj √ = 0. εj This function is extended by

0

vjn (x) = γj Then, the sequence

(unj )j

√ 1/ εj .

beyond

One then lets:

 dist(x, Ω+ )  hn . εj

is constructed by translating

an appropriate cut-o function, as in [8]. We rst choose and let

wjn (x) := vjn (x0 , xN − cn

p 2εj ).

vanishes shortly beyond. Then, we let

un ,

and multiplying by

cn ≥ max{1, k∇hn kL∞ (ω) }

1 on the support of vjn , p unj (x) = un (x0 , xN − 2cn 2εj )wjn (x).

This function is

23

and (As

in the end of the proof of Proposition 4.1, we have to modify slightly

n to ensure uj

=u

0

ω × (−1, 0),

in

unj

in order

however, this is easily done, and one checks that

j ) L∞

n this modied uj satises a uniform (in

bound.) In order to show that (28)

holds, we just need to check

Z

(ηεj + (vjn (x))2 )W (∇unj (x)) dx ≤

lim sup j→∞ Since

Z

∇unj (x) = wjn (x)∇un (x0 , xN − 2cn

W (∇un (x)) dx .

(29)

Ω+ hn

ω×(0,+∞)

p p 2εj ) + un (x0 , xN − 2cn 2εj )∇wjn (x), this

inequality is clear as soon as we have established that

Z

|un (x0 , xN − 2cn

lim sup ηεj j→∞

and since

u

n

p 2εj )∇wjn (x)|p dx = 0

ω×(0,+∞)

is bounded in

L∞ ,

we need to show

Z

|∇wjn (x)|p dx = 0 .

lim sup ηεj j→∞

ω×(0,+∞)

This integral is bounded by

|γj0 |p

Z √ {0