<span class="mathjax-formula">$\Gamma $</span>-convergence for stable states and local minimizers

0-convergence for stable states and local minimizers

ANDREABRAIDES ANDCHRISTOPHERJ. LARSEN

Abstract. We introduce new definitions of convergence, based on adding stabil- ity criteria to0-convergence, that are suitable in many cases for studying conver- gence of local minimizers.

Mathematics Subject Classification (2010):49J45 (primary); 49K40 (second- ary).

1. Introduction

0-convergence is a key tool in studying approximations of energies, based on com- paring global minimizers of an energy Eand minimizers of approximate energies

En. In particular, ifEn 0

!E, then

1. ifun!uand theunasymptotically minimizeEn, thenuminimizes E

2. if u minimizes E, then there exist un ! u such that the un asymptotically minimize En(wheneverEnare equicoercive).

This convergence is used in two main ways in applications: ifEis a physical stored energy, it can sometimes be more convenient to find approximate energies that are more easily analyzed (see, e.g., [4, 5, 7, 8]). On the other hand, it can be that the physical energy has a natural small scale, such as atomistic energies, and one seeks a continuum model without a scale. In this case, the “true” physical energy is some

En, withnlarge, and the limiting energyEis the approximation.

The physical motivation for these approximations is the study of stable states, and global minimizers of stored energies are important examples. But to study all stable states, one must also consider at least strict local minimizers. Unfortu- nately, (1) above typically fails when “minimize” is replaced by “locally minimize”

or “strictly locally minimize.” This is particularly a problem when the energies En

contain oscillations, and therefore many local minimizers, while the limit E does not. These oscillations are often quite natural physically, such as in atomistic mod- els and homogenization in the presence of material microstructure. (2) is also an This research began while the second author was visiting Universit`a di Roma “Tor Vergata”. This material is partially based on work supported by the National Science Foundation under Grant No. DMS-0505660.

Received October 23, 2009; accepted in revised form February 18, 2010.

issue, except when E can easily be localized near a strict local minimizer, so that what was a strict local minimizer is now a global minimizer. We should also note that there are cases in which it has been shown that0-convergence can work for lo- cal minimization (see,e.g., Kohn-Sternberg [11], Serfaty [14], Sandier-Serfaty [13], Jerrard-Sternberg [10]). In those cases, the energiesEndid not develop fine oscil- lations. Our primary interest is developing definitions of convergence that can be used for these oscillating energies.

A definition of stability was recently introduced in [12] such that, loosely speaking, a point is stableif it is not possible to reach a lower energy state from that point without crossing an energy barrier of a specified height. That notion im- plies local minimality (in the context in which it was introduced) and is implied by strict local minimality. Moreover, it is maintained in some natural examples

of 0-convergence. Our idea of convergence is to combine this stability with 0-

convergence. To that end, in Section 2 we give a notion ofstable convergenceof energies, which implies an analog of the convergence result (1) and (2) mentioned above for local minimizers (Theorem 1). Unfortunately, contrary to0-convergence, stable convergence is not maintained after addition of continuous perturbations, a key property of0-convergence. To overcome this drawback we provide a definition of strong stable convergence, which then gives the desired property for the addi- tion of continuous perturbations (Theorem 2). Despite its technical appearance, that definition can be easily checked for sequences of oscillating energies such as in the case of the homogenization of elliptic functionals. The main examples of the paper are contained in Section 3, and concern the elliptic homogenization already mentioned, as well as the homogenization of perimeter functionals.

Finally, in Section 4 we note that the notion of stable convergence can be adapted to the analysis of more general energies, where stability at a point is al- ways trivial, such as in the case of the passage from discrete-to-continuous. In that context the energies En are defined on discrete spaces, so that stability and local minimality are always trivial, but their limit E is defined on a continuous (non- discrete) space. The stable convergence of En can be then understood as taking into account discrete paths uniformly converging to a continuous one (in that sense stability at a point is substituted by stability at a converging sequence of points).

We show how this notion of stable convergence again implies convergence of stable sequences.

