Higher-order phase transitions with line-tension effect

DOI:10.1051/cocv/2010018 www.esaim-cocv.org

HIGHER-ORDER PHASE TRANSITIONS WITH LINE-TENSION EFFECT

Bernardo Galv˜ ao-Sousa

Abstract. The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincar´e, Anal. non lin´eaire 4(1987) 487–512], and in a different form by Albertiet al. in [Arch. Rational Mech. Anal. 144(1998) 1–46] for a first- order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies

Fε(u) :=ε³

Ω|D²u|²+1 ε

ΩW(u) +λε

∂ΩV(T u),

whereuis a scalar density function andW andV are double-well potentials, the exact scaling law is identified in the critical regime, whenελε²³ ∼1.

Mathematics Subject Classification.49Q20, 49J45, 58E50, 76M30.

Received April 13, 2009. Revised July 13, 2009 and November 8, 2009.

Published online April 23, 2010.

1. Introduction

In this paper we seek to estimate the asymptotic behavior of the family of energies ε³

Ω

|D²u|²dx+1 ε

Ω

W(u) dx+λε

∂Ω

V(T u) dH^N−1,

where u ∈ H²(Ω), Ω is a bounded open set in R^N of class C², T u is the trace of u on ∂Ω, W and V are continuous and non-negative double-well potentials with quadratic growth at infinity, and lim

ε→0⁺λε=∞.

It is known that the transition layer in the interior of the domain has width of orderε(see [2,9,14,16,20–22]).

To formally find the order of the width of the transition layer on the boundary, it suffices to study the caseN = 2.

Therefore, by focusing on a neighborhood of a point on the boundary (assuming the boundary is flat), consider

Keywords and phrases. Gamma limit, functions of bounded variations, functions of bounded variations on manifolds, phase transitions.

1 Department of Mathematics and Statistics, McMaster University, Hamilton ON L8S 4K1, Canada. [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2010

a 2−D energy in the half ball of radiusδ centered at that point x₀ of the boundary, and changing variables to a fixed domain,e.g. the unit ball, we obtain

ε³ δ²

B+|D²u|²dxdy+δ² ε

B+W(u) dxdy+λεδ

E

V(T u) dH¹.

Equi-partition of energy between the first and last terms leads toδ≈ελ⁻ε¹³ which, in turn, yields ^δ_ε² ≈ελ⁻ε²³, which vanishes with ε, which seems to indicate that the middle term will not contribute for the transition on the boundary. One also concludes that on the boundary, the energy will scale as ^ε_δ³₂ ≈λεδ≈ελε²³. Hence there are three essential regimes for this energy depending on how the quantityελε²³ behaves asε→0⁺.

In this paper we study the case in whichελε²³ converges to a finite and strictly positive value. The other two regimes will be treated in a forthcoming paper.

Consider the functional Fε(u) :=

⎧⎨

⎩ ε³

Ω

|D²u|²dx+1 ε

Ω

W(u) dx+λε

∂Ω

V(T u) dH^N−1 ifu∈H²(Ω),

∞ otherwise.

(1.1)

Theorem 1.1 (compactness). Let Ω⊂R^N be a bounded open set of classC² and letW :R→[0,∞)be such that

(H₁^W) W is continuous andW⁻¹({0}) ={a, b}for somea, b∈R, a < b;

(H₂^W) W(z)C|z|²− 1

C for allz∈Rand for someC >0.

Let V :R→[0,∞)be such that

(H₁^V) V is continuous andV⁻¹({0}) ={α, β} for someα, β∈R, α < β;

(H₂^V) V(z)C|z|²− 1

C for allz∈Rand for someC >0;

(H₃^V) V(z) 1 Cmin

|z−β|,|z−α|₂

for allz∈(α−ρ, α+ρ)∪(β−ρ, β+ρ) and for someC, ρ >0.

Assume thatελε²³ →L∈(0,∞)asε→0⁺and consider a sequence{uε} ⊂H²(Ω)such thatsup_ε>0Fε(uε)<∞.

