Riemannian approximation of Finsler metrics
Andrea Braides
Giuseppe Buttazzo
Ilaria Fragala
Dipartimento di Matematica
Universita di Roma \Tor Vergata"
Via della Ricerca Scientica
00133 Roma
ITALY
[email protected]
Dipartimento di Matematica
Universita di Pisa
Via Buonarroti, 2
56127 Pisa
ITALY
[email protected]
Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci, 32
20133 Milano
ITALY
[email protected]
Abstract. We show that the class of smooth and isotropic Riemannian metrics is dense in the class
of all lower semicontinuous Finsler metrics, with respect to the ;-convergence of energy integrals.
2000 Mathematics Subject Classication: 49J45, 58B20.
Key words: Riemannian and Finsler metrics, Gamma convergence.
1
1. Introduction
In this paper we prove a density result which answers positively a question about the approximation
of every Finsler metric by Riemannian metrics risen in [9, Remark 4.3].
By a Finsler metric, we mean a Borel function ' = '(x; z ) on IRN IRN , which is convex and 2homogeneous in z ; we say that ' is Riemannian if it is of the type
'(x; z ) = A(x)z z
for a suitable symmetric N N -matrix of coecients A . The convergence under consideration is the
;-convergence of the energies Ln associated to a sequence of metric integrands 'n through
(1:1)
Ln ( ) :=
Z 1
0
'n (; 0) ds ;
2 H 1 ((0; 1); IRN ) :
Such convergence, introduced in 1975 by De Giorgi and Franzoni, is nowadays a commonly-used tool of
the Calculus of Variations, and in this case it corresponds loosely speaking to the convergence of the inma
of Ln for curves with prescribed end-points. We refer to the volume [13] for a thorough introduction to
this topic (see also [6]).
In 1983, Acerbi and Buttazzo studied in [1] the asymptotic behaviour of a sequence fLng as in (1.1) in the
homogenization case, namely when 'n (x; z ) = '(nx; z ), being ' periodic in its rst variable (see also [2,
5, 7, 10, and 14]). They proved that in this case the ;-limit of fLng can be represented by an integral of
the same type, with a convex homogenized integrand = (z ) independent of x . Moreover, they gave an
example when '(x; z ) is Riemannian and nevertheless the corresponding function is not. Such example
clearly shows that, in contrast with the case of quadratic forms, which are closed by ;-convergence [13,
Proposition 13.11], the class of Riemannian metrics is not closed in the class of all Finsler metrics with
respect to the ;-convergence of energy integrals. In particular, the following problem arises in a natural
way:
\Is the class of Riemannian metrics dense in the class of all Finsler metrics?"
Here we prove that the answer is yes: we indeed obtain a stronger result, namely the density of isotropic
Riemannian metrics in the class of Finsler metrics locally equivalent to the Euclidean metric. Such class has
been recently considered by Buttazzo, De Pascale and Fragala in [9], where it is shown that, if the metrics
'n are obtained by dierentiation from a sequence of geodesic distances dn , then the ;-convergence of
fLng is equivalent to the uniform convergence of fdn g on compact subsets of IRN IRN . Also, it is shown
that such convergence is in turn equivalent to the ;-convergence of suitable energy functionals related
to mass-transfer problems. Thus our density result can be read in each of these topologically equivalent
contexts.
Let us conclude with some comments about the proof of our theorem. It is rather articulated so that it is
divided in several steps, while the preliminary tools are summed up in Section 2. Given a Finsler metric
' = '(x; z ), the key idea consists in freezing the variable x = x0 , and in approximating '(x0 ; z ) in a
countable dense set of directions z . The dependence on x , that we assume to be lower semicontinuous, can
be treated essentially by localization, while the approximation of '(x0 ; z ) in every nite set of directions
can be made homogenizing a suitable sequence of Riemannian metrics, exploiting an explicit formula for
their homogenized integrand, and using a crucial convexity argument.