2. "-stability

Xwill be a metrizable topological space, equipped with a distance denoted by dist.

Definition 2.1 ("-slide and"-stability). Let E : X ! [0,+1] and" > 0. An

"-slideforEatuis a continuous function : [0,1]! Xsuch that (0)=u,

sup

0s<t1[E( (t)) E( (s))]<"

andE( (1)) < E(u).

We say thatu is "-stablefor E if no"-slide atu exists, andstableif it is"- stable for"> 0 small enough (see [12] for motivation in the context of evolutions based on local minimization).

LetEn : X![0,+1]. A sequence{un}isuniformly stablefor{En}if, forn large enough and all">0 small enough, eachun is"-stable for En.

Note that in general stable points might not be local minimizers, since for stability, the values of Eat points close by a given point are relevant only if these points are connected by an"slide (for"arbitrarily small) to the given point. As an example one can take the real function

f(x)= 8<

:

0 ifx=0

sin

✓1 x

◆

otherwise.

In this case 0 is not a local minimizer, but it is"-stable for"<1. A similar example is

f(x)= 8<

:

0 ifx =0

x²+sin²

✓1 x

◆

otherwise.

Here also 0 is not a local minimizer, but it is"-stable for"<1.

Note, however, that for the function

f(x)= 8<

:

0 ifx =0

xsin

✓1 x

◆

otherwise, 0 is not a local minimizer, but neither is it"-stable for any">0.

Definition 2.2 (relative stability). We say that{En} isstable relative to E if the following hold:

1. Ifuhas an"-slide for Eandu_n !u, then eachu_n has an"-slide for E_n(forn large enough)

2. Ifuis a strict local minimizer ofE, then there existu_n ! uuniformly stable for{En}(and so, by (1),uis stable forE).

Note that conditions (1) and (2) above can be empty if no u has an"-slide for E or E has no strict local minimizer. A trivial example is when E is constant, in which case every {E_n} is stable relative to E. Note also that for a given E, we can consider E_n := E, and it is not generally true that{E_n}is stable relative toE.

To some degree, this is analogous to the fact that a constant sequence E_n := Eas above does not0-converge toE(unless Eis lower semicontinuous).

To make this stability more meaningful, it will be coupled with0-convergence in the sequel.

Definition 2.3 (stable convergence). If {En} is stable relative to E and 0-con- verges to E, we say that it stable converges to E. Stable convergence will be denotedEn s

!E.

Theorem 2.4. Let En s

!E.

(i)Let each stable point forEbe a local minimizer. Then, ifun !uand the unare uniformly stable forEn, thenuis a local minimizer ofE;

(ii)Let each strict local minimizer for E be a stable point. Then, each strict local minimizeruofEis the limit of a sequence of uniformly stable points forEn. Proof. (i) We first suppose thatu_n !uand eachu_nis"-stable forE_n. Then, from condition (1) in Definition 2.2,umust be -stable for E, for every 2(0,"), and hence a local minimizer by hypothesis.

(ii) We suppose thatu is a strict local minimizer of E. Then u is stable for E. So, from condition (2) in Definition 2.2, it follows that there existun !uand

">0 such that eachunis"-stable forEn.

While it is true that if{En}0-converges toE, then{En+f}0-converges toE+ f for continuous f, the following example illustrates that this is not necessarily true for stable convergence. Consequently, we need additional information to conclude the convergence of continuous perturbations, and this is the subject of Definition 2.6 and Theorem 2.7 below.

Example 2.5. Consider X = R ^{and set E}n(x) := 1+sin(nx). Then {En} 0- converges to E = 0, and, as remarked above, {En}is stable relative to E (as is every other sequence of energies!). But, {En(x)+x}does not stably converge to E(x)+xsince (1) is violated: eachx 2R^{has an}"-slide forE+x, for all">0, but if we take a sequence of local minimizersu_n of E_n+x converging tou they have no"-slide for"<2 andnlarge enough.