Then there exist a subsequence {uε} (not relabeled), u ∈ BV

Ω;{a, b} , and v ∈ BV

∂Ω;{α, β} such that uε→uin L²(Ω) andT uε→v in L²(∂Ω).

The next theorem concerns the critical regime whereε and λε are “balanced”, i.e. ελε²³ ∼1, and all terms play an important role. Hereλε is large enough to render the energy sensitive to the transition that occurs on the boundary, but not too big as to force the value on the boundary to converge to a constant.

We define

(1) Ea :={x∈Ω :u(x) =a}for allu∈BV

Ω;{a, b} ;

(2) m is the energy density per unit area on the transition interfaces between the interior potential wells, precisely,

m:= inf R

−R

W(f(t)) +|f(t)|² dt:f ∈H_loc² (R), f(−t) =a, f(t) =bfor alltR, R >0

; (1.2)

(3) σis the interaction energy on the transition interface between bulk wells and boundary wells,i.e., σ(z, ξ) := inf

R 0

W(f(t)) +|f(t)|² dt:f ∈H_loc²

(0,∞) , f(0) =ξ, f(t) =z for alltR, R >0

; (1.3) (4) Fα:={x∈∂Ω :v(x) =α} for allv∈BV

∂Ω;{α, β} ;

(5) c is a lower bound to the energy on a transition interface between the wells of the boundary potential,

c:= inf

1 8

R

−R

R

−R

|f(x)−f(y)|²

|x−y|² dxdy+ R

−R

V

f(x) dx:f ∈H_loc³² (R),

f∈H¹²(R), f(−t) =α, f(t) =β for alltR, R >0

; (1.4) (6) cis an upper bound to the energy on a transition interface between the wells of the boundary potential, c:= inf

7 16

_∞

−∞

_∞

−∞

|f(x)−f(y)|²

|x−y|² dxdy+ _∞

−∞V

f(x) dx:

f ∈H_loc³² (R), f(−t) =α, f(t) =β for alltR, R >0

. (1.5) Theorem 1.2 (critical case). Under the same hypotheses of Theorem 1.1the following statements hold:

(i) (Lower bound) For everyu∈BV(Ω;{a, b})and v∈BV(∂Ω;{α, β})and for every sequence {uε} ⊂ H²(Ω)such that uε→uinL²(Ω),T uε→v inL²(∂Ω), we have

lim inf

ε→0⁺ Fε(uε)mPer_Ω(Ea) +

z=a,b

ξ=α,β

σ(z, ξ)H^N−1

{T u=z} ∩ {v=ξ} +cLPer∂Ω(Fα);

(ii) (Upper bound)For everyu∈BV(Ω;{a, b})andv∈BV(∂Ω;{α, β}), there exists a sequence{uε} ⊂ H²(Ω)such that uε→uinL²(Ω),T uε→v inL²(∂Ω), and

lim sup

ε→0⁺ Fε(uε)mPer_Ω(Ea) +

z=a,b

ξ=α,β

σ(z, ξ)H^N−1

{T u=z} ∩ {v=ξ} +cLPer∂Ω(Fα).

The main results, Theorems1.1and1.2, imply, in particular, that min

a<-

Ωudx<b α<-

∂ΩvdH^N−1<β

Fε=O(1) asε→0⁺,

where we impose a mass constraint to avoid trivial solutions which yield no energy. Note that these conditions pose no difficulties to the Γ-convergence due to the strong convergence of uε and T uε. Thus we identify the precise scaling law for the minimum energy in the parameter regimeελε²³ ∼1.

Observe that, although Theorem 1.2 does not prove that the sequence {Fε}ε>0 Γ-converges as ε → 0⁺, since the constants of the lower and upper bounds for the last transition term do not match, we can apply Theorem 8.5 from [10] to prove that there exists a subsequenceεn →0⁺such that the corresponding subsequence of functionals Γ-converges.

Hence Theorem1.2shows that the limiting functional concentrates on the three different kinds of transition layers: an interior transition layer of dimension N −1, where the limiting value of u makes the transition betweenaand b; the boundary of the domain, also of dimensionN −1, where there is the transition between the interior phases aand b and the boundary phases αand β; and a transition interface on the boundary, of dimensionN−2, where the limiting value of the traceT umakes the transition betweenαandβ.

