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DAVID CHIRON - profil-zyak-2012
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On the de nitions of Sobolev and BV spaces into
singular spaces and the trace problem
David CHIRON
Laboratoire J.A. DIEUDONNE,
Universite de Nice - Sophia Antipolis,
Parc Valrose, 06108 Nice Cedex 02, France
e-mail : chiron@math.unice.fr
Abstract
The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces,
due to N. Korevaar and R. Schoen on the one hand, and J. Jost on the other hand. We prove
that these two notions coincide and de ne the same p-energies. We review also other de nitions,
due to L. Ambrosio (for BV maps into metric spaces), Y.G. Reshetnyak and nally to the notion
of Newtonian-Sobolev spaces. These last approaches de ne the same Sobolev (or BV) spaces,
but with a di eren t energy, which does not extend the standard Dirichlet energy. We also prove
a characterization of Sobolev spaces in the spirit of J. Bourgain, H. Brezis and P. Mironescu in
s;pterms of \limit" of the space W as s ! 1, 0 < s < 1, and nally following the approach
1s ;p s;ppproposed by H.M. Nguyen. We also establish the W regularity of traces of maps in W
(0 < s 1 < sp).
Keywords: Sobolev spaces; BV maps; metric spaces; traces.
AMS Classi c ation : 46E35.
1 Introduction
The aim of this paper is to relate various de nitions of Sobolev and BV spaces into metric spaces.
The rst one is due to N. Korevaar and R. Schoen (see [12]), which follows the pioneering work of
M. Gromov and R. Schoen [9]. The second one is the approach of J. Jost ([11]). We will next focus
on other approaches. For the domain, we restrict ourselves to an open bounded and smooth subset
N
of R , N 2 N . Many results extend straightforwardly when
is a smooth riemannian manifold.
Extensions are also possible when
is a measured metric space (see [11], [20], [10]). However, we do
not aim such a generality. For the target space, we will work with a complete metric space (X;d).
pFirst, we recall the de nition of L ( ;X), for 1 p 1. A measurable map with separable
pessential range u :
! X is said to be in L ( ;X) if there exists z 2 X such that d(u();z) 2
pL ( ;R). If
is bounded (and more generally if j
j < 1), then the point z is not relevant: if
p p pu 2 L ( ;X), then for any z 2 X, d(u(:);z) 2 L ( ;R). The space L ( ;X) is complete for the
distance Z 1
pp
pd (u;v) d(u(x);v(x)) dxL
1if 1 p <1 and for p =1, is complete for the distance
1d (u;v) supess d(u(x);v(x)):L x2
1;pWe recall the de nitions of Sobolev spaces W ( ;X), 1 < p <1 and BV ( ;X) spaces proposed
by these authors. These spaces naturally appeared in the theory of harmonic functions with values
into spaces coming from complex group actions, or into in nite dimensional spaces. This is the reason
why [12] developped in the context of metric spaces targets the well-known theory of Sobolev maps:
compact embeddings, Poincare inequality, traces, regularity results for minimizing maps for non-
positively curved metric spaces (even though X is only a metric space, the notion of \non-positively
curved metric space" does make sense, and we refer to [12] for instance for the de nition)... Since
we allow the target space X to be singular, one can not reasonably de ne Sobolev spaces of higher
order. Our interest for these spaces was motivated by the study of topological defects in ordered
media, such as liquid crystal, where an energy of the type Dirichlet integral plus a potential term
appears naturally for maps with values into cones (see [6]). It is then important that this Sobolev
theory extends naturally the usual Dirichlet integral.
1;p1.1 De nition of W ( ;X) and BV ( ;X).
