From discrete to continuum: A variational approach

79 Pages·2000·0.353 MB·English

by  Braides A., Gelli M.S.

FROM DISCRETE TO CONTINUUM:
A VARIATIONAL APPROACH
The one-dimensionalcase
Andrea Braides and Maria Stella Gelli
Abstract TheseLectureNotescoverthecoursegivenatSISSAbyABinSpring
2000 (Chapters 1 and 2) and some general results just hinted at in the course
(Chapter 3).They include mainlyresults by the authors andby Chambolle,Dal
Maso,GarroniandTruskinovsky,butsomeresults arenew;e.g.,Sections1.4{1.8
(except 1.4.1 and 1.7.2),and Section 3.3.
CONTENTS
Introduction iv
1 Discrete problems with limit energies de(cid:12)ned on
Sobolev spaces 1
1.1 Discrete functionals 1
1.2 Equivalent energies on Sobolev functions 1
1.3 Convex energies 3
1.3.1 Nearest-neighbour interactions 3
1.3.2 Long-range interactions 4
1.4 Energies with superlinear growth 6
1.4.1 Nearest-neighbour interactions 6
1.4.2 Next-to-nearest neighbour interactions 7
1.4.3 Long-range interactions 9
1.5 A general convergence theorem 14
1.6 Convergence of minimumproblems 16
1.6.1 Limitcontinuumminimumproblems 16
1.6.2 Next-to-nearestinteractions:phasetransitionsand
boundary layers 17
1.7 More examples 24
1.7.1 Weaknearest-neighbourinteractions:multiple-density
limits 24
1.7.2 Very-long interactions: non-local limits 26
1.7.3 Homogenization 29
1.8 Energies depending on second di(cid:11)erence quotients 30
2 Limit energies on discontinuous functions: two ex-
amples 32
2.1 The Blake Zisserman model 32
2.1.1 Coerciveness conditions 32
2.1.2 Limitenergies for nearest-neighbour interactions 34
2.1.3 Equivalent energies on the continuum 35
2.1.4 Limitenergies for long-range interactions 35
2.1.5 Boundary value problems 37
2.1.6 Homogenization 41
2.1.7 Non-local limits 41
2.2 Lennard Jones potentials 47
2.2.1 Coerciveness conditions 48
2.2.2 Nearest-neighbour interactions 48
2.2.3 Higher-orderbehaviourofnearest-neighbourinter-
actions 50
Contents iii
2.2.4 Convergence of minimumproblems 52
2.2.5 Long-range interactions 52
3 General convergence results 54
3.1 Functions of bounded variation 54
3.2 Nearest-neighbour interactions 56
3.2.1 Potentials with local superlinear growth 56
3.2.2 Potentials with linear growth 62
3.2.3 Potentials of Lennard Jones type 65
3.2.4 Examples 66
3.2.5 A remark on second-neighbour interactions 68
3.3 Long-range interactions 69
References 72
INTRODUCTION
Intheselecturenoteswetreattheproblemofthedescriptionofvariationallimits
of discrete problems in a one-dimensional setting. Given n N we consider
2
energies of the general form
n n(cid:0)j
j ui+j ui
En( ui )= (cid:21)n n (cid:0)
f g j(cid:21)n
j=1i=0
XX (cid:16) (cid:17)
de(cid:12)nedon(n+1)-tuples ui .Wemayview ui asadiscretefunctionde(cid:12)nedon
f g f g n
alatticecoveringa(cid:12)xedinterval[0;L]byintroducingpointsxi =i(cid:21)n((cid:21)n =L=n
n
isthe lattice spacing) Ifwe picture the set xi as the reference con(cid:12)gurationof
f g
anarray ofmaterialpoints interactingthrough someforces, and let ui represent
j
the displacementofthe i-thpoint,then n canbe thoughtas theenergy density
oftheinteractionofpointswithdistancej(cid:21)n (j latticespacings)inthereference
j
lattice. Note that the only assumption we make is that n depends on ui
f g
through the di(cid:11)erences ui+j ui, but we (cid:12)nd itmoreconvenient tohighlightits
(cid:0)
dependence on the ‘discrete di(cid:11)erence quotients’
ui+j ui
(cid:0) :
j(cid:21)n
One mustnotbe distracted fromthis notationandshould note the generalityof
the approach.
