The effect of a singular perturbation on nonconvex ... - CMU Math

The Effect of a Singular Perturbation on Nonconvex Variational Problems PETER STERNBERG Communicated

by

M. E. GURTIN

Abstract We study the effect of a singular p e r t u r b a t i o n on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation p a r a m e t e r e ~ 0. The technique utilizes the notion of "_P-convergence" of variational problems developed by DE GtOR6L The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. F o r the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.

Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . Section 1 Scalar Dependent Energy . . . . . . . . . . . . . . . . A. Functions of Bounded Variation . . . . . . . . . . . . B. The Result for W: R -+ R . . . . . . . . . . . . . . C. Generalization to an Integrand with Spatial Dependence Section 2 Vector Dependent Energy . . . . . . . . . . . . . . . . A. Generalization to W: R 2 -+ R . . . . . . . . . . . . . B. Properties of the Degenerate Metric d r . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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Introduction We are concerned with the effect of a singular p e r t u r b a t i o n on a nonconvex v a r i a t i o n a l problem. The goal is to characterize the asymptotic behavior of minimizers in the limit as some p e r t u r b a t i o n p a r a m e t e r e ~ 0; this goal is achieved by showing that the minimizers converge to a limit which solves a new variational

210

P. STERNBERG

problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics. Our approach uses a tool developed by DE GIORGI called "F-convergence" of variational problems ([1], [7]). The fundamental idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In doing so one characterizes the desired limit of minimizers as a solution of a new variational problem, the "F-limit" of the perturbed sequence of functionals. Before perturbation, the variational problems we study are mathematically trivial. Beginning with a functional F : LI(f2) --~ R (.q Q R", open, bounded) given by r(u) = f W(u) dx g2

with W=> 0 and

W ( t ) = 0 at more than one t, consider the problem:

(P)

inf F(u)

for u possibly subject to a constraint such as f u dx = c, and for a variety of a nonconvex W. Problem (P) has a chronic failure of uniqueness for such W: a piecewise constant absolute minimizer is determined by any partitioning of the domain into regions so as to accommodate the constraint. If, for example, minimization of F models a physical problem, then this nonuniqueness might be due to the neglect of some small effect. Restoring the effect through the addition of a singular perturbation might then resolve this failure of uniqueness. Choosing e 2 IVu [2 as perhaps the simplest possible perturbation, we are led to the functional F.(u) ---- f W(u) + e 2 lVul 2 dx D

and the problem

(P3

inf F.(u). u constrained as in (P)

Our goal is to characterize Uo = lim~_~o u. (in LI(~)), where u~ is a solution of (P~). Since minimizers of (P) have a purely geometric characterization, one might expect the same of the criterion which selects a limit Uo. We shall show that this is indeed the case by establishing that Uo solves a new variational problem (Po)

inf

u~BV(~)

Fo(u),

where (inf F~) = e (ins Fo) + o(e). Solutions o f (Po) typically involve a partition of ~ into regions separated by minimal surfaces or surfaces of constant curvature. Often this partition problem (Po) is easy to solve directly. In that case, the technique also yields information on the structure of constrained minimizers and the existence of local minimizers of (P~).

Singular Perturbation on Nonconvex Variational Problems

211

The analysis of the problem by this method clearly differs from the more classical approach of matched asymptotic expansions: the focus here is on the asymptotic behavior of the energy of (P~) rather than on an expansion for u~ itself. Furthermore, in the classical approach one knows (or presumes) the location of a boundary layer, whereas one of our tasks is to determine its locatiom The two viewpoints, however, are not unrelated. The identification of (Po) requires the construction of a sequence of functions {~} which efficiently traverse this boundary layer in bridging the zeros of W, in close analogy with the notion of an "inner expansion". For a rigorous analysis of (P~) with a Dirichlet condition using matched asymptotic expansions, see the work of BURGER & FRAE~KEL ([2]). Many others have studied similar problems by this approach. (See e.g. CAGINALP [4] and HowEs [17].) When it applies, the advantages of the F-convergence technique are numerous: the problem (Po) determines the location of the interior boundary layer, the analysis is considerably easier, and, as will be discussed later (see remark (1.14)), the results are immediately adaptable to continuous perturbations of

(P~). An earlier application of this technique to (P~) was carried out by MoDIcA & MORTOLA ([22], [23]), who obtained the F-limit for the unconstrained problem with various choices of scalar-dependent 14/".Our results generalize this work and the approach borrows many ideas from these authors. In Section 1, we consider W: R--~ R having exactly two zeroes, a and b, and we attach the constraint f u dx = e,

where a [f2 l < c < b l O ]

t2

(t'1 = n-dimensional Lebesgue measure). A typical minimizer of the unperturbed problem (P) might then take the form of Figure 1. GURTIN([13], [14]) raised the question of describing limits of minimizers of (P~) with these conditions as a model for obtaining the stable density distributions u for a fluid confined to a container /2, within the context of the Van der Waals-Cahn-HiUiard theory of phase transitions. Recent contributions to this problem include the work of NOVICK-COHEN

Fig. 1. A Typical Minimizer of (P)

Fig. 2. Solution of (Po): Uo = lim~_~o u~.