Let 1 p <1. We recall in this subsection the de nition of Sobolev spaces of N. Korevaar and
pR. Schoen given in [12], naturally based on limits of nite di erences in L . For " > 0, we set
"
fx2
; d(x;@ ) > "g:
p 1For u2 L ( ;X) and " > 0, we introduce
8 Z p1 d(u(x);u(y))> N 1 " dH (y) if x2
;< N 1 N+p 1jS j "S (x)"e (u)(x)" >>:
0 otherwise:
pHere, S (x) is the sphere in
of center x and radius R > 0. For u2 L ( ;X), e (u) has a meanningR "
1 uas an L ( ;R) function. We then consider the linear functional E on C ( ) de ned for f 2C ( )c c"
by Z
uE (f) f(x)e (u)(x) dx:""
Next, we set
p uE (u) sup limsup E (f) 2 R [f1g:+"
"!0f2C ( ) ; 0f1c
p 1;pDe nition 1 ([12]) Let 1 p <1. A map u2 L ( ;X) is said to be in W ( ;X) for 1 < p <1
por in BV ( ;X) for p = 1 if and only if E (u) <1.
1;pWe summarize now some of the main properties of the spaces W ( ;X) and BV ( ;X), which
come from the theory developped in [12] (section 1 there).
1 N 1Notice that we have chosen to divide the original density measure e (u) of [12] by the factor jS j so that all"
approximate derivatives are based on averages.
2p pProposition 1 ([12]) Let 1 p <1. Then, for every u2 L ( ;X) such that E (u) <1, there
exists a non-negative Radon measure jruj in
such thatp
e (u) *jruj" p
pweakly as measures as "! 0 (hence E (u) =jruj ( ) ). Moreover, if 1 < p <1,p
1jruj 2 L ( ;R ):p +
Remark 1 We lay the emphasis on the fact that the weak convergence of e (u) to jruj as "! 0" p
holds for the real parameter "! 0 and not for a sequence " ! 0. This is due to a \monotonicity"n
property (see Lemma 5 below).
The last statement in Proposition 1 is valid only for p > 1, which motivates the de nition of the
1;1space W ( ;X).
1;1 1De nition 2 ([12]) We de ne W ( ;X)fu2 BV ( ;X); jruj 2 L ( ;R )g.1 +
Let us de ne for 1 p <1, the constants 0 < K 1 byp;N
Z
1 p N 1K j! ~ej dH (!); (1)p;N N 1jS j N 1S
N 1;p~e denoting any unit vector in R . De nitions 1 and 2 extend the classical Sobolev spaces W ( ;R)
and BV ( ;R).
Theorem 1 ([12]) Assume X = R is endowed with the standard distance. Then, for 1 p <1,
1;p 1;pW ( ;X) = W ( ;R) and BV ( ;X) = BV ( ;R):
1;pMoreover, if u2 W ( ;R) for 1 p <1 or u2 BV ( ;R), then
Z
p p 1E (u) = K jruj or E (u) = K jruj( ) ;p;N 1;N
n Remark 2 If p = 2 and X = R (euclidean), n2 N arbitrary, then, by Theorem 1, we have
Z
2 2E (u) = K jruj ;2;N
2hence E coincides, up to a constant factor K with the usual Dirichlet energy. This remains true2;N
nif X is a smooth complete riemannian manifold. However, for X = R but p = 2,jruj orjruj (inp 1
the sense of measures) may not be equal (even up to a constant factor) to the standard quantities
pjruj orjruj (in the sense of measures).
p pThe second result is the lower-semicontinuity of E for the L ( ;X) topology.