The difficulties in proving a Γ-convergence result arise mainly from the nature of the functional under consideration. On one hand, the energy involves second-order derivatives, which prevents us from following the usual techniques in phase transitions, such as truncation and rearrangement arguments to obtain monotonically increasing test functions for the constant c. In [2], these techniques are crucial to find a test function that matches both the lifting constant and the optimal profile problem for the boundary wells. On the other hand, for the boundary term, the functionals are also nonlocal. Thus the estimates for the recovery sequence have to be sharper, since the nonlocality extends its contribution beyond the characteristic length of the phase transition.

The usual methods for localization make use of truncation arguments, which do not apply in this setting due to the fact that the fractional seminorm is of higher-order.

Similar difficulties can also be found in the papers [5–8] where, similarly, the Γ-convergence is not established.

The difference between the constantscandcarises from two factors. First, from Proposition2.9it does not follow that the lifting constant is independent of the value of the trace g. And second, when estimating the upper bound for the recovery sequence, the transition betweenαandβ is accomplished on a layer of thickness δε=o(ε). So we rescale the integrals byδε, but because of the non-locality of the fractional energy, it obtains a contribution from a layer of thicknessε, which after rescaling becomes of thicknessε/δε→ ∞. This accounts for the fact that the integration limits of the constantcextend to infinity, while for cthey are bounded.

The proofs of Theorems 1.1 and 1.2 are divided through the next sections. We begin by studying two auxiliary one-dimensional problems. More precisely, letI, J⊂Rbe two open intervals and define the following functionals

Fε(u;I) :=

⎧⎨

⎩ ε³

I

|u(x)|²dx+1 ε

I

W

u(x) dx ifu∈H²(I),

∞ otherwise,

(1.6) and

Gε(v;J) :=

⎧⎪

⎨

⎪⎩ ε³

8

J

J

v(x)−v(y)²

|x−y|² dxdy+λε

J

V

v(x) dx ifv∈H³²(J),

∞ otherwise.

(1.7) In Sections4.1and4.2we prove a compactness result and a lower bound forFεwhich follows the techniques developed in [14]. In Section4.3 we will prove a compactness result forGε, while in Section4.4 we will prove a lower bound by finding “good points” x^±_i such that most of the transition energy is concentrated between x⁻_i andx⁺_i and we modify the original sequence{un}on a small set to be admissible for c. In Section5.1 we will prove Theorem1.1in the critical regime using a slicing argument to reduce the compactness in the interior to the auxiliary problem studied in Section 4.1, and analogously, we reduce the compactness on the boundary to the one-dimensional problem for Gεstudied in Section 4.3. In Section 5.2we prove the lower bound result for Theorem 1.2 using the fact that the energy concentrates in different mutually singular sets. Finally, in Section 5.3we prove the upper bound for Theorem1.2.

From Theorem1.2, we deduce the following corollary.

Corollary 1.3. Under the same hypotheses of Theorem 1.1, and assuming that α = β, then the sequence {Fε}ε>0 Γ-converges asε→0⁺ to

F0(u) :=

⎧⎨

⎩

mPer_Ω(Ea) +

z=a,b

σ(z, α)H^N−1

{T u=z} ifu∈BV(Ω;{a, b}),

∞ otherwise,

wherem is defined as in (1.2)andσis defined as in (1.3).

From the result of Theorem1.2, we know that the Γ-limit of the functionals Fεas ε→0⁺ will concentrate its energy on three surfaces: the discontinuity surface ofu, the boundary∂Ω, and the discontinuity surface ofv.

Moreover, we know the precise energy of the first two terms. For the last term, we expect it to be the product of the perimeter of the surface times the value c of the transition between the two boundary preferred phases αandβ. Since the fractional norm on the boundary is non-local, the definition ofcshould span the whole real line and the lifting constant should be independent of the function g, as in the first-order case (see [2]). We offer the following conjecture.