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P. STERNBERO

SEGEL ([24]) a n d CARR, GURTIN & SLEMROD ([5]). T h e l a t t e r g r o u p o f a u t h o r s prove that in one dimension, stable minimizers of (P,) are monotone, and their limit is a step function with only one discontinuity. Our Theorem 1 generalizes this result to /2 C R'. It says that any limit point o f (u~} must minimize

inf

u~ B V( ~9) W(u(x)) = 0a.e. f udx =c Q

Pera (u : a},

where Per a A : perimeter of A in /2, and BV(/2): space of functions of bounded variation, defined e.g. in ([11]). Thus, as e - + 0 the minimizers of (P,) select a function Uo that minimizes the area of the interface separating the "states u:a and u = b (see Figure 2). Essentially the same result has been proved recently also by MODICA ([20]). Section 1 also includes, in Theorem 2, a generalization of Theorem 1 to a spatially dependent W. The associated limiting problem (Po) which Uo solves is then a weighted partitioning problem. In Section 2 we consider generalizations to vector-dependent W. For W: R 2 -+ R, zero on two disjoint simple closed curves, and positive elsewhere, Theorem 3 uses the techniques of _P-convergence to show that a limit of minimizers of (P,) must satisfy the minimal interface criterion which arose in the scalar case. Theorem 3 also characterizes the cost--per unit area along the interface--of the transition made by the minimizers u, :/2--~- R 2; we show that it tends asymptotically to the distance between the two zero curves of IV, measured with respect to a degenerate Riemannian metric in the plane derived from W.

1. Scalar Dependent Energy

A. Functions of Bounded Variation We describe first some o f the basic definitions and properties of functions of bounded variation; we will need these to arrive at the partitioning problem (Po). For a more complete description, see ([llD. Throughout the paper/2 will be an open, bounded subset of R n with Lipschitzcontinuous boundary. For uE L1(/2), define:

f lVu[ :--

sup

f u(x) (V . g(x)) dx.

(1.1)

g~Cl ( ~9,Rn)

[gl~1

The space of functions of bounded variation, BV(/2), consists in those u E Ll(/2) for which f [~TuI < ~ ; BV(f2) is a Banach space under the norm: ~2

fluldx+ t2f lVul.

/2

Notice that [Vu I is not an L 1 function, but rather the total variation of the vectorvalued measure ~Tu. (See [9], p. 349.) If u E BV(/2), the integral of any positive,

Singular Perturbation on Nonconvex Variational Problems

213

continuous function h with respect to the measure 1~Tul can be expressed as f h(x) lVTul= D

sup

f u(x) (~7 . g(x)) dx.

(1.2)

g~ci(D,R n) D

Ig(x)l~hCx) An important example is the case when u : ZA, the characteristic function of a subset ,4 of R n. Then

f i~Tul = Q

sup

f (V " g(x))dx.

geCl(~,Rn) A [gl~l

If this supremum is finite, A is called a set of finite perimeter in ~2. If ~A is smooth, then by the Divergence Theorem

f [Vu 1 = H,,-I (~A A O), where H "-I is (n - 1)-dimensional Hausdorff measure (surface area measure). It is therefore natural to define the perimeter of any subset of D by: Pera A = perimeter of A in Q---- f [~7ZaI. D

The following two properties, easily proved, will be useful later.

Proposition 1. (Lower Semicontinuity) ([11]) I f u,--~ u in LI(D), then l i m i n f f [~Tu~l ___ f 1~Tul. D

D

Proposition 2. (Compactness of B V in L 1) ([11 ]) Bounded sets in the B V norm are compact in the L x norm. We now present two technical lemmas; the first is an approximation theorem for sets of finite perimeter by sets with smooth boundary.

Lemma 1. Let ~ be an open, bounded subset of R n with Lipschitz-continuous boundary. Let A ~ D be a set of finite perimeter in D with O < IAl < [D[. Then there exists a sequence of open sets {Ak} satisfying the following five conditions: (i) (ii) (iii) (iv)

(v)

~A k (h ~2 E C 2, t(Ak A D) d A 1--* 0 as k --~ oo, Pei-a Ak ~ Pera A as k -+ oo, H'-I(~Ak I% 8D) = O, l ak A o l -- Ial for all k su~ciently large.

Here I'1 refers to n-dimensional Lebesgue measure.

Proof. First extend Za to a function fi E BV(R n) such that ~(x) : ZA(x)

for x E D,

f l~Tt/I ~---O. OD

(See [11], 2.8, 2.16.)