2
2. Notation and preliminaries
Let 0 < . We denote by M = M; the class of all lower semicontinuous Finsler metrics
' : IRN IRN ! [0; +1) controlled from above and below respectively by and times the Euclidean
norm j j (see [4]). More precisely, every ' in M satises
(2:1)
(2:2)
(2:3)
for every z 2 IRN ; x 7! '(x; ) is lower semicontinuous ;
for every x 2 IRN ; z 7! '(x; z ) is convex and 2-homogeneous ;
jz j2 '(x; z ) jz j2 ; 8 (x; z ) 2 IRN IRN :
We notice that, as a consequence of (2.2), every ' in M satises the symmetry condition '(x; z ) =
'(x; ;z ). An element ' of M is called Riemannian if it is of the type
'(x; z ) =
N
X
i;j =1
aij (x)zi zj ;
in particular we say that such ' is isotropic when aij (x) = a(x)ij , i.e., '(x; z ) = a(x)jz j2 , and that ' is
smooth when its coecients aij are smooth functions.
To every ' 2 M , we associate the energy functional L' dened by
Z
L' ( ) := '(; 0 ) ds ;
(2:4)
I
2X ;
where X := H 1 (I; IRN ) is the space of Sobolev curves dened on the real interval I . We note that, by
(2.1)-(2.3), L' turns out to be lower semicontinuous with respect to the strong L2 -convergence on X (see
for instance [8, Chapter 4]).
We shall deal with the asymptotic behaviour, in the sense of ;-convergence, of a sequence of functionals
Ln associated to varying integrands 'n according to (2.4). We recall that a sequence of functionals
Fn : E ! IR [ f+1g dened on a metric space (E; d) ;-converges to F at x if both the ;{ liminf and
the ;{ limsup inequalities hold, that is
;{ liminf
Fn (x) := inf liminf
Fn (xn ) : d(xn ; x) ! 0 F (x) ;
n
n
;{ limsup Fn (x) := inf limsup Fn (xn ) : d(xn ; x) ! 0 F (x):
n
n
Let 'n ; ' 2 M , and let Ln ; L be the functionals associated respectively to 'n ; ' according to (2.4). The
;-convergence of the sequence fLng will be always considered with respect to the strong L2 -convergence
on X or equivalently, due to the growth condition from below in (2.3), with respect to the weak H 1 convergence on X .
;
Proposition 2.1. There exists a distance function d; such that: Ln ;!
L if and only if d; (Ln ; L) ! 0.
R
;
; e
e
e
Proof. We note that Ln ;! L if and only if Ln ;! L , where Ln ( ) := Ln ( ) + I j j2 ds and Le( ) :=
R
R
R
L( ) + I j j2 ds . All the functionals Len are minorized by ( ) := I j 0j2 ds + I j j2 ds , which is
lower semicontinuous and coercive on the separable space X endowed with the L2 -metric. Therefore, by
Corollary 10.23 of [13], we infer that the ;-convergence for sequences of functionals of the form (2.4) is
metrizable.
3
;
Proposition 2.2. We have Ln ;!
L if one of the following conditions holds:
(i) 'n (x; z ) converge increasingly to '(x; z ) ;
(ii) j'n (x; z ) ; '(x; z )j !n (x)jz j2 , with !n ! 0 uniformly on compact subsets of IRN .
Proof. (i) It follows from [13, Proposition 5.2] and the monotone convergence theorem (see also [3, Theorem
1.21 (a)]).
(ii) Let fng X , n ! in L2 . Possibly passing to a subsequence, we may assume that liminf
Ln(n ) =
n
R
0
2
lim
L ( ) < +1 . In particular, since Ln(n ) I jn j ds , we deduce that fn g is bounded in H 1 , so
n n n
(a subsequence of) it converges uniformly on I , and we have
Z
lim
jLn (n ) ; L(n )j lim
j'n (n ; n0 ) ; '(n ; n0 )j ds
n
n
I
lim
sup !n (n (s))
n
Z
s2I
I
jn0 j2 ds = 0 :
Thus, using also the lower semicontinuity of L , we obtain
;
;
;
liminf
Ln(n ) ; L( ) liminf
Ln (n ) ; L(n ) + liminf
L(n ) ; L( )
n
n
n
;
= liminf
L(n ) ; L( ) 0 ;
n
which proves the ;{ liminf inequality. To prove the ;{ limsup inequality, it is enough to take n , and
apply the Lebesgue dominated convergence theorem.