Conjecture 1.4. Under the same hypotheses of Theorem1.1, then the sequence{Fε}ε>0Γ-converges asε→0⁺ to

F0(u, v) :=

⎧⎨

⎩

mPer_Ω(Ea) +

z=a,b

ξ=α,β

σ(z, ξ)H^N−1

{T u=z} ∩ {v=ξ} +cLPer∂Ω(Fα) if (u, v)∈ V,

∞ otherwise,

where V :=BV(Ω;{a, b})×BV(∂Ω;{α, β}),m is defined as in (1.2), σis defined as in (1.3), andc is defined by

c:= inf

ζ _∞

−∞

_∞

−∞

|f(x)−f(y)|²

|x−y|² dxdy+ _∞

−∞V

f(x) dx:f ∈H_loc³² (R), lim

x→∞f(−x) =α, lim

x→∞f(x) =β

, (1.8) andζ is defined by

ζ:= inf

⎧⎪

⎨

⎪⎩

R×R⁺D²u(x, y)²dxdy

R

R

g(x)−g(y)²

|x−y|² dxdy

:u∈H²(R×R⁺), T u(·,0) =g inR

⎫⎪

⎬

⎪⎭, (1.9)

which is independent ofg∈H_loc³² (R) such that limg(−x) =αas x→ ∞and limg(x) =β asx→ ∞.

2. Preliminaries 2.1. Slicing

We now show a slicing argument introduced by [2] and improved in [14]. First we fix some notation. Given a bounded open setA⊂R^N, a unit vectorein R^N, and a functionu:A→R, we denote by

M the orthogonal complement ofe, Aethe projection ofAontoM,

A^y_e:={t∈R:y+te∈A}, for ally∈Ae,

u^y_e the trace ofuonA^y_e,i.e.,u^y_e(t) :=T u(y+te), for ally∈Ae.

Definition 2.1. For everyδ >0, two sequences {vε},{wε} ⊂L¹(E) are said to be δ-close if for everyε >0 vε−wεL¹(E)< δ.

Proposition 2.2. Assume thatE is a Lipschitz, bounded and open subset of R^N−1. If {wε} ⊂L¹(E)is equi- integrable and if there are N−1 linearly independent unit vectors ei such that for every δ >0 and for every fixedi= 1, . . . , N−1, there exist a sequence{vε}(depending oni) that is δ-close to{wε} with{vε^y}precompact in L¹(E_e^y_i) forH^N−2-a.e. y∈Ee_i, then{wε} is precompact inL¹(E).

2.2. Fractional order Sobolev spaces

We will use the norms and seminorms of several fractional order spaces, introduced by Besov and Nikol’skii and summarized in [1,27]. Consider the following norms and seminorms for the spaceW^32,2(J) whereJ⊂Ris an open interval:

|u|²

H¹2(J):=

J

J

u(x)−u(y)²

|x−y|² dxdy,

|u|²

H³2(J):=

J

J

u(x)−2u_x+y

2 +u(y)²

|x−y|⁴ dxdy, u²

W³2,2(J):=u²H1(J)+|u|²

H¹2(J), u²

H³2(J):=u²L²(J)+|u|²

H³2(J).

We will need to compare the two seminorms and for that we invoke an auxiliary result (see [13,25]).

Proposition 2.3. Let r >1 and letu: (a, b)−→[0,∞] be a Borel function. Then b

a

1 (x−a)^r

x a

u(y) dy

dx 1

r−1 b

a

u(x) (x−a)^r−1dx.

Lemma 2.4. Let J ⊂Rbe an open interval and let u∈H³²(J). Then

|u|²

H³2(J) 1 8|u|²

H¹2(J).

Proposition 2.5 (Gagliardo-Nirenberg-type inequality). Let J ⊂ R be an open interval. Then there exists C=C(J)>0 such that

uH¹(J)C

u_L¹³2(J)|u|²³

H¹2(J)+uL²(J)

for allu∈H³²(J).

We recall two inequalities due to Gagliardo and Nirenberg (see [15,24]).

Proposition 2.6. Let Ω⊂R^N be a bounded open set satisfying the cone property. If u∈L²(Ω) and ∇²u∈ L²(Ω), then u∈H²(Ω)and

∇uL²(Ω)CL^N(Ω)

u_L²¹2(Ω)∇²u_L¹²2(Ω;R^N×N)+uL²(Ω)

, whereC >0 is independent ofuandΩ.