(1.3) (1.4)

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P. STERNBERG

Summarizing the argument of GIUST! ([11], 1.24, 1.26), we see that a standard mollification of ~ provides a sequence of C ~ functions (f~} satisfying f~ ~ ~

in L 1 ,

lim f IVf, I = f IVh].

e~O

12

Then define sets C~,t --{f~(x)> t}. By use of the co-area formula ([11]) and Saao's Theorem it can be shown that there exist a value of t E (0, 1) and a sequence ek -+ 0 such that ~Cek,t ~ C ~ , ZC,k,t --~ ~A

in

L1(/2),

Pera Cek, t ~ Pera A, and •

8/2) = O.

Such a sequence, which we denote simply by {Ck}, will not, in general, satisfy the condition (v). It therefore remains to be shown that the sequence of sets (Ck} satisfying (i)-(iv) can be altered so as to satisfy (v) as well; that is, one must remove some measure from either Ck f~/2 or ~Q\ Ck (whichever is too big) and give it to the other without disrupting the smoothness of the boundary or distorting too drastically the perimeter of the boundary in /2. To this end, we let E k := Ck f~ /2, and assume without loss of generality that I E k [ - t A I > O . Define =

Igkl-

IAI

which, by (ii), goes to 0 as k - + c~. Also define Lk =

2k

,

(1.5)

and impose o n / 2 a grid Gk of hypercubes [t)k.~Nk (.:~t Ji= 1 of side length Lk with Q~ Q/2 for all i. Since #/2 is Lipschitz-continuous, there exists a sequence of grids {Gk} such that: lim I/2 \ Gkl = 0.

(1.6)

It follows from (1.5) that the measure of any cube in the lattice exceeds the amount of volume which we need to transfer from Ek to /2 \ Ek. In fact, L~, > 22k. Selecting the cube 0 k which maximizes

{IONIAEkI : Q~ E Gk}, we split the argument into two cases, depending on whether or not QkQ Ek.

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215

Case 1. Ok Q Ek. Since 2k < 89IQ k I, one can remove a smooth subset of 0 k, say Sk, having volume ;tk and perimeter which goes to zero as k--> o0. Placing this set Sk in the complement of Ek yields the desired set Ak : = Ck \ Sk.

Case 2. (~k \ Ek =l= fJ. Then [Q*/5 Ekl O, d is a C 2 function in (] d(x)] < s) with

IV dl :

1.

(1.9)

Furthermore,

lim H " - l ( ( d ( x ) = s)) = H"-J(OA).

(1.10)

s-->O

Proof. When restricted to {0 < d(x) < s) or ( - s < d(x) < 0), d will be C k provided 8A E C k ([10], App. A, and [19]). The triangle inequality yields Id(x) - d(y) l _ 0. (c) IV has exactly two roots, which we label a and b, with a O. (See Fig. 4).

a b Fig. 4. Graph of I,V

s

Restating the unperturbed problem (P) for this W, we begin with the variational problem: (p)

inf

u~L I( F~) ~ UdX=C ,f~

f W(u) dx

, -:

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P. STERNBERG

where c is any number satisfying arOl v

in

LI(.Q)

implies lim infF,(v~) ~

Fo(v).

(1.11)

(ii) For each v E Ll(g2), there exists a sequence {gq} in L1(I2) satisfying

LI(~Q)

(1.12)

jlim F,j (@9) = Fo(v).

(1.13)

~V-+ v

in

and

Notation. If {F,}, Fo satisfy (1.11)-(1.13), we write F(LI([2) -) !ira~ F,(~) = Fo(v). Q--~t)

Remark 1.14. The real advantage of proving F-convergence, rather than simply the convergence of minimizers, is that the results adapt immediately to continuous perturbations of F~. This is clear from (1.11)-(1.13). Thus one can characterize the asymptotic behavior of minimizers of a whole family of problems obtained from F~ by the addition of a functional continuous with respect to LI(~2) (e.g.

f+ w(.)+

ug(x) + e [Vul 2 dx for gE L~(Q)).

Proof of Theorem 1. For the moment we delay the proof of inequality(1.11) and the construction of a sequence yielding (1.12) and (1.13) and show how Theorem 1 follows from these claims.

Let Wo E BV(Q) be a minimizer of Fo. Existence of such a function follows from the direct method using the compactness and lower semicontinuity of BV(O) with respect to LI(s (i.e. Propositions 1 and 2). In fact, minimizers will have an interface which is analytic and of constant mean curvature for dimension n < 8. For a more complete description of minimizers of Fo see the work of GONZALEZ, MASSARI & TAMANINI ([12]). Let {w,j} be the sequence satisfying (1.12), (1.13) for w0. Assuming that the minimizers {u,j} converge in L ' ( O ) to a limit uo, it follows from (1.11) that lim inf Using that

F,j(u,) ~ Fo(uo).

F,j(u~) ~ F,j(w~j), one has Fo(uo) < lim inf F,j(u,) < lim F,. (w,) = V(wo). =

=

j---> o ~

J

Thus Uo must be a minimizer of Fo and Theorem 1 follows.