Theorem 2 ([12]) Let 1 p < 1. Then, the p-energy is lower semicontinuous for the strong
p pL ( ;X) topology. In other words, if u ! u in L ( ;X) as n! +1, thenn
p pE (u) liminf E (u )2 R [f+1g:n +
n!+1
3
61.2 Alternative de nition (J. Jost)
We recall now the alternate de nition proposed by J. Jost ([11]). We mention that this work
Nconsiders a metric measured space as a domain instead of an open subset of R or a riemannian
Nmanifold as in [12]. We will however not consider this general setting. Let, for x2 R and " > 0,
" (y) dy B (x); (2)"x
and consider the functionals for 1 p <1
RZ p "d(u(x);u(y)) d (y)p x
RJ (u) dx:" p "jx yj d (y)
x
Note that the initial point of view of J. Jost was to deal with harmonic maps, and thus he only
considered the case p = 2. The de nition relies on -con vergence (see [7]), and is as follows. First,
pp pwe de ne J : L ( ;X)! R [f1g to be the -limit of J as "! 0, or for a subsequence " ! 0+ " n
pas n! +1, for the L ( ;X) topology. We recall that this means that for any sequence u ! u as"
p"! 0 (or u ! u as n! +1) in L ( ;X),n
p p p pliminf J (u ) J (u) or liminf J (u ) J (u) ;" n" "n"!0 n!+1
pand there exists a sequence u ! u as "! 0 (or u ! u as n! +1) in L ( ;X) such that" n
p p p plim J (u ) = J (u) or lim J (u ) = J (u) :" n" "nn!+1"!0
pThe existence of this -limit for some subsequence " ! 0 as n ! +1 is guaranteed if L ( ;X)n
satis es the second axiom of countability (see [7]).
p 1;pDe nition 3 ([11]) Let be given 1 p <1 and u2 L ( ;X). Then, u is said to be inW ( ;X)
for 1 < p <1 or in BV( ;X) for p = 1 if and only if
pJ (u) <1:
1;pWe denote for the momentW ( ;X) andBV( ;X) since we do not know yet that these spaces
1;pare actually W ( ;X) and BV ( ;X). Notice that the approach of J. Jost does not allow to de ne
1;1directly W ( ;X).
Let us point out that the notion of -con vergence is in general stated for a countable sequence
" ! 0, and not for a real parameter " ! 0. One main problem with De nition 3 is that we don
not know at this stage if the -limit has to be taken for the full family "! 0, or for a subsequence
" ! 0. In particular, it is not shown in [11] that the functional J does not depend on the choicen
of the subsequence " ! 0. This will be however a consequence of our result, and in fact that then
-con vergence holds for the full family "! 0.
Theorem 3 Let 1 p <1. As "! 0 (resp. for any sequence " ! 0 as n! +1), the functionaln
p pp pE is the pointwise and the -limit, in the L ( ;X) topology, of the functionals J (resp. J ). The" "n
pfunctional J is now well-de ne d and
p pJ = E :
In particular, for 1 < p <1,
1;p 1;pW ( ;X) =W ( ;X) and BV ( ;X) =BV( ;X):
41;pThis theorem clari es the notion of W ( ;X) (for 1 < p <1) and BV ( ;X) (for p = 1) of J.
pJost: J does not depend on the choice of some subsequence.
Although natural, we have not been able to nd in the literature a result concerning the fact
that these two de nitions coincide. If the de nition of J. Jost allows to derive existence results more
general than in [12] for harmonic maps into homotopy classes in non-positively curved metric spaces,
[12] gives regularity results (namely lipschitzian). Fortunately, these two notions coincide for the
1 1;2Sobolev space H ( ;X) W ( ;X), with the same energy thus, one can apply, for harmonic maps,
the existence results of [11] and the regularity results in [12]. Notice however that [13] establishes
existence results similar to those of [11] in a slightly di eren t context.
p +Our second result concerns the pointwise and -limit of the functional E : L ( ;R) ! R ,"
de ned as Z
u uE (u) e (u)(x) dx = E (1) = sup E (f);" e " "
f2C ( ) ; 0f1c
pas "! 0 in the L ( ;X) topology.
Proposition 2 Let 1 p < 1 and " be a positive sequence, " ! 0 as n ! +1. Then, then n
p pfunctional E is the pointwise and the -limit for the L ( ;X)-topology of the functionals E (resp."
E ) as "! 0 (resp. n! +1)."n
We turn now to other approaches to the de nition of Sobolev spaces into metric spaces, based on
a characterization through post-composition with lipschitz maps.
1.3 Characterizations of Sobolev and BV maps by post-composition
In this Section, we are interested in characterizations of BV and Sobolev maps with values in X
by composition with 1-lipschitzian maps ’ : X! R.