Definition 2.6 (strong stability). We say thatEn s-s

!E(the convergence isstrong- ly stable) if the following hold:

1. En s

!E

2. If is a path fromu(i.e., : [0,1]! X, (0)= u, and is continuous) and un !u, then there exist paths n fromunand n from n(1)such that

i)⌧ 7!E_n( _n(⌧))is decreasing up too(1)asn!+1;i.e., sup

0⌧1<⌧21

⇣E_n( _n(⌧₂)) E_n( _n(⌧₁))⌘

!0 asn! 1 ii) sup_⌧2[0,1]dist( _n(⌧), (⌧))=o(1)

iii) there exist 0=⌧₁ⁿ<⌧₂ⁿ < ... <⌧_nⁿ =1 with maxi[⌧_iⁿ ⌧_iⁿ ₁] =o(1) such that maxi|En( _n(⌧_iⁿ)) E( (⌧_iⁿ))| =o(1)andEn( _n(⌧))is betweenEn( _n(⌧_iⁿ))andEn( _n(⌧_iⁿ₊₁))for⌧ 2(⌧_iⁿ,⌧_iⁿ₊₁), up too(1);

i.e., there exist infinitesimal n>0 such that min En( _n(⌧_iⁿ)),En( _n(⌧_iⁿ₊₁)) _nEn( _n(⌧))

max En( _n(⌧_iⁿ)),En( _n(⌧_iⁿ₊₁)) + n

3. Eand each E_nare lower semicontinuous

4. "-stability forEimplies local minimality for all">0.

We remark that while (2) iii) above may seem unwieldy, and instead a condition like “kEn( _n(⌧)) E( (⌧))k1 = o(1)” might seem more natural, it seems that in practice (2) iii) will be more useful. Note moreover that condition (2) i) allows the energies En to be discontinuous. This will not be used in the analysis and the examples below, but it may be useful for other types of applications; e.g., for problems within the variational theory of Fracture Mechanics (see [12]).

Theorem 2.7. IfEn s-s

!E, then(En+G)!^s (E+G)for every continuousGsuch that(En+G)is coercive.

Proof. That (E_n + G) 0-converges to (E + G) follows from the fact that 0- convergence is stable under continuous perturbations.

For condition (1) of Definition 2.2, suppose thatuhas an"-slide forE+G (and therefore a(" )-slide for some >0) andun!u. Then we choose n, _n satisfying property 2 in Definition 2.6 and set _n⁰(⌧):= n(⌧)for⌧ 2[0,1], and

n0(⌧):= n(⌧ 1)for⌧ >1. We then consider⌧₁<⌧₂2[0,T]. If⌧₁,⌧₂2[0,1], then

En( _n⁰(⌧₂)) En( _n⁰(⌧₁))= En( _n(⌧₂)) En( _n(⌧₁))o(1).

If⌧₁,⌧₂>1, then

En( _n⁰(⌧₂)) En( _n⁰(⌧₁))=En( _n(⌧₂)) En( _n(⌧₁))E( (⌧ⁿ_j)) E( (⌧_iⁿ))+o(1) for some⌧_iⁿ ⌧ⁿ_j. If⌧₁<1<⌧₂, then

En( _n⁰(⌧₂)) En( ⁰_n(⌧₁))=En( _n(⌧₂)) En( _n(⌧₁)) E( (⌧_iⁿ)) E( (0))+o(1) for some⌧_iⁿ, so that in any case

En( _n⁰(⌧₂)) +G( _n⁰(⌧₂)) En( _n⁰(⌧₁))+G( _n⁰(⌧₁))

 E( (⌧_j))+G( (⌧_j)) E( (⌧_i))+G( (⌧_i)) +o(1)

<" +o(1)

(2.1)

for some⌧_i ⌧_j, where we used the continuity ofG together with (2) ii) and iii) in Definition 2.6, as well as the fact that is an"-slide foru. The same argument gives

(En+G)( _n⁰(1)) (En+G)( _n⁰(0))(E+G)( (1)) (E+G)( (0))+o(1), so that _n⁰ is an"-slide forEn+G, fornsufficiently large.