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P. STERNBERG

We now return to the task of proving /'-convergence: (1.11)-(1.13). Before proving (1.11), we should make some preliminary observations about the kinds of L~-convergent sequences {v,} and limits v that needs be considered. If W(v(x)) :# 0 on a set of positive measure, then Fo(v) = + co. But 1

lim inf F,(v,) ~ lim i n f - E

f

w(~,(x))

dx =

+

o~

D

as well, so that (1.11) is immediate. Equally simple is the case in which

fvdx#c, D

for here

f v~dx#c D

for all small e, again yielding lira inf F~(v~) = + oz. Therefore, consider only those v E Lt(Q) satisfying

W(v(x)) = 0

a.e.,

f u dx = c.

Proof of Inequality (1.11). First we assume that the sequence {v,} satisfies

a < v, < b.

(1.15)

Applying the Cauchy-Sehwarz inequality to F,(v,), we obtain

F,(v,) >=2 f ~ / ~ )

[Vv,(x)[ dx.

D

Let $ : R - + R

be defined by t

4'(0 ----2

f 1/W(s),is,

(1.16)

u

so that

Faro > f IV*(v,(x))l dx. ~2

Then, from (1.15) and the L t convergence of v, to v, it follows that r

-+ $(v) in L'(Q).

By the lower semicontinuity shown in Propositon 1, we conclude that lim inf F~(v,) >=lim inf f ]Vck(v,) t dx >= f IV4,(v) [. D

D

Singular Perturbation on Nonconvex Variational Problems Now

[

4,(v(x))

o

{v = a)

2 / I/-~-s)ds

{v : b},

221

since W(v(x)) = 0 a.e., and therefore f IV4,(v)[ =

IV(s)

2

Pero {v = a} = Fo(v),

D

which establishes (1.11). To justify assumption (1.15), we compare {v,) to the truncated sequence {v*} defined by:

v~* =

Ii "(x)

{v,(x) < a)~ b} {a ~ v~(x) {v,(x) > b}.

First note that v,-> v in LI(/2) implies that v~*--> v in L'(/2). Also,

F~(v,) ~

f'

~ W(v,) --k e IVv~l2 dx

D

-- F~(v*)+

.

1

~{]o~>b} T

{v~__F~(v*). Since the proof of (1.11) made no use of the constraint

f v~dx : c, this last inequality shows that it suffices to consider only sequences bounded as in (1.15). The proof of (1.12), (1.13) involves the construction of a sequence of functions such as to traverse efficiently a boundary layer while bridging the values a and b. Before presenting the proof, we discuss some properties of the solution z(s) of the following ordinary differential equation, which will be used in the construction:

dz _ I/-W---~ , ds

(1.17)

z(O) : 89(a + b). Local existence is clear since ] / W - ~ will be Lipschitz-continuous in a neighborhood of 89(a + b). However, by writing z(s)

f 89

1/IV(r/)~d~7= s

(1.18)

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P. STERNBERG

and noting that W0/) > 0 for a < ~ / < b, one sees that local solutions may be extended to all of R. Furthermore, a .< z(s) < b

for all s,

(1.19)

lim z(s) = a.

(1.20)

and lira z(s) = b, S'-~

OO

8-q ~ --

In fact, since W"(a) > 0 and that 1

oO

W"(b) > O, it follows from Taylor's Theorem c1

:< ]~7 - a-'----~ for [~ - a f small,

1

c2

=< I~ - b]

b l small,

for I~/ -

where cl and c2 are positive constants depending on W. This implies the decay estimates: Ib-z(s)l~c3e

-o's

as s - + o o ,

I a - z ( s ) [ < = c a e c's as s--~ - oo,

(1.21)

where c3 and c4 are again positive constants depending on W.

Construction of {~,i} satisfying (1.12), (1.13). Let v E LI(~Q). We may immediately assume that

vE SV(~),

W(v(x))=O

a.e.,

f v,tx= c D

(otherwise Fo(v) = oo and the choice Q~ = v for each e achieves (1.12), (1.13)). Therefore we may write v(x) = {~

xEA x E B,

where A and B are sets of finite perimeter in L~ and

alAl+blSl=c. Let _P : = 8A f~ 8B and assume F E C 2. At the conclusion of the proof we will show that this represents no loss of generality. Recalling Lemrna 2, consider the function d: f2 --> R, given by d(x)

f

dist ( x , / )

[ "dist ( x , / ' )

xE B x E A,

which represents the signed distance to /1. Now define a sequence of functions g, : R --~ R which effect the transition

Singular Perturbation on Nonconvex Variational Problems

223

between the zeroes of W:

s > 21/7b-z

1

(s - 21/7-) + b

1/7 21t~ }

so that, by (1.9),

f.= f

!w(g)§

{Id(x)l ~ F~(v,) and ve-+ v in La(~). Now define f : ~ •

f(x, s) = 2(s - g~(x)) (g~(x) - s) and

~p, : .(2 --~ R

by %(x)

~,(x) -----g,(fx)f(x, s)ds. An application of the Cauchy-Schwarz inequality leads to

O~(v~) >= f f(x, v,) IVv~[ dx = , ~ u p R , ' f f(x, v,) ax. o( ,

Jot~l

) ~J

R by

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231

For fixed a E Col(Q, Rn), lal ~ 1, it follows that

O,(v,) ~ ff ~V~p~(x), o(x)} - \ , f )

Vxf(X, s) ds, a(x) dx,

and an integration by parts yields re(x)

GXv3 >= - f

f

if(x, s) (V 9 ~(x))) + (Vxf(x, s), (~(x)) as dx.