Proposition 2.7. Let J ⊂Rbe an open bounded interval. Ifu∈L¹(J) andu∈L²(J) thenu∈H²(J)and u_L⁴_3(J)C

u_L²¹1(J)u_L¹²2(J)+uL¹(J)

, for some constant C >0.

2.3. Lifting inequalities

We need to relate theL² norm of the Hessian with its equivalent on the boundary,i.e., theH¹² fractional seminorm of the derivative of the trace. In this section, we estimate the ratio between these two seminorms.

We start with an auxiliary lemma from [11].

Lemma 2.8. Let 1p <∞, letE⊂R^N andF ⊂R^mbe measurable sets and letu∈L^p(E×F). Then

F

E

|u(x, y)|dx p

dy _p¹

E

F

|u(x, y)|^pdy ¹_p

dx.

Proposition 2.9. Letg∈H³²(0, R)and consider the triangleT_R⁺:={(x, y)∈R²: 0< y < ^R₂, y < x < R−y}.

Then, 1

8 ζR,g := inf

⎧⎪

⎨

⎪⎩

T_R⁺D²u(x, y)²dxdy R

0

R 0

g(x)−g(y)²

|x−y|² dxdy

:u∈H²(T_R⁺), T u(·,0) =gin (0, R)

⎫⎪

⎬

⎪⎭ 7

16· (2.1) Proof. We divide the proof in two steps.

Step 1.Upper bound.

Define the diamond

TR:=

(x, y)∈R²: 0xR, |y|min{x, R−x}

. (2.2)

Given a functiong∈H³²(0, R), we lift it to the diamondTR by u(x, y) := 1

2y x+y

x−y

g(t) dt.

We are only interested in the lifting on the positive part of the diamond,i.e., on the triangleT_R⁺, but observe that u(x,·) is even, and we will take advantage of that fact for some estimates. Since g is continuous, one deduces immediately thatuis continuous and

T u(x,0) = lim

y→0⁺

∂u

∂x(x, y) = lim

y→0⁺

g(x+y)−g(x−y)

2y =g(x).

Moreover,

∂²u

∂x²(x, y) =g(x+y)−g(x−y)

2y ,

∂²u

∂x∂y(x, y) = g(x+y) +g(x−y)

2y −g(x+y)−g(x−y)

2y² ,

∂²u

∂y²(x, y) =g(x+y)−g(x−y)

2y −g(x+y) +g(x−y)

y² + 1

y³ x+y

x−y

g(t) dt.

We can easily deduce that ^∂∂x^2u2²

L²(T_R⁺)= ¹₄|g|²

H¹2(0,R), and note that

∂²u

∂x∂y(x, y) = 1 2y²

y 0

(g(x+y)−g(s+x) +g(x−y)−g(s+x−y)) ds.

Use Hardy’s inequality from Proposition2.3to obtain ∂²u

∂x∂y ²

L²(T_R⁺)

1

16|g|²

H¹2(0,R). Finally, notice that

∂²u

∂y²(x, y) = 1 y³

y 0

f₂(r;x, y) dr,

where f₂(r;x, y) :=

x+y r+x

(g(x+y)−g(s)) ds+

r+(x−y) x−y

(g(s)−g(x−y)) ds. Using Hardy’s inequality in Proposition2.3again, we deduce that

∂²u

∂y^{2 2}

L2(T_R⁺)

1

16|g|²

H¹2(0,R).

We finally put the three estimates for the partial derivatives of uof second order together to obtain

T_R⁺

|∇²u|²dxdy 7 16|g|²

H¹2(0,R). Step 2.Lower Bound in (2.1).

Case 1. Assume thatv∈L¹(T_R⁺;R²)∩C^∞(T_R⁺;R²) is such that∇v∈L²(T_R⁺;R^2×2).

First it is easy to prove that v(x+y,0)−v(x−y,0)

2y

² 1 2

₁

0

∇v(x+y−ty, ty)dt+ ₁

0

∇v(x−y+ty, ty)dt 2

.