In order to state the following application of Proposition 2.2 (i), it will be useful to recall that, if f is a
lower semicontinuous function on a metric space (E; d) with values in [0; +1], its Yosida transforms are
by denition
f (x) = yinf
ff (y) + d(x; y)g ;
2E
where is a parameter in IR+ .
Proposition 2.3. For every 2 IR+ , the function f is -Lipschitz continuous, and the sequence ffg
converge increasingly to f as ! +1 .
Proof. We refer to [11, Lemma 2.8], [13, Chapter 9], or [7, Section 1.2].
We nally recall a result about the homogenization of Riemannian metrics. We denote by Y the unit cube
of IRN .
Proposition 2.4. Let f'ng M be of the type 'n (x; z) := PNi;j=1 aij (n x)zi zj , where the functions
aij are Borel measurable, bounded, and Y -periodic. Then fLng ;-converges to L , being L the energy
associated to the convex function ' (independent of x ) given by
(2:5)
N
n1 Z t X
'(z ) = t!lim
inf t
+1
0 i;j =1
;
o
aij ( )i0 j0 ds : 2 H 1 (0; t); IRN ; (0) = 0 ; (t) = tz :
Proof. We refer to [5] and to [7, Section 16.1].
4
3. The density result
We state below our main result. For convenience, we divide the proof in some steps.
Steps 1 and 2 provide by easy arguments a simplied version of our approximation problem. Such preliminary simplications are however useful, as they contain the basic idea of approximating ' in a countable
and dense set of directions. This allows to explicitely dene in Step 3 the required sequence of coecients.
In order to prove that the associated Riemannian metrics converge in energy to ' , we rst show that,
as mentioned in the Introduction, the dependence on x does not play an essential role in the problem,
namely that we can reduce to the case when ' is independent of x . Accordingly, we have to modify our
sequence of coecients, by freezing locally the variable x . The new coecients thus obtained turn out
to be periodic, so that we fall into the classical homogenization setting; in particular, we can conclude in
Step 4 enforcing formula (2.5).
Theorem 3.1. For every ' 2 M , there exists a sequence f'ng of isotropic and smooth Riemmanian
elements of M which approximate ' ,
such that fL'n g ;-converges to L' .
i.e.
Proof. We rst note that, by Proposition 2.1, a diagonal argument can be applied to ;-converging
sequences in M . So we can approximate ' by a sequence f'j g , successively approximate each 'j , and
iterate. For simplicity, throughout the proof we set L := L' , and Lj := L'j whenever f'j g is a sequence
in M indexed by j .
Step 1. We claim that it is enough to consider the case when ' = '(x; z ) is Lipschitz continuous in x ,
and to nd for such ' a sequence f'n g of isotropic Riemannian elements of M , not necessarily smooth,
for which the statement holds.
Indeed, since ' is by assumption lower semicontinuous in x , by Proposition 2.3 it can be written as the
supremum of a sequence of Lipschitz continuous elements of M . Thus, in view of Proposition 2.2 (i), we
may assume directly that x 7! '(x; z ) is Lipschitz continuous for every z .
In a similar way, setting 'n (x; z ) = an (x)jz j2 , we can deal with the smoothness of the coecients an . More
precisely, since each an is lower semicontinuous, by Proposition 2.3 we may write it as the supremum of its
Yosida transforms an; , and apply again Proposition 2.2 (i) to nd out that the corresponding functionals
Ln; ;-converge to Ln as ! +1 . Finally, since each an; is continuous, the smooth functions
an;;h = an; h , obtained by convolution with a sequence of molliers h , converge uniformly to an;
on compact subsets of IRN as h ! +1 . Therefore we can conclude by applying Proposition 2.2 (ii).