-q g~(x)

Using the L ~ bounds on gl, v~,f, Vxf, a and V 9a, we pass to the limit as e-+ 0 Thus, v(x)

lim inf G,(v,) ~ -- f

f f(x, s) (V. a(x)) + (Vxf(X, s), a(x)) ds dx

~2 gt(x) g2(x)

= -

.o

= -

Vx" (f(x, s) a(x)) ds dx

f Z(v=g2(x)} f *'t(x)

f z~o=~(x,~ v . (h(x) ~(x)) dx.

Finally, taking the supremum over all admissible a, we obtain an expression equivalent to (1.2). Therefore, lim inf G~(v,) >= Go(v). To construct a sequence {O,i} satisfying the analogues of (1.12), (1.13), i.e. ~j--~ v

in L1(/2),

(1.40)

lim G~.(0~.) = Go(v), 3 J

(1.41)

j-~ oo

we again first suppose that v E BV(O) takes the form

v(x) = t{g~(x) Ig2(x)

xE

A

x EB

with

f

v d x ~ r

t2

and 8A I'~ 8B smooth. In constructing the transition layer sequence, the differential equation (1.17) of Theorem 1 is replaced by ?z

~ s (x, s) = (z - g~(x)) ( g z ( x ) -

z),

(1.42)

z(x, o) = 89(gl(x) + g~(x)) = : ~(x).

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P. STERNBERG

R->R with zE

Since g , , g2 are C ~ functions we obtain a solution z : .Q • ([7]). Arguing as before we find that

g~(x) < z(x, s) < g2(x) lim

s~>oo

z(x, s) = g2(x),

lira

s-+ -- oo

for all s,

CI(,-Q

X

R)

(1.43)

z(x, s) = gl(x),

(1.44)

the limits on the right being a p p r o a c h e d at an exponential rate. We also need an L~(-Q • R) b o u n d on zx(x, s) (here z x denotes the spatial gradient of z). To this end we note that z(x,s)

d't

j'l (~ ~(x)

= s.

(1.45)

gl(x)) (g2(x) -- B)

_

Differentiating both sides with respect to x and solving the resulting equation for z x, we obtain

zAx, s) = (z(x, s) - g,(x)) (ge(x) - z(x, s)) •

i~ - g , ) 0

- g2) + Vg2_.

(~ --

g(x)

(1.46)

g t ) ( ~d~ : ] -- g 2 ) 2 z(x,s)

- Vg, _f

g(x)

.

(~ _ g,)2 (g~ _ ~i

]"

Since gx(x) < z(x, s) < g 2 ( x ) by (1.43) and gt and g2 are b o u n d e d in the C1(/2) topology, it follows that Zx is b o u n d e d for any finite s. Passing to the limit as s - + ~ oo in (1.46) and using L ' H o s p i t a l ' s Rule, we conclude from (1.44) that lim

8~)" OO

Zx(X, s) = Vge(X)

and

lim

S-~" - - OO

Zx(X, s) = - V g l .

We thus infer that sup

t Zx(X, s) [ < oo.

(1.47)

12•

Reintroducing the distance function d given by

{t-

dist (x, 8A A c~B)

d(x) = /dist (x, 8A F~ ~B)

xEA x E B,

one can define a b o u n d a r y layer sequence {~} through g2(x)

I

~Xx) = z(x, d(x)/O [g,(x)

{d > 2 l/e}

{/at < l/2} {d < - 2 r

where ~ is linear in d(x) on { I / e < ]d I < 2 l / ~ so as to be continuous for all x. F r o m here on the p r o o f of (1.40), (1.41) follows in the same m a n n e r as did (1.12), (1.13), except that one must use (1.47) to estimate ] g ~ l 2 in proving the analogue o f inequality (1.28).

Singular Perturbation on Nonconvex Variational Problems

233

2. Vector Dependent Energy In this section, we consider a generalization of Theorem 1 to a variational problem in which the nonconvex integrand is vector-dependent. In order to preserve the "two-phase" nature of minimizers, we consider a nonnegative integrand W: R 2 - + R which is zero on two disjoint closed curves F~ a n d F2, where F~ lies in the interior of F 2 . " As in Section 1, one goal is a characterization of the limit of minimizers of the perturbed problem. Theorem 3 shows that such a limit will again minimize interfacial surface area in .(2. As before, ~ is an open, bounded subset of R n with Lipschitz-continuous boundary. However, this characterization is incomplete since the limit problem does not determine where on F t kg /'2 the limit takes its values. We also characterize the cost per unit area along the interface of the transition made by the perturbed minimizers from Fx to F 2. The latter is measured (asymptotically) by the length of a geodesic that minimizes distance with respect to a degenerate Riemannian metric on the plane. This is accomplished in Part A by identifying the F-limit and by proving/'-convergence in this setting. In Part B we establish certain properties of the degenerate metric which were needed in Part A, including the existence of geodesics that minimize distance.