Finally, for condition (2) of Definition 2.2, we begin by supposing thatuis a strict local minimizer ofE+G, and therefore a strict minimizer in some closed ball

B(u,r). Letv_n !ube such thatE_n(v_n)! E(u). Then we have E(u)+G(u)= lim

n!+1(En(v_n)+G(v_n)) lim sup

n!+1

inf

B(u,r)(En+G).

Letun be minimizers of(En+G)|_B(u,r) (which exist by the coerciveness and the lower semicontinuity of En+G). Up to subsequences they converge to someu 2

B(u,r), and we have E(u)+G(u) = inf

B(u,r)(E+G)E(u)+G(u)

 lim inf

n!+1(En(un)+G(un))=lim inf

n!+1 inf

B(u,r)(En+G).

We then obtain E(u)+G(u) = E(u)+G(u), and thenu = uby the strict mini- mality ofu. This shows thatun !u, and thatunare local minimizers forEn+G. Note now that min(E+G)|@B(u,r) E(u)+G(u)+"⁰for some"⁰>0. This implies that theunare"⁰/2-uniformly stable for{En}. To check that, note first that

lim inf

n!1 min(En+G)|@B(u,r) E(u)+G(u)+"⁰.

(Otherwise, choose v_n 2 @B(u,r)that, up to a subsequence, converge to some v 2 @B(u,r)and lim infn!1(En(v_n)+G(vn)) < E(u)+G(u)+"⁰. But then E(v)+G(v) < E(u)+G(u)+"⁰, a contradiction.) Let n be an"⁰/2-slide for En +G atun. Then in particular En( _n(t))+G( _n(t)) < En(un)+G(un)+

"⁰/2, which implies that n(t) 2 B(u,r)for allt, fornlarge enough. We finally

obtain a contradiction since the conditionEn( _n(1))+G( _n(1)) <En(un)+G(un) contradicts the minimality ofun.

3. Examples of strongly stable convergence

In this section we examine in some detail two examples of oscillating functionals.

The first is elliptic homogenization, and even though the approximating functionals have oscillations, there do not exist local minimizers unrelated to local minimizers of the limit functional. Still, the proof of stable convergence requires constructing

"-slides for the approximating energies in a way that mimic a given"-slide for the

limit functional, even though there exist small oscillations for these energies.

The second example also involves functionals with oscillations, and here the oscillations cause the existence of many local minimizers unrelated to local mini- mization of the limit functional. Superficially, it would seem that size of the oscilla- tions is bounded away from zero, but in fact, due to the definition of"-slide, we can show that the size tends to zero, so that for any given"> 0, these artificial (strict) local minimizers eventually become un-"-stable.

Example 3.1 (elliptic homogenization). Consider En(u)=

Z 1 0

a(nx)|u⁰(x)|²dx, u2H¹(0,1),

whereais a bounded positive coefficient, bounded away from zero and periodic of period 1. Here X= H¹(0,1)endowed with theL²-distance. It is well known (see, e.g., [6]) that the0-limit ofEnis

En(u)=a Z 1

0 |u⁰(x)|²dx, u2 H¹(0,1), whereais the harmonic mean ofa: a ¹ =R1

0 a(y) ¹dy. Note thatE_n andEare strictly convex, up to translations, so that they are lower semicontinuous, and have only global minimizers corresponding to specified Dirichlet data. Hence, (1), (3) and (4) of Definition 2.6 are satisfied.

We now check (2). Considerun!uwith sup_nEn(un) <+1, and suppose is a path fromu. We will set↵_n=kun uk1, which is infinitesimal asn!+1. The numberk=kn⌧1 will be chosen in such a way that

1⌧kn⌧ 1

↵_n. (3.1)

For notational simplicity we still assume that ¹_k 2N^.

The paths nwill bringunto an interpolationIn(u)ofuon the lattice_kn¹Z^with minimal energy En with respect to functions with the same values on the nodes of the lattice. To that end, define

z_iⁿ=nk

✓ u

✓ i nk

◆ u

✓i 1 nk

◆◆

, i =1, . . . ,nk.