Step 2. Let fi gi2IN be a dense sequence in S N ;1 := fz 2 IRN : jz j = 1g , and set
S
'
(x; z ) if z 2 ki=1 IRi
(3:1)
; 'k (x; z ) := k (x; z ) ;
k (x; z ) :=
jz j2 otherwise
where k (x; ) denotes as usual the convex envelope of k (x; ). It is easy to check that
(3:2)
'k (x; z ) '(x; z ) ;
(3:3)
'k (x; i ) = '(x; i ) ; i = 1; : : : ; k :
We claim that Lk ;-converge to L . Indeed, the ;{ liminf inequality follows from (3.2) and the lower
semicontinuity of L , as, if k ! , we have
liminf
Lk (k ) liminf
L(k ) L( ) :
k
k
5
On the other hand, taking into account that, by (2.2) and (2.3), 'k (x; ) turn out to be equi-Lipschitz on
compact subsets of IRN (see for instance [12, Lemma 2.2 of Chapter 4]), condition (3.3) and the density of
fi g entail that 'k converge pointwise to ' on IRN IRN . Therefore the ;{ limsup inequality is satised
by dominated convergence taking the constant sequence k .
Summarizing, the proof of the theorem is achieved provided we show that, for every xed k , there exist
suitable lower semicontinuous coecients ak;n such that
(3:4)
;{ n!lim
+1
Z
I
ak;n ( )j 0 j2 ds =
Z
'k (; 0) ds :
I
Step 3. Let ' = '(x; z ) belong to M and depend continuously on x . Let fi g be the dense sequence of
directions chosen in Step 2. For every n 2 IN, let Gn;i := n1 ZZ N + IRi : We set
if x 2 Gn;i \ Gn;l ; i 6= l ; i; l = 1; : : : ; k ;
S
ak;n (x) = : '(x; i ) if x 2 Gn;i n l6=i Gn;l ; i = 1; : : : ; k ;
otherwise .
8
<
Note that each ak;n is a lower semicontinuous function of x on IRN .
The remaining part of the proof is devoted to show that (3.4) holds with such denition of ak;n .
Let 2 X . For a given > 0, let > 0 be such that the following conditions are satised:
(3:5)
jx ; yj < ) max
(3:6)
setting G := ZZ N +
sup j'(x; ) ; '(y; )j ; sup j'k (x; ) ; 'k (y; )j < ;
2S N ;1
N
[
=1
2S N ;1
;
IRe ; it holds meas ;1 (G ) = 0 :
Such exists respectively thanks to the uniform continuity of ' and 'k on compact subsets of IRN IRN ,
and since the portion of the curve contained into the hyperplane fx = tg , = 1; : : : ; N , has a negligible
one-dimensional Hausdor measure for a.e. t 2 IR.
The support of intersects a nite number of open cubes delimited by G . We denote by Qj such cubes,
for j = 1; : : : ; J , and by xj their centres; moreover, we set Ij := ;1 (Qj ).
For every j = 1; : : : ; J , we dene a Y -periodic coecient bjk by
8
<
bjk (x) = : '(xj ; i )
if x 2 G1;i \ G1;l ; i 6= l ; i; l = 1; : : : ; k ;
S
if x 2 G1;i n l6=i G1;l ; i = 1; : : : ; k ;
otherwise .
By Proposition 2.4, we have
;{ n!lim
+1
Z
Ij
bjk (n )j 0 j2 ds =
Z
Ij
'jk ( 0 ) ds ;
for a suitable convex function 'jk which can be represented through formula (2.5). Suppose for a moment
to know that
(3:7)
'jk (z ) = 'k (xj ; z ) ;
6
being 'k dened by (3.1). Equality (3.7) will be proved in Step 4. We now show that (3.4) holds under
assumption (3.7).
We Rbegin by proving the ;{ liminf inequality. Let n ! ; it is not restrictive to suppose that
sup I jn0 j2 ds < +1 . Note that, setting bjk;n (x) := bjk (nx), by (3.5) we have
n
sup jak;n (x) ; bjk;n (x)j =
(3:8)
x2Qj
sup
x2Qj \Gn;i
j'(x; i ) ; '(xj ; i )j < :
Moreover, by (3.6) and by the uniform convergence of n to , we have
J
[
(3:9)
meas
j =1
Ij = meas(I ) ;
n (Ij ) Qj for n large enough :
and
Then, using (3.9), (3.8), (3.7), and (3.5), we infer
liminf
n
Z
J Z
X
ak;n (n )jn0 j2 ds
j =1 Ij
Z
Z
J
J
X
X
j
0 2
0
2 ds liminf
a
(
)
j
j
liminf
b
(
)
;
jn j ds
k;n
n
n
n
k;n
n
n
Ij
Ij
j =1
j =1
=
J
X
I
ak;n (n )jn0 j2 ds = liminf
n
liminf
n
j =1
J Z
X
j =1 Ij
Z
Ij
J Z
X
bjk;n (n )jn0 j2 ds ; C 'k (xj ; 0 ) ds ; C Z
I
j =1 Ij
'jk ( 0 ) ds ; C
'k (; 0 ) ds ; C ;
where C is a positive constant possibly varying from line to line. Letting tend to zero, we obtain the
required inequality.