A. Generalization to W: R2-+ R Consider first a model in which W is only radially dependent. Let u :, f2 ~ R 2 and W ( u ) = ( l u l - a ) 2 ( I u l - 6 ) 2 with 0 < a < b . Then the unperturbed problem is inf

((lul

uELI(~,R2) ~ ' "

- a)2(lul - b)2dx,

(2,1)

lul=c

g2

where a 1•1 < c < b IOI. Clearly any function with range on the circles of radii a and b that satisfies the constraint will minimize (2.1). Now introduce the perturbation 1~2 tVUl2 ( = E2 IVUl] 2 -~E: 2 ]VU2I 2)

and let u, denote a solution of inf f ( t u l I i.!=c ~,

u~Hl( ~) d

-

a)2 (lul - b) 2 + e 2 1~TulZdx.

Proposition4. Suppose there exists a scalar function lu~]-->" Ro in La(s Then Ro solves inf

REBV(~) R(x)E{a,b}a.r f Rdx=c D

Per a {R = a}.

RoELX(~)

(2.2)

such that (2.3)

234

P. STERNBERG

Proof. If one writes

u(x) = R(x) (cos O(x), sin O(x))

with R :~ 0,

(2.2) becomes inf

f (R(x) - a) 2 ((R(x) - b) 2 + e 2 IVR 12 + e2R 2 IV012 dx.

u~H~(O) t 2 Rdx:c D

(2.4)

Then from

u,(x) : R,(x) (cos O~(x), sin O~(x)), it is evident that a minimizer must satisfy V0, : 0. The value of the constant 0~ is arbitrary without any further boundary conditions or constraint. Since the infimum in (2.4) must be achieved by a function of the form

u(x) : R(x) (cos 0, sin 0), with 0C R, R E H~(E2), the problem reduces to the scalar case of Section 1. The proposition follows from Theorem 1. Thus, the moduli of the minimizers of (2.2) converge in L1($2) to a solution of the partition problem, and the phase is such as to effect the transition between the two zero states of W along a radial path in the plane.

Remark. The existence of a subsequential limit Ro follows from Proposition 3. Note that since the value of the constant 0, is arbitrary, one cannot expect any determination of the constant phase 0o of the limit of minimizers Uo----Ro(cos 0o, sin 0o). Generalization. Our model problem reduced to the scalar case because Wwas only radially dependent and the phase 0~ of u~ was constant. To generalize the problem we distort the radial dependence and consider W = T 2, where T : R 2 --* R has the following properties:

T 6 C 2,

T:

0

only on FI L / / 2 ,

where / ' l , Fz are two disjoint simple closed curves on the plane that admit C 3 regular parametrizations 0~: [0, 1]--> FI, fl: [0, 1 ] - - > / 2 , respectively. Furthermore, we assume FI C interior of F2 and

T > 0 in ~ , where ~ denotes the subset of we suppose

[~TT(y) t ~: mo

R2

(2.5)

lying exterior to F~ but interior to F2. Finally,

for y E t ~ ( ~ F~ k J / 2 )

for some mo > 0.

(2.6)

The unperturbed problem is now inf

u~L~(.Q,R 2)

( T 2 ( u ) dx.

(2.7)

Singular Perturbation on Nonconvex Variational Problems

235

Ul

Fig. 5. T = 0 o n

F~ U / ' 2

Its solutions u are in one-to-one correspondence with the partitions of g2 into sets A and B such that u(x)E Fx on A and u(x)E F2 on B. Choosing the usual perturbation, we obtain the perturbed problem: inf

u~H I(.D,R 2)

fT2(u)

+ e 2 [Vu[ 2 dx.

(2.8)

As in Section 1, any Ll-convergent subsequence of minimizers must converge to a solution of the F-limit problem, so that the task is to evaluate this F-limit. However, in contrast to the scalar case, we are as yet unable to prove the compactness of minimizers (see Remark (2.30)). We begin by defining the rescaled sequence of functionals H~:L'(f2)--> R through

H~(u)=l(lT2(u)-~Te[Vu[2dx

otherwise.UEHl(f2'R2)

The proposed limit functional Ho : L I ( ~ ) -+ R is given by

Ho(u)

12L(_y) Ver~ {u E/'1}

T(u(x)) =

I+ oo

otherwise,

1

0 a.e.,

Z{,+rl)EBV(g2)

where t2

L(r)

f

I (t)l dt.

tt

L(y) is defined for 7 : [t,, t2] --~ ~ of

Lipschitz-continuous, and y is a minimizer inf L(y). (2.9)

$'(tt)Ert

y(t2)~F~

The existence of y is proved below, in Lemma 9. Theorem3. =

236

P. STERNBERG

Thus (i) for each vE La(Q), and for each sequence {v,} in La(Q), v, -+ v

in L 1( Q) implies lim inf H,(v~) ~ Ho(v);

(2.10)

(ii) for each v E La(Q) there exists a sequence {~j} in LI(Q) satisfying ~j--~ v in L1(Q), lim Hs.(e,.) =

j->- ~

J

J

(2.11)

Ho(v).