By estimating the right-hand side using Lemma2.8and Minkowski inequality, we obtain

|v(·,0)|²

H¹2(0,R)8∇v²_L2(T_R⁺). Case 2. Assume thatv∈L¹(T_R⁺;R²) is such that∇v∈L²(T_R⁺;R^2×2).

First by reflection, extend the function to v ∈ L¹(TR;R²) with ∇v ∈ L²(TR;R^2×2). Let ϕε be the stan- dard mollifiers and consider vε := v ϕε defined in T_R^ε :=

(x, y)∈TR:d

(x, y), ∂TR > ε

. Then vε → u inL¹_loc(A;R²),∇vε→ ∇v inL²(A;R^2×2) andvε(·,0)→T v inL¹

A∩(R× {0});R² for any open setATR. We can find a subsequence (not relabeled) such thatvε(x,0)→T v(x) forL¹-a.e. x∈A∩(R× {0}). Then by Case 1, we have

A∩(R×{0})

A∩(R×{0})

T v(x)−T v(y) x−y

² dxdy lim inf

ε→0⁺

A∩(R×{0})

A∩(R×{0})

vε(x,0)−vε(y,0) x−y

² dxdy 8 lim

ε→0⁺

A∩T_R⁺

|∇vε|²dxdy= 8

A∩T_R⁺

|∇v|²dxdy.

LetAn⊂An+1TR be such thatTR=

An. Then one deduces that R

0

R 0

T v(x)−T v(y) x−y

²dxdy8

T_R⁺

|∇v|²dxdy.

Apply this result tov:=∇uto deduce R

0

R 0

g(x)−g(y) x−y

² dxdy8

T_R⁺

|∇²u|²dxdy,

which proves the lower bound in (2.1).

2.4. Slicing on BV

We use here the same notation as in Section2.1.

Theorem 2.10(slicing ofBV functions). Letu∈L¹(Ω). Thenu∈BV(Ω)if and only if there existN linearly independent unit vectors ei such thatu^ye_i∈BV(Ω^ye_i)for L^N−1-a.e. y∈Ωe_i and

Ω^y_ei

|Du^ye_i|(Ω^ye_i) dy <∞

for alli= 1, . . . , N.

We state an immediate corollary of Theorem 1.24 from [17].

Proposition 2.11. Let Ω⊂R^N be a bounded open Lipschitz set and letE⊂Ωbe a set of finite perimeter.

Then there are setsEn⊂Ωof class C² such that

L^N(EEn)→0,

H^N−1(∂E∂En)→0. (2.3)

Proposition 2.12 (see Sect. 5.10 in [12]). Let A⊂R^N be an open set, let E⊂A be a Borel set, let e be an arbitrary unit vector, and E has finite perimeter in A. Then E_e^y has finite perimeter in A^y_e and∂E_e^y∩A^y_e = (∂E∩A)^y_e, and

A_e

H⁰(∂Ee^y∩A^ye) dy=

A∂E∩A

νE, edH^N−1.

Conversely, E has finite perimeter in Aif there exist N linearly independent unit vectors ei,i= 1, . . . , N such

that

A_ei

H⁰(∂E_e^y_i∩A^y_ei) dy <∞ for alli= 1, . . . , N.

2.5. Functions of bounded variation on a manifold

We consider several spaces of functions with domainsA⊂R^N which are not open. Specifically,Awill be the boundary of an open and bounded set Ω of classC² and so it will be a compact Riemannian manifold (without boundary) of class C² and dimension N −1 in R^N. Such a manifold is endowed with a unit normal field ν which is continuous and defined for everyx∈A. In this section we give a brief definition of these spaces. For more details see [3,12,18].

The space of integrable functions on a manifold. LetA⊂R^N be a compact Riemannian manifold (without boundary) of classC¹ and dimensionN−1 and define the restriction measureH^N−1A(E) :=H^N−1(E∩A).

A function v is said to be integrable on A, and we write v ∈ L¹(A;H^N−1A), if and only if v is H^N−1A- measurable andH^N−1A-summable, precisely

v⁻¹(J) isH^N−1A-measurable for every open setJ ⊂R;

A

|v(x)|dH^N−1(x)<∞.