Our goalis to describe the behaviour of problems of the form
n
min En( ui ) uifi : u0 =U0; un =UL
f g (cid:0)
i=0
n X o
(andsimilar),andtoshowthatforaquitegeneralclassofenergiestheseproblems
havealimitcontinuouscounterpart.Here fi represents theexternalforces and
f g
U0;UL are the boudary conditions at the endpoints of the interval (0;L). More
general statement and di(cid:11)erent problems can be also obtained. To make this
asymptoticanalysisprecise, we use the notation and methods of(cid:0)-convergence,
for which we refer to the lecture notes by A. Braides (cid:0)-Convergence for Be-
ginners (a more complete theoretical introduction can be found in the book by
G. Dal Maso An Introduction to (cid:0)-convergence). We will show that, upon suit-
ably identifying discrete functions ui with suitable (posssibly discontinuous)
f g
interpolations, the free energies En ‘(cid:0)-converge’ to a limit energy F. As a con-
sequence we obtain that minimizers of the problem above are ‘very close’ to
minimizersof
Introduction v
L
min F(u) fudt: u(0)=U0; u(L)=UL :
(cid:0) 0
n Z o
The energies F can be explicitly identi(cid:12)ed by a series of operations on the
j
functions n.InordertogiveanideaofhowF canbedescribed,we(cid:12)rstconsider
the case when only nearest-neighbour interactions are taken into account:
n(cid:0)1
ui+1 ui
En( ui )= (cid:21)n n (cid:0) :
f g (cid:21)n
i=0
X (cid:16) (cid:17)
In this case, the limit functional F can be described by introducing for each n
a ‘threshold’ Tn such that Tn + and (cid:21)nTn 0, and de(cid:12)ning a limit bulk
! 1 !
energy density
f(z) =lim(convex envelope of ~n(z));
n
and a limitinterfacial energy density
g(z)=lim(subadditive envelope of (cid:21)n ~~n z );
n (cid:21)n
(cid:16) (cid:17)
where
~n(z)= n(z) if jzj(cid:20)Tn ~~n(z)= n(z) if jzj(cid:21)Tn
+ otherwise, + otherwise.
(cid:26) 1 (cid:26) 1
Note the crucial separation of scales argument: essentially, the limit behaviour
of n(z) de(cid:12)nes the bulk energy density, while (cid:21)n n(z=(cid:21)n) determines the in-
terfacialenergy.The limitF isde(cid:12)ned (uptopassingtoitslowersemicontinuity
envelope) on piecewise-Sobolev functions as
0
F(u)= f(u)dt+ g(u(t+) u(t ));
(0;L) (cid:0) (cid:0)
Z S(u)
X
where S(u) denotes the set of discontinuity points of u. Hence, we have a limit
energy with two competing contributions of a bulk part and of an interfacial
energy. In this formwe can recover fracture and softening phenomena.
Thedescriptionofthelimitenergygetsmorecomplexwhennotonlynearest-
neighbour interactions come into play. We (cid:12)rst examine the case when interac-
tions up to a (cid:12)xed order K are taken into account:
K n(cid:0)j
j ui+j ui
En( ui )= (cid:21)n n (cid:0)
f g j(cid:21)n
j=1i=0
XX (cid:16) (cid:17)
j
(or, equivalently, n = 0 if j > K). The main idea is to show that (upon some
controllable errors) we can (cid:12)nd a lattice spacing (cid:17)n (possibly much larger than
vi Preface
(cid:21)n) such that En is ‘equivalent’ (as (cid:0)-convergence is concerned) to a nearest-
neighbour interaction energy on a lattice of step size (cid:17)n, of the form
m(cid:0)1
ui+1 ui
En( ui )= (cid:17)n n (cid:0) ;
f g (cid:17)n
i=0
X (cid:16) (cid:17)
and to which then the recipe above can be applied.
The crucial points here are the computation of n and the choice of the
scaling (cid:17)n. In the case of next-to-nearest neighbours this computation is partic-
ularly simple, as it consists in choosing (cid:17)n = 2(cid:21)n and in ‘integrating out the
contribution of (cid:12)rst neighbours’: in formula,
2 1 1 1
n(z)= n(z)+ min n(z1)+ n(z2):z1+z2 =2z :
2 f g
In a sense this is a formulaof relaxation type. If K >2 then the formulagiving
n resemblesmoreahomogenization formula,andwehavetochoose(cid:17)n =Kn(cid:21)n
with Kn large. In this case the reasoning that leads from En to En is that the
overallbehaviour ofa systemofinteractingpointwillbehave as clustersoflarge
arrays of neighbouring points interacting through their ‘extremities’
When the number of interaction orders we consider is not bounded the de-
scription becomes more complex. In particural additional non-local terms may
appear in F.
Note that (cid:12)rst order (cid:0)-limits may not capture completely the behaviour of
minimizers for variational problems as above. Additional information,as phase
transitions,boundary layer e(cid:11)ects and multiple cracking, maybe extracted from
the study of higher order (cid:0)-limit.