(2.12)

As a prerequisite for the proof, we remark on the kinds of La-convergent sequences {vs} and limits v that need to be considered in demonstrating the inequality (2.10). One may immediately assume

T(v(x)) = 0

a.e. in Q,

since otherwise vs-+ v implies that lim H,(v,) = c~. Hence we assume v takes the form ~-+o

ta(x) v(x) = {~b(x)

x EA x E B,

where A ~ B = Q and a, bELa(g2) with a : A - - > l " l , b : B - - * / " 2 . Concerning the sequence {v,}, one may suppose v, E Ha(O) since otherwise H,(v,) = c~. In fact, one may suppose v, E C ~ since C a ( Q ) is dense in Ha(O). In proving (2.10) we make use of properties established in Part B of this section concerning the degenerate Riemannian metric dr defined by t2

dr(Y a, Y2) =

inf

~ Lipschitz. continuous ~'(t 1)=Yl y(t2) =Y2

(T@(t))

[~)(t)] dt

for Yl, Yz E ~

(2.13)

II

and the associated "distance to /"~", given by

h(y) = inf dr(yo, y). Yo~Ft

In particular, we note that h is Lipschitz-continuous in ~ (and therefore differentiable a.e.), and that

IVh(y)l = T(y)

for a.e. y E 9 .

(2.14)

These facts are confirmed below in L e m m a 11. N o w define ~h(v~(x)) for {v~ E ~}

g~(x)

L

~0

elsewhere.

Then the restriction of g~ to {v, E ~} is a Lipschitz-continuous function satisfying Vg~(x) = Vyh(v~(x))" Vxv~(x) a.e.,

(2.15)

Singular Perturbation on Nonconvex Variational Problems

237

so that, in view of (2.14);

]Vg~ I
L(y)

(2.16)

in Ll(O), where in B.

(2.17)

As a final preliminary to the proof, observe that, for fixed tE (O,L(y)), (2.17) implies g~(x)-h(v(x))>t in { g , > t ) \ B ,

h(v(x)) - g,(x) > L(y) - t

in B \ {g, > t } .

Thus

f Ig~(x) - h(v(x))[ dx = ~g~._ Ig~(x) - h(v(x))[ dx >= min {t,L(y) - t} ]{g,(x) > t}AB[ = min {t,L(y) - t) f ]X{g,>t} - Zsl dx. --

--

.Q

e--~0 for all t E ( O , L ( y ) ) .

Consequently, X(g,>t}---~ZB i n L l ( O ) a s

Proof of (2.10). To establish (2.10), we note first, using (2.16), that

>= f

__IT2(v,(x) ) +

e IVv, I2 ,Ix

{o 0 , we define the map T a : R 2-->R by

y~

iT(y) + a

To(y) = 189a with T e e

C2(R2)

(2.32)

yCR2\~"

and satisfying for y r ~ ' \

89a < T~(y)
/ ' l 7o(0) -= ao.

k(so, t) = 7o(t),

(2.40)

k(s, O) = a(s),

(2.41)

k(s, to) = ?n(to),

(2.42)

is a parametrization o f / ' 2 and o~(~) =

Since ?o minimizes Lo,

~---~Lo(k(s, t))ls=~~ = O. Use of (2.40) shows that ~k_

~)o ~ ( ~

fo I~l VT(7~) " ~s (SO, t) -}- (T(To) -}- 6) ~

)

-~- ~s k(Sn, t) dt = O.

0

Integrating by parts yields

o

--

' -~s (so, t)) dt +

(2.43)

(T(Ta(t)) + O) ( ~k )]'o [ro(t)[ ~)gt),~ (~o, t) o = o.

Since 7~ solves (2.35), the integral in (2.43) vanishes. From (2.41) and (2.42) follows ~k_ (so, o) = ~'(~)

and

~k_ (so, to) = Us (r~(to)) = o.

Singular Perturbation on Nonconvex Variational Problems

247

Then (2.43) implies t - % ( O') R given by

/q(s) 2e(s) -- be(s) ,

(2.48)

in which the superior dots indicate differentiation with respect to s. The quantity 2e measures the tangential speed of ~'e relative to the normal speed with respect t o / ' 1 . Thus 2e small means that ~,~ is progressing efficiently away from/'~ towards / 2 - The following lemma enables us to bound the arc-length near / i . Lemma 7. There exists a number c2 E (0, it), independent of t~, such that 12e(s) I ~ 89

for 0 _< s ~ min {s : ve(s) = c2}.