The ;{ limsup inequality can be obtained in a similar way. For every j = 1; : : : ; J , by (3.7), there exist
nj ! uniformly on Ij as n ! +1 such that
limsup
Z
n
Ij
bjk;n (nj )jnj 0 j2 ds Z
Ij
'k (xj ; 0 ) ds :
Moreover, it is well known that it is not restrictive to assume that nj coincide with on @Ij for every
n (see [13, Chapter 21]). We claim that the ;{ limsup inequality is satised taking the sequence fng
dened by
n (s) = nj (s) if s 2 Ij :
Indeed, using (3.9), (3.5), (3.7), and (3.8), we obtain
Z
I
J Z
X
'k (; 0 ) ds =
j =1 Ij
J
X
j =1
limsup
n
limsup
n
Z
Ij
J Z
X
j =1 Ij
J Z
X
'k (; 0 ) ds bjk;n (nj )jnj 0 j2 ds ; C
ak;n (nj )jnj 0 j2 ds ; C
j =1 Ij
limsup
n
'k (xj ; 0 ) ds ; C
J Z
X
j =1 Ij
Z
bjk;n (nj )jnj 0 j2 ds ; C
= limsup ak;n (n )jn 0 j2 ds ; C :
n
I
7
By the arbitrariness of > 0, the ;{ limsup inequality is also achieved.
Step 4. It now remains to prove equality (3.7), being k and j xed. We observe that, by denition (3.1)
and Proposition 2.4, (3.7) reads
(3:10)
n1 Z t
lim inf t
t!+1
0
o
;
bjk ( )j 0 j2 ds : 2 H 1 (0; t); IRN ; (0) = 0 ; (t) = tz = k (xj ; z ) :
We begin by proving the inequality in (3.10). Since the left hand side of (3.10), namely 'jk (z ), is a
convex function of z , it is enough to prove that 'jk (z ) k (xj ; z ), whence the thesis by convexication.
Moreover by homogeneity we may assume
that jz j = 1. We claim that, for every xed t 2 (0; +1), given
;
N
;
1
1
2 S , there exists a curve 2 H (0; t); IRN , joining 0 to t , such that
Z t
(3:11)
0
bjk ( )j 0 j2 ds t k (xj ; ) :
Indeed, set (s) = s , s 2 (0; t). We then have j 0 (s)j = 1, and it is immediate to check that bjk ( (s)) =
k (xj ; ) for a.e. s 2 (0; t), which gives the equality in (3.11). Therefore, for every xed t 2 (0; +1), the
inmum at the left hand side of (3.10) is less than or equal to k (xj ; z ). Passing to the limit as t ! +1 ,
we obtain the required inequality.
Finally, we prove the inequality in (3.10). It is enough to show that, for every xed t 2 (0; +1) and
every z = with unitary modulus, the inmum at the left hand side of (3.10) is larger than or equal to
N ;1 and for every curve 2 H 1 ;(0; t); IRN joining
k (xj ; z ). Actually we claim that, for a given 2 S
0 to t , it holds
Z t
(3:12)
0
bjk ( )j 0 j2 ds t k (xj ; ) :
Let us dene
n
o
(3:13)
A0 := s 2 (0; t) : bjk ( (s)) = ;
(3:14)
Ai := s 2 (0; t) : bjk ( (s)) = '(xj ; i ) ;
n
o
i = 1; : : : ; k :
It is easy to verify that each of the following modications of can be done without increasing the energy
02
0 bk ( )j j ds , and without changing the endpoints (0) and (t). For a given curve as in Figure (a),
the modications below are illustrated in Figures (b) and (c).