Letvbe the minimizer of min

(Z ₁

0

a(y)|v⁰(y)|²dy: v(0)=0, v(1)=1 )

(=a),

interpolated to the whole ofRin such a way thatv(x) xis 1 periodic (in particular v(z) = z if z 2 Z). Upon considering translations, we also suppose that v⁰ is continuous at zero. Recall thatvsatisfies

(a(y)v⁰(y))⁰=0. (3.2)

The interpolated function will be defined by In(u)(x)=u

✓i 1 nk

◆ + 1

nzⁿ_i v

✓ n

✓

x i 1 nk

◆◆

ifi 1

nk x i nk,

and the path n : [0,1]!X will be defined by

n(⌧)=⌧In(u)+(1 ⌧)un = In(u)+(1 ⌧)(un In(u)).

Then we get En( _n(⌧))=

Z 1 0

a(nx)(In(u)⁰+(1 ⌧)(u⁰_n In(un)⁰))²dx

= Xnk i=1

Z _nkⁱ

i 1 nk

a(nx)(I_n(u)⁰+(1 ⌧)(u⁰_n I_n(u)⁰))²dx

= Xnk i=1

Z _nkⁱ

i 1 nk

a(nx)

✓ z_iⁿv⁰

✓ n

✓

x i 1 nk

◆◆

+(1 ⌧)(u⁰_n In(u)⁰)

◆2

dx

= Xnk i=1

1 n

Z 1/k 0

a(y)(zⁿ_iv⁰(y)+(1 ⌧)w⁰_n,i(y))²dy

= Xnk i=1

1 n

Z 1/k 0

a(y)⇣

(zⁿ_iv⁰(y))²+(1 ⌧)²(w⁰_n,i(y))²

+2(1 ⌧)zⁿ_iv⁰(y)w⁰_n,i(y)⌘ dy, where

w_n,i(y)=n un

✓i 1+yk nk

◆

z_iⁿv(y)for 0y 1 k.

Note that, integrating by parts and using (3.2) and the periodicity ofaandv, we get Z 1/k

0

a(y)v⁰(y)w_n,i⁰ (y)dy =

Z 1/k 0

(a(y)v⁰(y))⁰w_n,i(y)dy +a

✓1 k

◆ v⁰

✓1 k

◆ w_n,i

✓1 k

◆

a(0)v⁰(0)wn,i(0)

= a(0)v⁰(0)

✓ w_n,i

✓1 k

◆

w_n,i(0)

◆

= a(0)v⁰(0)n

✓ un

✓ i nk

◆ un

✓i 1 nk

◆

u

✓ i nk

◆ u

✓i 1 nk

◆◆

, so that

En( _n(⌧)) = Xnk i=1

1 n

Z 1/k 0

a(y)⇣

(zⁿ_iv⁰(y))²+(1 ⌧)²(w⁰_n,i)²⌘

dy+Cn

= a Xnk i=1

1

nk|z_iⁿ|²+(1 ⌧)²C_n⁰ +Cn,

where C_n⁰ =

Xnk i=1

1 n

Z 1/k 0

a(y)(w⁰_n,i(y))²dy

= Xnk i=1

1 n

Z 1/k 0

a(y)

✓ u⁰_n

✓i 1+yk nk

◆

zⁿ_iv⁰(y)

◆2

dy

 C Xnk i=1

1 n

Z 1/k 0

a(y) u⁰_n

✓i 1+yk nk

◆2

+ |zⁿ_i|²|v⁰(y)|²

!2

dy

 Ckak1

Z ₁

0 |u⁰_n|²dy+ Xnk i=1

1 nk|z_iⁿ|²

!

C, and

|C_n|2 Xnk i=1

|zⁿ_i|ku u_nk1Cnk↵_n.