The space of functions of bounded variation on a manifold. We give a short introduction to the space of functions of bounded variation on a manifold. For more details we refer to [19].

LetTAbe the cotangent bundle ofAand let Γ(TA) be the space of 1-forms onA. Then, given a function v∈L¹(A), define the variation ofv by

|Dv|(A) := sup

A

vdivwdH^N−1:w∈Γc(TA), |w|1

. (2.4)

Then v ∈L¹(A) is said to be a function of bounded variation, i.e., v∈BV(A) if |Dv|(A)<∞. Moreover, if v=χE for some setE⊂A, thenE has finite perimeter if and only ifv∈BV(A), and

PerA(E) =|Dv|(A) =H^N−2(E∩A)<∞.

Proposition 2.13. LetΩ⊂R^N be an open bounded set of class C² and letE⊂∂Ωbe a set of finite perimeter with respect toH^N−2. Then there are setsEn⊂∂Ωof class C² such that

H^N−1(EEn)→0, H^N−2(∂∂ΩE∂∂ΩEn)→0.

3. Characterization of constants

Lemma 3.1. Assume thatV :R→[0,∞)satisfies(H₁^V)−(H₃^V). Then the constantcdefined in (1.4)belongs to(0,∞).

Proof. Assume by contradiction thatc= 0. Then there exist two sequences{fn} ⊂H_loc³² (R) and{Rn} ⊂(0,∞) satisfying

fn(−x) =α, fn(x) =β for allxRn, (3.1)

1 8

R_n

−R_n

R_n

−R_n

|fn(x)−fn(y)|²

|x−y|² dxdy+ R_n

−R_n

V

fn(x) dx−−−−→^n→∞ 0. (3.2) Let 0<2δ < β−α. Sincefn(−Rn) =α,fn(Rn) =β, andfn is continuous, there exists an interval (Sn, Tn) such that

fn(Sn) =α+δ < β−δ=fn(Tn), fn

[Sn, Tn] = [α+δ, β−δ]. (3.3) By (H₁^V) and the continuity of V we have thatCδ := min

z∈[α+δ,β−δ]V(z)>0. Then by (3.2), 0 = lim

n→∞

R_n

−R_n

V(fn(x)) dx lim

n→∞

T_n S_n

V(fn(x)) dxlim inf

n→∞ Cδ(Tn−Sn), and so Tn−Sn→0. For any t∈[0,1], define

gn(t) :=fn

Tnt+Sn(1−t) .

Thengn(0) =α+δ andg(1) =β−δ. Changing variables in (3.2) yields T_n

S_n

T_n S_n

|f_n(x)−f_n(y)|²

|x−y|² dxdy= 1 (Tn−Sn)²

₁

0

₁

0

|g_n(s)−g_n(t)|²

|s−t|² dsdt→0.

This implies that T_n^g−ⁿS_n

H¹2(0,1) → 0, and so, up to a subsequence (not relabeled), _T ^gn

n−S_n → constant in L²(0,1). SinceTn−Sn→0, this implies thatg_n →0 inL²(0,1).

On the other hand,

0< β−δ−(α+δ) =gn(1)−gn(0) = ₁

0

g_n(t) dt.

Letting n→ ∞, we obtain a contradiction. This shows thatc >0.

To prove thatc <∞, take any functionf ∈C²such that f(t)αfor t−1 andf(t) =β fort 1. It is

easy to verify that the energy is finite.

Remark 3.2. From the proof of the previous lemma, it follows that for every 0< δ <^β−₂^α, the constant

cδ:= inf 1

8 T

S

T S

|f(x)−f(y)|²

|x−y|² dxdy+ T

S

V

f(x) dx:f ∈H_loc³² (R), f(S) =α+δ, f(T) =β−δ, fn

(Sn, Tn) = [α+δ, β−δ], for someS, T ∈R

(3.4) also belongs to (0,∞).