(2.49)

This estimate is needed to prove Lemma 8. There is a positive number c3 independent of ~ such that dist (ye(s), I'1) ~ ca,

(2.50)

provided s ~ min {s : dist (ye(s), I'1) = c2}, where "dist" refers to Euclidean distance. Proof of Lemma 7. The proof of Lemma 7 is split into three steps: first, we derive

a differential equation satisfied by 2e; then we use this to obtain a differential inequality; finally, we integrate this inequality to obtain (2.49).

Step 1. To derive a differential equation for 2e, note first that 2e(0) = 0 since (2.44), (2.46) imply r~(s) = ue(s) od(u6(s)) + re(s) be(s) n'(ue(s)) -}- be(s) n(ue(s)).

(2.51)

By (2.39), re(o) is orthogonal to ~'(ue(0)), while re(0) = 0, so that

o = (re(o), ~'(ue(0))) = ue(0) (~'(ue(0)), ~'(ue(0))) = ug0), the primes denoting differentiation with respect to u. It then follows from (2.48) that 2e solves the initial value problem 2e

be/Je- /lebe ue (be)2 -- be

2 ~e ve

(2.52)

2e(0) = 0. Now, since ~'e is an extremal for the functional Le, ze must be an extremal for the functional

z-+ f (T(z) -}- ~) ~ M(z(t)) dt. tl

Singular Perturbation on Nonconvex Variational Problems

249

Setting the first variation equal to zero, we obtain the Euler-Lagrange equation

d

where M ( z ~ ) : ~ M(z~(s)) and JM is the Jacobian matrix of the coordinate map M. Since I~t(z~)l -- I~1-- 1, the equation reduces to

V~(z~)= d ((7~(z~) + a) M(z~)) JM(z~) or

VT(z6) = (VT"(zo), ~ ) M(zo) JM(zn)+ (f(z~) + 8) M(z~) JM(Zo).

(2.53)

Use of (2.46) and (2.51) yields )~/(z~) =/i~a' + t;an + (v~ii~ -}- 2/t~ba) n' -F (tJ~)2 a " -~ v~(/ts)~ n", while (2.44) gives M,(zn) = od(u~) q- von'(uo), Mv(zn) = n(un). Applying the Frenet equations ([9]) oJ' = kn,

n" = - k o d ,

,(2.54)

we can calculate

~I(z~) S~(z~) =

(, ).

2~I(z~) JM(Z~) = (()l~/(Zo), M,(za)), (~r(zn), Mv(za)))i In this manner we arrive at

~t(z~) J.(z~)

= (u~(1 - kv~) ~, ~ ) ,

(2.55)

M(zo) JM(z~) : (~(1 -- kve) 2 - (1 - kvn) (2kun/'~ + k'vn(it~)2), i)o-F k(/q) z (1 - kv~)),

(2.56)

where k" = ~ u (k(u)) =. .

"

L

Note that a ~ C ~ assures that the signed curvature k defined by (2.54) is C x. Substituting (2.55), (2.56)into th~ Euler-Lagrange equation (2.53)and solving for/ia and iJn, one finds that ~'. - (1 -- kvo) 2 h~ ~a it~ ---- (7"(z,~)+ t~) (1, -- kva) ~ + (1 - kvo--~)' ba =

I"(z~) + t)

- k(itn)z (1 - kvo),

(2.57) (2.58)

250

e. STERNBERG

where al" v)] 7:. = Vu(U,

,

I(u.v) (u.~.v ~)

7:o = ~a T( u ,

v)[

,

I(u.v) = (u ~.v ,)

=

and ~ . : = 2kit,b, + k'va(t~,) 2 .

(2.59)

Note that the (u, v) coordinates are only defined for v < / z , so that 1 - kvo > 0 by (2.45). Thus /i, is finite. Substitution from (2.57) and (2.58) yields

Since 2.

=

it./b., one arrives at

L=-

+ o~. + 2~k(h.) 2 (1 - kvn) 2

~u (T(zo) + ~) (1 -- kv.) 2 i:.

(2.60) This completes step 1.

Step 2. We now estimate the terms on the right side of (2.60) to obtain differential inequalities that control 26. F r o m here on we shall restrict our attention to (s : v~(s) =f T@#)) as t~

__> --

min

T(y) (t~' -- t'~).

min

T(y)~J 89c2

c2/2~dist(y, FD~_c2

Thus, because of (2.78) and (2.79), Lo(y~) > ( =

~ ~dist(y,/'D ~c2

since the arc-length of y~ between y~(t~) and

v~(t;).

F(t'~') cannot

be less than

v~(t'~')

-

-

Singular Perturbation on Nonconvex Variational Problems

255

Comparing (2.77) to this last inequality, one concludes that ( ~/2~,(y.r~)~,_ min _ T (y) ) 89 c2 --= dis,t~,~u)_~c~ k

ye~

I

256

P. STERNBERG

Thus Ln(7~)