By (3.1) we haveCn =o(1)and we obtain thatEn( _n)is decreasing for⌧ 2[0,1] up too(1). Hence nsatisfies (i). Moreover n(1)= In(u). Note that

En(In(u))=a Xnk i=1

1

nk|zⁿ_i|²=E(Pn(u)),

where Pn(u)is the piecewise-affine interpolation ofu on the nodes of the lattice

1 nkZ^.

Let be a path with bounded energy forE;i.e., sup{E( (⌧)):⌧ 2[0,1]}<

+1. Note that this implies that{ (⌧)}⌧ is bounded inH¹(0,1), and by the Rellich embedding theorem that{ (⌧)}⌧ is continuous with values inL¹. With fixednwe can choose 0=⌧₁ⁿ <⌧₂ⁿ < ... <⌧_nⁿ=1 such that, having set _n =max_i| (⌧_iⁿ)

(⌧_iⁿ ₁)|, we have

n⌧ 1

kn, (3.3)

where k = kn has been chosen as above. The path n is then defined by taking

n(⌧_iⁿ)= In( (⌧_iⁿ)), and by linear interpolation between those points. The same proof as above shows that En( _n(⌧))is betweenEn( _n(⌧_iⁿ))andEn( _n(⌧_iⁿ₊₁))for

⌧ 2(⌧_iⁿ,⌧_iⁿ₊₁), up too(1). Hence, also condition (iii) is satisfied.

Finally, note that since In( (⌧_iⁿ))and (⌧_iⁿ)have the same value at all points i/nk, we can apply Poincar´e’s inequality on all intervals((i 1)/nk,i/nk)to get

Z 1

0 |In( (⌧_iⁿ)) (⌧_iⁿ)|²dx  1 nk

Z 1

0 |In( (⌧_iⁿ))⁰ (⌧_iⁿ)⁰|²dx

 C 2

nksup{En( (⌧))}. The inequality extends to all⌧by interpolation, and also (ii) holds.

Example 3.2. We consider theManhattan metricfunction':R²!{1,2} '(x1,x2)=

(1 ifx₁2Z^orx22Z 2 otherwise,

and the related scaled-perimeter functionals with forcing term f En(A)=

Z

A

f(x)dx+ Z

@A

'(nx)dH¹

defined on Lipschitz setsA. We assume that f is a (smooth) bounded function (we may assume thatkfk1 1), so that the first integral is continuous with respect to the convergence Aj ! A, understood as theL¹convergence of the corresponding characteristic functions.

By reasoning as in [3], it can be proved that the energiesE_n 0-converge to an energy of the form

E(A)= Z

A

f(x)dx+ Z

@^⇤A

g(⌫)dH¹

defined on all sets of finite perimeter (⌫ denotes the normal to@^⇤A). A direct computation (following for example a similar homogenization problem for curves as in [1], see also [6]) shows that actually

g(⌫)=k⌫k1= |⌫₁| + |⌫₂|.

Furthermore, it is easily seen (a proof can be derived from [2]) that the same E is equivalently the0-limit of

eEn(A)= Z

A

f(x)dx+H^1(@A),

defined on Awhich are the union of cubesQⁿ_i := ¹_n(i+(0,1)²)withi 2Z^{2. We} denote byAnthe family of suchA. Note thateEnis the restriction ofEn toAn.

We now prove that actuallyEn s

!E. We first show that ifAhas an"-slide for EandAn ! A, then each Anhas an("+o(1))-slide for En(and so, as above, an

"-slide fornsufficiently large).

We first observe that an arbitrary sequenceAnof Lipschitz domains converging to a set Acan be substituted by a sequence inAn with the same limit. To check this, consider a connected component of@An. Note that for n big enough every portion of @An parameterized by a curve : [0,1] ! R^{2such that'(n (0))} = '(n (1)) = 1 and'(n (t)) = 2 for 0 < t < 1 can be deformed continuously to a curve lying on' ¹(1)and with the same endpoints. If otherwise a portion of

@An lies completely inside a cubeQⁿ_i is can be shrunk to a point or expanded to the whole cube Qⁿ_i. In both cases this process can be obtained by a O(1/n)-slide,

since either the lengths of the curves are bounded by 2/n, or the deformation can be performed so that the lengths are decreasing.