Lemma 3.3. Define the constant c as before by

c:= inf 7

16 _∞

−∞

_∞

−∞

|f(x)−f(y)|²

|x−y|² dxdy+ _∞

−∞V

f(x) dx:

f ∈H_loc³² (R), f(−t) =α, f(t) =β, for alltR, R >0

,

whereV satisfies the properties of Theorem1.1.

Then c∈(0,∞).

Proposition 3.4. Under the conditions of Theorem 1.1,c=c, wherec is defined by

c:= inf 3

2³⁵ ₁

−1

₁

−1

|g(x)−g(y)|²

|x−y|² dxdy

¹_{3 1}

−1V

g(x) dx ²₃

:

g∈H_loc³² (R), g∈H¹²(R), g(−t) =α, g(t) =β for allt1

.

Proof. First we prove thatcc. Letη >0, andf ∈H_loc³² (R), R >0 be such that f∈H¹²(R), f(−t) =α, f(t) =β, for alltR,

1 8

R

−R

R

−R

|f(x)−f(y)|²

|x−y|² dxdy+ R

−R

V

f(x) dxc+η.

Then

c+η 1 8R²

₁

−1

₁

−1

|(f(Rx))−(f(Ry))|²

|x−y|² dxdy+R ₁

−1V

f(Rx) dx

1

8S_R² ₁

−1

₁

−1

|gR(x)−g_R (y)|²

|x−y|² dxdy+SR

₁

−1V

gR(x) dxc

wheregR(x) =f(Rx) which is admissible forc, and SR= arg min

S>0

1 8S²

₁

−1

₁

−1

|gR(x)−g_R (y)|²

|x−y|² dxdy+S ₁

−1V

gR(x) dx

=

⎛

⎜⎜

⎜⎝ ₁

−1

₁

−1

|gR (x)−g_R(y)|²

|x−y|² dxdy 4

₁

−1V

gR(x) dx

⎞

⎟⎟

⎟⎠

13

·

Letη→0⁺ to deduce thatcc. The converse inequality follows trivially from following the first part of the

proof from the end to the beginning.

Proposition 3.5. Under the conditions of Theorem 1.1,c=c, wherec is defined by

c:= inf

3·7¹³ 4

_∞

−∞

_∞

−∞

|g(x)−g(y)|²

|x−y|² dxdy

¹_{3 ∞}

−∞

V

g(x) dx ²₃

:

g∈H_loc³² (R), g(−t) =α, g(t) =β, for all t1

.

4. Two auxiliary one-dimensional problems 4.1. Compactness for F

Theorem 4.1. Assume thatW :R→[0,∞)satisfies(H₁^W)−(H₂^W). LetI⊂Rbe an open, bounded interval, let{εn} be a positive sequence converging to0, and let{un} ⊂H²(I)be such that

sup

n

Fε_n(un;I)<∞. (4.1)

Then there exist a subsequence (not relabeled) of {un} and a functionu∈BV

I;{a, b} such thatun →u in L²(I).

Proof. Given a sequence{un} ⊂H²(I) satisfying (4.1), by the compactness result in [14] and (H₂^W), we obtain a subsequence{un} (not relabeled) and a functionu∈BV

I;{a, b} such thatun→uinL²(I).

4.2. Lower bound for F

Theorem 4.2(lower bound estimate forFε). LetI⊂Rbe an open and bounded interval and letW :R→[0,∞) satisfy (H₁^W)−(H₂^W). Let u ∈ BV

I;{a, b} , let v ∈ BV

∂I;{α, β} , and let {uε} ⊂ H²(I) be such that sup

ε

Fε(uε;I) =:C <∞,uε→uinL²(I)andT uε→v inH⁰(∂I). Then

lim inf

ε→0⁺ Fε(uε;I)mH⁰(S(u)) +

∂I

σ

T u(x), v(x) dH⁰(x), wherem andσare defined in (1.2)and (1.3), respectively.

Proof. Passing to a subsequence (not relabeled), we can assume that lim inf

ε→0⁺ Fε(uε;I) = lim

ε→0⁺Fε(uε;I).

Since uε → u in L¹(I) and W(uε)L¹(I) Cε, by the growth condition (H₂^W), we have that, up to a subsequence (not relabeled),uε→uin L²(I), and sup_εuεL²(I)C.