We translate the connected components of jA0 , jAi in order that the sets A0 , Ai associated to the
new curve thus obtained become intervals (see Figure (b)).
We replace jA0 by the segment of straight line the endpoints of jA0 , and we let 0 2 S N ;1 be the
direction of such line (see Figure (c)).
We reparametrize in order that it has a constant speed ci on each interval Ai (cf. [1, proof of
Theorem IV.1]), namely, for some positive constants ci ,
Rt j
(3:15)
0 (s) = ci i
8s 2 Ai ; i = 0; : : : ; k :
8
3
2
1
0
Figure (a)
Figure (b)
Figure (c)
We claim that the curve obtained by the above modications satises the required inequality (3.12).
R
Indeed, since (t) = 0t 0(s) ds = t , using (3.15) and setting meas (Ai ) = i , i = 0; : : : ; k , we nd that
P
t = ki=0 i ci i . Then, by the homogeneity of k (xj ; ) and the convexity of k (xj ; ), we deduce
Z t
0
bjk ( )j 0 j2 ds =
k
X
i k (xj ; i )c2i = t
i=0
k
X
=t
k
X
i
(x ; )c2
k j i i
i=0 t
k
i (x ; c ) t x ; X
i c = t (x ; ) :
k j i i
j
i i
k
k j
i=0 t
i=0 t
Thus claim (3.12) is true, and our proof is now complete.
Remark 3.2. An analogous statement as Theorem 3.1 holds for p -homogeneous and symmetric convex
integrands. More precisely, for a given p > 1, we can replace M = M; by the class Mp = Mp; of all
integrands ' = '(x; z ) satisfying
(3:16)
(3:17)
(3:18)
for every z 2 IRN ; x 7! '(x; ) is lower semicontinuous ;
for every x 2 IRN ; z 7! '(x; z ) is convex and it holds
'(x; tz ) = jtjp '(x; z ) ; (t; z ) 2 IR IRN ;
jz jp '(x; z ) jz jp ; (x; z ) 2 IRN IRN :
In this case, by minor changes in the proof of Theorem 3.1, one obtains the density in Mp of the isotropic
integrands of the type a(x)jz jp , with a smooth.
Remark 3.3. Theorem 3.1 can be transposed into an approximation result for Hamilton-Jacobi equa-
tions, via the ;-convergence of functionals related to the Legendre transforms of the Hamiltonians. More
precisely, let ' 2 M , and consider the evolution problem
(3:19)
8
<
@u + '(x; ru) = 0 in IRN [0; +1) ,
@t
:
u(x; 0) = g(x)
in IRN ,
where g is a given bounded function uniformly continuous on IRN . The unique viscosity solution to (3.19)
can be expressed through the Lax formula
n
(3:20)
u(x; t) = inf g(y) + inf
;
Z s
t
' (u; u0 ) d : u(s) = y ; u(t) = x ;
u 2 W 1;1 (s; t); IRN
o
: y 2 IRN ; 0 s < t
9
where ' (x; z ) := supfz z 0 ; '(x; z 0 ) : z 0 2 IRN g is the Legendre transform of ' . Applying Theorem
3.1, one nds a sequence of isotropic smooth Riemannian metrics n (x; z ) = an (x)jz j2 which provides the
;-convergence of the length functionals L n to L' . As a consequence, the function u in (3.20) can be
approximated uniformly on compact sets by the unique viscosity solutions un to
(3:21)
8
<
@u + (x)jruj2 = 0 in IRN [0; +1) ,
@t n
:
u(x; 0) = g(x)
in IRN ,
where n = (4an );1 .
The analogue statement holds, in view of Remark 3.2, for Hamiltonians belonging to the class Mp for
general p > 1. A detailed proof can be derived with minor changes from [5, Theorem 16.6] (see also [5,
14, 15]).
Acknowledgments. We aknowledge the hospitality of the Issac Newton Institute in Cambridge, where
this paper was initiated during the program `Models of Fracture' in November 1999. This work is part of
the European Research Training Network "Homogenization and Multiple Scales" under contract RTN11999-00040.
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