We can therefore assume that An 2 An and that there exist an"-slide for E at Aobtained by continuous family A(t)with 0 t 1. We fix N 2 N^{and set} t^N_j = j/N. For all j 2 {1, . . . ,N}let A_n^N,j be a recovery sequence in An for A(t^N_j ). Furthermore we set A_n^N,0= An. Note that, since A_n^N,j ! A(t_j^N)andA(t) is continuous, we have|An^N,j4An^N,j+1| = o(1)as N ! +1. We may suppose that the set An^N,j+1 is the union of An^N,j and a family of cubes Q_i^N,j. We may order the indices i and construct a continuous family of sets A^N,j,i(t)such that

A^N,j,i(0)= A^N_n^,j[S

k<i Q_k^N,j, A^N,j,i(1)= A^N_n^,j [S

ki Q_k^N,j,

⇣H^1(AN,j)^H^{1(AN,j+1)⌘ c}

n H^1(AN,j,i(t))

⇣

H^1(AN,j)_H^1(AN,j+1)⌘+ c n. Since also |A^N,j,i(t)|differs from |A^N,j|and|A^N,j+1|by at mosto(1)as N ! +1, by gluing all these families, upon reparametrization we obtain a familyA_n^N(t) such that A_n^N(0) = An, A_n^N(1) = An(1), and, ifs < t then we have, for some

j <k

En(A_n^N(s)) E(A(t^N_j )) c

n o(1), En(A_n^N(t))E(A(t_k^N))+ c

n +o(1).

Since A(t)is an"-slide forEwe have

E(A(t_k^N))E(A(t^N_j ))+", so that

En(A_n^N(t)) En(A_n^N(s))+"+ c

n+o(1).

By choosingN andnlarge enough we obtain the desired("+o(1))-slide.

We now prove the second condition; namely, that given a strict local minimum A₀forEthere exist a sequence(A_n)of"-stable sets forE_nconverging to A₀.

Let >0 be such thatE(A0) < E(A)if|A4A0| andA6= A0. In partic- ular, sinceEis coercive and lower semicontinuous with respect toL¹-convergence, we have

min{E(A): |A4A0| = }> E(A0).

We therefore have, for somec>0

min{En(A): |A4A0| = }> E(A0)+c.

In fact, otherwise there exist Aen with lim infnEn(eAn)  E(A0). By the equico- erciveness of En we may suppose that eAn ! A, with|A4A0| = , so that we get

E(A)lim inf

n En(eAn)E(A0), and a contradiction.

We can therefore choose Anas a minimizer of min{En(A): |A4A0| }.

By0-convergence we have An! A0, and clearly no"-slide exists forEnfromAn

if"<min{En(A): |A4A0| = } En(An).

4. Discrete-to-continuous^"-slides

In a discrete environment (i.e., when the spaceXis composed of discrete points, or composed of functions defined on a discrete set) the concept of local minimizer may make little sense; however, in many occasions we encounter sequences of “discrete spaces” Xn “approximating” a “continuous” space X (e.g., finite-element spaces approximating continuous function spaces). We can therefore extend the definition

of"-slide and"-stability to be coupled with this discrete-to-continuous process. The

spaces Xn will all be identified as subsets of X, on which we consider the distance dist , and we will consider functionals En : Xn ! [0,+1]. Note that the usual extension

Fn(u)=

(En(u) ifu2 Xn

+1 otherwise

would give functionals for which all isolated points in Xnare local minimizers and have no"-slide for all"> 0. In particular this would make Definition 2 hold only whenEhas no"-slide.

We say thatu2 Xhas an"-slide along En if there exist integersNn !+1 and functions n : {0, . . . ,Nn}! Xn such that

(i) sup_kdist( _n(k), n(k 1))!(1/Nn), with!(0+)=0;

(ii) sup_jk(E_n( _n(k)) E_n( _n(j))) <";

(iii) lim sup_n(En( _n(Nn) En( _n(0))) <0;

(iv) limn n(0)=u.