1
DISCRETE PROBLEMS WITH LIMIT ENERGIES DEFINED
ON SOBOLEV SPACES
1.1 Discrete functionals
Wewillconsiderthelimitofenergiesde(cid:12)nedonone-dimensionaldiscretesystems
of n points as n tends to + . In order to de(cid:12)ne a limitenergy on a continuum
1
we parameterize these points as a subset of a single interval (0;L). Set
L n i
(cid:21)n = ; xi = L=i(cid:21)n; i=0;1;:::;n: (1.1)
n n
n n
We denote In = x0;:::;xn and by n(0;L) the set of functions u : In R.
f g A !
If n is (cid:12)xed and u n(0;L) we equivalently denote
2A
n
ui =u(xi):
j
Given K N with 1 K n and functions f : R [0;+ ], with j =
2 (cid:20) (cid:20) ! 1
1;:::;K, we will consider the related functional E : n(0;L) [0;+ ] given
A ! 1
by
K n(cid:0)j
j
E(u)= f (ui+j ui): (1.2)
(cid:0)
j=1i=0
XX
n
Note that E can be viewed simplyas a function E :R [0;+ ].
! 1
An interpretation with a physical (cid:13)avour of the energy E is as the internal
interactionenergyofachainofn+1materialpointseachoneinteractingwithits
K-nearestneighbours,undertheassumptionthattheinteractionenergydensities
depend only on the order j of the interaction and on the distance between the
two points ui+j ui in the reference con(cid:12)guration. If K = 1 then each point
(cid:0)
interacts withitsnearest neighbouronly,whileifK =nthen each pairofpoints
interacts.
Remark 1.1 Fromelementarycalculuswe havethatE islowersemicontinuous
j
if each f is lower semicontinuous, and that E is coercive on bounded sets of
n(0;L).
A
1.2 Equivalent energies on Sobolev functions
We will describe the limitas n + of sequences (En) with En : n(0;L)
! 1 A !
[0;+ ]of the general form
1
2 Discrete problems with limit energies de(cid:12)ned on Sobolev spaces
Kn n(cid:0)j
j
En(u)= fn(ui+j ui): (1.3)
(cid:0)
j=1i=0
XX
Since each functional En is de(cid:12)ned on a di(cid:11)erent space, the (cid:12)rst step is to
identifyeach n(0;L)withasubspaceofacommonspaceoffunctionsde(cid:12)nedon
A
(0;L).In order toidentifyeach discrete function witha continuous counterpart,
we extend u by u~:(0;L) R as the piecewise-a(cid:14)ne function de(cid:12)ned by
!
ui ui(cid:0)1
u~(s)=ui(cid:0)1+ (cid:0) (s xi(cid:0)1) if s (xi(cid:0)1;xi): (1.4)
(cid:21)n (cid:0) 2
1;1
Inthiscase, n(0;L)isidenti(cid:12)edwiththosecontinuousu W (0;L)(actually,
1;1 A 2
in W (0;L)) such that u is a(cid:14)ne on each interval (xi(cid:0)1;xi). Note moreover
that we have
0 ui ui(cid:0)1
u~ = (cid:0) (1.5)
(cid:21)n
on (xi(cid:0)1;xi). If no confusion is possible, we willsimplywrite u in place of u~n.
Aswewilltreatlimitfunctionalsde(cid:12)nedonSobolevspaces,itisconvenientto
rewrite the dependence ofthe energy densities in(1.3)withrespect to di(cid:11)erence
quotients rather than the di(cid:11)erences ui+j ui. We then write
(cid:0)
Kn n(cid:0)j
j ui+j ui
En(u)= (cid:21)n n (cid:0) ; (1.6)
j(cid:21)n
j=1i=0
XX (cid:16) (cid:17)
where
j 1 j
n(z)= fn(j(cid:21)nz):
(cid:21)n
With the identi(cid:12)cation of u with u~, En maybe viewed as an integralfunctional
1;1
de(cid:12)ned on W (0;L). In fact, for (cid:12)xed j 0;:::;K 1 , k 0;:::;n 1
2 f (cid:0) g 2 f (cid:0) g
and i such that i k<i+j we have
(cid:20)
i(cid:0)k+j(cid:0)1 i(cid:0)k+j(cid:0)1
ui+j ui 1 uk+m+1 uk+m 1 0
(cid:0) = (cid:0) = u~(x+m(cid:21)n)
j(cid:21)n j (cid:21)n j
m=i(cid:0)k m=i(cid:0)k
X X
n n
for all x (xk;xk+1), so that
2
i+j(cid:0)1 xnk+1 i(cid:0)k+j(cid:0)1
j ui+j ui 1 j 1 0
(cid:21)n (cid:0) = u~(x+m(cid:21)n) dx:
(cid:16) j(cid:21)n (cid:17) j Xk=i Zxnk (cid:16)j mX=i(cid:0)k (cid:17)
We then get
n(cid:0)j j(cid:0)1 L(cid:0)(j(cid:0)1(cid:0)l)(cid:21)n j(cid:0)1(cid:0)l
j ui+j ui 1 j 1 0
(cid:21)n n (cid:0) = n u~(x+k(cid:21)n) dx:
Xi=0 (cid:16) j(cid:21)n (cid:17) j Xl=0Zl(cid:21)n (cid:16)j kX=(cid:0)l (cid:17)
Convex energies 3
and the equality
En(u)=Fn(u~); (1.7)
where
Kn j(cid:0)1 L(cid:0)(j(cid:0)1(cid:0)l)(cid:21)n j(cid:0)1(cid:0)l
1 j 1 0
n v (x+k(cid:21)n) dx
Fn(v) =8>>>>><Xj=1Xl=0 j Zl(cid:21)n (cid:16)j kX=(cid:0)l (cid:17) if v 2An(0;L)
+ otherwise.