1.3.1 BV maps into metric spaces
In [1], L. Ambrosio has proposed another approach to de ne the space BV ( ;X) when X is
1a locally compact and separable metric space. First, let us de ne for u2 L ( ;X) the measure (of
possibly in nite mass)jDuj on
to be the least measure such that for every borelian set B
and
every 1-lipschitzian map ’ : X ! R,
j’ uj (B)jDuj(B):BV ( ;R)
Here,j’ uj is the usual measure in the BV sense. From [1], such a measure exists.BV ( ;R)
1De nition 4 ([1]) We de ne BV( ;X)fu2 L ( ;X); jDuj( ) <1g.
Here, we do not know at this stage that BV( ;X) = BV ( ;X) in the sense of De nition 1, which
justi es that we denote BV( ;X) this space to avoid confusions.
Proposition 3 Assume X is locally compact and separable. Then,
BV( ;X) = BV ( ;X):
Moreover, for u2 BV ( ;X), we have in the sense of measures
K jDujjruj jDuj:1;N 1
51Remark 3 In the case N = 1 (since then K = 1) or X = R, then E (u) =jruj( ) and the two1;N
BV energies are equal. However, in general, the two constants K and 1 are optimal in the sense1;N
N Nthat for X the euclidean space R and with v , v :
! R de ned by v (x) (x ;0;:::;0) and1 2 1 1
v (x) x, then2
K jDv j = K dx =jrv j > 0 and jrv j = dx =jDv j > 0:1;N 1 1;N 1 1 2 1 2
Therefore, the two BV energies are only equivalent and not equal up to a constant factor even in the
1usual euclidean (vectorial) case: the minimizers ofjDuj( ) or E (u) (in some subset of BV ( ;X))
may then be di eren t. To overcome this di cult y, when X has dimension n, we should choose maps
n’ with values into R instead of R.
L. Ambrosio has developped in this framework the well-known results concerning BV maps, in
particular the de nition of the regular part of the measurejDuj, the de nition of the jump set of of
N 1a BV map as a recti able set of locally nite H measure and the notion of approximate limit
along the orthogonal direction to this jump set. By Proposition 3, one can use equivalently any of
these two de nitions, when X is locally compact and separable.
1.3.2 Reshetnyak’s characterization of Sobolev spaces
In [19], Y.G. Reshetnyak proposed a similar approach to de ne the Sobolev maps for a metric
pspace target. Let 1 p <1 and u2 L ( ;X). Set
p
R(u) inf jjwjj ; 8z2 X; jr(d(u();z))j w a:e: in
2 R [f1g:p +L ( ;R)
De nition 5 ([19]) Let 1 p <1 be given. The Reshetnyak-Sobolev space is de ne d to be the set
1;p pR ( ;X)fu2 L ( ;X); R(u) <1g.
Paralleling the proof of Proposition 3, we have the following result (see also [10] for a similar
result when X is a Banach space).
Proposition 4 Let 1 p <1. Then, we have
1;p 1;pR ( ;X) = W ( ;X):
1;pMorever, for u2 W ( ;X), we have
pK R(u) E (u) R(u):p;N
Remark 4 When X is locally compact and separable, we could also have de ned the Reshetnyak-
Sobolev space as the set of maps u2 BV( ;X) in the sense of L. Ambrosio such that the measure
ppjDuj belongs to L ( ;R), with p-energy jj jDuj jj . This gives, in this case, the existence of apL ( )
minimizer w for R(u) for any 1 p <1 (this is not obvious for p = 1).
pRemark 5 As in Remark 3, if N = 1 or X = R, then E = R. Moreover, the constants K and 1p;N
Nare optimal (consider the maps v , v :
! R de ned in Remark 3). Therefore, if we are interested1 2
in the minimization of the Dirichlet energy into a riemannian manifold with singularities, then R
does not extend the standard Dirichlet integral (up to a constant factor).
61.4 Link with the Newtonian-Sobolev and Cheeger-Sobolev spaces
The paper [10] generalizes a de nition of Sobolev spaces introduced in [20] as Newtonian-Sobolev
spaces, for an arbitrary metric space as target space.
pDe nition 6 ([10]) Let 1 p < 1 be given and u 2 L ( ;X). Then, u is said to be in the
1;p pNewtonian-Sobolev space N ( ;X) if there exist , w 2 L ( ;R) such that for any 1-lipschitzian
curve : [0;‘]!