We say thatu_n ! u are "-stable for E_n ifu 2 X has no"-slide along E_n with

n(0)=u_n.

Definition 4.1 (stable inE). We say that{En}isstable inEif the following hold:

1. Ifuhas an"-slide forEandun !u, then fornlarge enough,uhas an"-slide alongEn with n(0)=un.

2. Ifuis a strict local minimizer ofE, then there existun !uand">0 such that eachunis"-stable forEn.

We say that Enconverge stably inEif the{En}is stable in EandEn 0

!E, and we write En s

!E.

With this definition Theorem 2.4 holds true with the same statement.

Example 4.2. We consider Xn as the set of all subsets Aof _n¹Z². We consider a smooth function f, positive outside a bounded set, and

En(A)= 1 n#

⇢

(i, j):i 2A,j 2 1

nZ²\A,|i j| = 1

n +X

i2A

1 n² f(i) . Each set A2Xnis identified with

[ ⇢i+ 1

n[0,1]²:i 2 A ,

so thatXnis seen as a subset of the space of all subsets ofR²with finite perimeter with the distance

dist(A,A⁰)= |A4A⁰|. With respect to this distance En0-converge to the same

E(A)= Z

A

f(x)dx+ Z

@^⇤A

g(⌫)dH¹

as in the example in the previous section. Moreover we have E_n!^s E. This is easily seen repeating the same argument as above, since the argument in Section 3 is precisely to compare continuous paths in the space of sets with finite perimeter with discretized paths in Xn.

References

[1] E. ACERBIand G. BUTTAZZO,On the limits of periodic Riemannian metrics,J. Anal. Math.

43(1983), 183–201.

[2] R. ALICANDRO, A. BRAIDESand M. CICALESE,Phase and anti-phase boundaries in bi- nary discrete systems: a variational viewpoint, Netw. Heterog. Media1(2006), 85–107.

[3] L. AMBROSIOand A. BRAIDES,Functionals defined on partitions of sets of finite perimeter, II: semicontinuity, relaxation and homogenizationJ. Math. Pures. Appl.69(1990), 307–333.

[4] A. BRAIDES, “0-convergence for Beginners”, Oxford University Press, Oxford, 2002.

[5] A. BRAIDES,A handbook of0-convergence, In: “Handbook of Differential Equations. Sta- tionary Partial Differential Equations”, Vol. 3, M. Chipot and P. Quittner (eds.), Elsevier, 2006.

[6] A. BRAIDESand A. DEFRANCESCHI, “Homogenization of Multiple Integrals”, Oxford Uni- versity Press, Oxford, 1998.

[7] A. BRAIDES and L. TRUSKINOVSKY, Asymptotic expansions by Gamma-convergence, Contin. Mech. Thermodyn.20(2008), 21–62.

[8] G. DALMASO, “An Introduction to0-convergence”, Birkh¨auser, Boston, 1993.

[9] L. C. EVANSand R. F. GARIEPY, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

[10] R. L. JERRARDand P. STERNBERG, Critical points via Gamma-convergence: general theory and applications, J. Eur. Math. Soc.11(2009), 705–753.

[11] R. V. KOHNand P. STERNBERG,Local minimizers and singular perturbations, Proc. Roy.

Soc. Edinburgh Sect. A111(1989), 69–84.

[12] C. J. LARSEN,Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl.

Math.63(2010), 630–654.

[13] E. SANDIERand S. SERFATY,Gamma-Convergence of Gradient Flows and Application to Ginzburg-Landau, Comm. Pure Appl. Math.57(2004), 1627–1672.

[14] S. SERFATY,Stability in 2D Ginzburg-Landau passes to the limit, Indiana Univ. Math. J.

54(2005), 199–222

Dipatimento di Matematica Universit`a di Roma “Tor Vergata”

Via della Ricerca Scientifica 00133 Roma, Italia [email protected]

Department of Mathematical Sciences Worcester Polytechnic Institute Worcester, MA 01609, USA [email protected]