>
>> 1 (1.8)
Note tha>>:t in the particular case Kn =1 we have (set n = n1)
L
0
n(v )dx if v n(0;L)
Fn(v) =8Z0 2A (1.9)
><+ otherwise.
1
>
De(cid:12)nition 1.2. (Converg:ence of discrete functions and energies) Withthe
1
identi(cid:12)cations above we will say that un converges to u (respectively, in L , in
1;1 1
measure, in W , etc.) if u~n converge to u (respectively, in L , in measure,
1;1
weakly in W , etc.), and we will say that En (cid:0)-converges to F (respectively,
1 1;1
withrespect tothe convergence inL ,inmeasure,weaklyinW ,etc.) ifFn (cid:0)-
1
converges to F (respectively, with respect to the convergence in L , in measure,
1;1
weakly in W , etc.).
1.3 Convex energies
j
We (cid:12)rst treat the case when the energies n are convex. We willsee that in the
case of nearest neighbours, the limit is obtained by simply replacing sums by
integrals, while in the case of long-range interactions a superposition principle
holds.
For simplicitywe suppose that the energy densities do not depend on n; i.e.,
j j
n = :
1.3.1 Nearest-neighbour interactions
We start by considering the case K =1, so that the functionals En are given by
n(cid:0)1
ui+1 ui
En(u)= (cid:21)n (cid:0) : (1.10)
(cid:21)n
i=0
X (cid:16) (cid:17)
The integral counterpart of En is given by
L
0
(v )dx if v n(0;L)
Fn(v)=8Z0 2A (1.11)
><+ otherwise.
1
>
:
4 Discrete problems with limit energies de(cid:12)ned on Sobolev spaces
Note that Fn depends on n only through its domain n(0;L).
A
The following result states that as n approaches the identi(cid:12)cation of En
1
with its continuous analog is complete.
Theorem 1.3 Let :R [0;+ ) be convex and let En be given by (1.10).
! 1 1;1
(i) The (cid:0)-limit of En with respect to the weak convergence in W (0;L) is
given by F de(cid:12)ned by
0
F(u)= (u)dx: (1.12)
(0;L)
Z
(ii) If
(z)
lim =+ (1.13)
jzj!1 z 1
j j
1
then the (cid:0)-limit of En with respect to the convergence in L (0;L) is given by F
de(cid:12)ned by
0 1;1
(u)dx if u W (0;L)
F(u)= (0;L) 2 (1.14)
8Z
<+ otherwise
1
1
on L (0;L). :
Proof (i) The functional F de(cid:12)nes a weakly lower semicontinuous functional
1;1
on W (0;L) and clearly Fn F; hence also we have (cid:0)-liminfjFj(u) F(u).
1;1 (cid:21) n (cid:21) n
Conversely,(cid:12)xed u W (0;L)letun n(0;L)be such thatun(xi)=u(xi).
2 2A
By convexity we have
n n
xi+1 0 1 xi+1 0 u(xni+1) u(xni)
(u)dt (cid:21)n u dt =(cid:21)n (cid:0) ;
Zxni (cid:21) (cid:16)(cid:21)n Zxni (cid:17) (cid:16) (cid:21)n (cid:17)
hence, summingup,
L
0
(u)dt En(un):
0 (cid:21)
Z
This shows that (un) is a recovery sequence for F.
(ii) If (1.13) holds then the sequence (En) is equi-coercive on bounded sets
1 1;1
of L (0;L) with respect to the weak convergence in W (0;L), fromwhich the
2
thesis is easily deduced.
1.3.2 Long-range interactions
Let now K N be (cid:12)xed. The energies En take the form
2
K n(cid:0)j
j ui+j ui
En(u)= (cid:21)n n (cid:0) : (1.15)
j(cid:21)n
j=1i=0
XX (cid:16) (cid:17)