,
Z Z‘ ‘
d(u (‘);u (0)) or w =1:
0 0
pThe Newtonian p-energy of u is then the in mum of jjjj for all the possible ’s:pL ( )
Z
pN(u) inf :
The maps are called upper gradients for u. Theorem 3.17 in [10] establishes that the Newtonian-
Sobolev energy is the same as the Reshetnyak-Sobolev energy. In particular, for p = 2, this energyR
2R(u) is not equal to the standard Dirichlet energy jruj dx (up to a constant factor).
pProposition 5 ([10]) Let 1 p <1. Then, in L ( ;X), we have R = N. In particular,
1;p 1;p 1;pN ( ;X) =R ( ;X) = W ( ;X):
We emphasize that neither [19] nor [10] (extending [20]) allow to de ne the space BV ( ;X).
Moreover, neither compactness result (as Theorem 1.13 in [12], when X is locally compact) nor
lower-semicontinuity (see Theorem 2) result for the energy are given. In fact, R = N is not lower
1semi-continuous for p = 1 and for the L ( ;X) topology.
Finally, [16] extends the notion of Cheeger type Sobolev spaces (see [5]) to a metric space target.
pLet 1 p <1 be given and u2 L ( ;X). We set
Z
p
H(u) inf liminf g 2 R [f1g;+j
j!+1
p pwhere the in m um is computed over all sequences (u ) 2 L ( ;X) and (g ) 2 L ( ;R) such thatj j
pu ! u in L ( ;X) as j! +1 and for any j2 N and any 1-lipschitzian curve : [0;‘]! ,j
Z
‘
d(u (‘);u (0)) g : (3)j j j
0
De nition 7 ([16]) Let 1 p < 1. The Cheeger type Sobolev space is de ne d to be the set
1;p pH ( ;X)fu2 L ( ;X);H(u) <1g.
Remark 6 We would like to emphasize the role played by the target space for all these de nitions
of Sobolev and BV maps. Let (X; d) be a complete metric space and let X be a closed subset of X
endowed with the induced distance, so that X is also a complete metric space. Then, if we view u as
p p p pa map u in L ( ; X) instead of L ( ;X) the quantities E (u), J (u), jDuj( ), R(u), and N(u) do
2not change . However, it is not clear whether H(u) = H(u) or not since the computation of H(u)
uses sequences (u ) X-valued instead of X-valued. In view of Proposition 6 below, we have at leastj
H(u) C H(u), for some constant C 1, when X is a length space.0 0
2This follows from the de nitions. ForjDuj, we use the well-known fact that any 1-lipschitzian map ’ : X! R can
be extended in a 1-lipschitzian map ’ : X! R, for instance by the formula ’(x) sup (’() d(x;)).2X
7Actually, Proposition 5 is a direct consequence of Remark 6 and Theorem 3.17 in [10], using
1an embedding of X into ‘ (X). In the statement of the following Proposition, we will need the
de nition of a length space.
De nition 8 Let (X;d) be a complete metric space. Then, (X;d) is said to be a length space if, for
every points x; y2 X, there exists a 1-lipschitzian map : [0;d(x;y)]! X such that (0) = x and
(d(x;y)) = y.
The result in the next Proposition is already known for 1 < p <1 and X a Banach space in [20]
(Theorem 2.3.2.). However, for sake of completeness and in view of Remark 6, we have included a
proof here when X is an arbitrary metric space. Furthermore, the following Proposition takes into
account the case p = 1 when X is a length
pProposition 6 Assume 1 < p <1. Then, in L ( ;X), we have H = N = R. In particular,
1;p 1;p 1;p 1;pH ( ;X) =N ( ;X) =R ( ;X) = W ( ;X):
1;1Assume p = 1 and, moreover, that X is a length space. Then H ( ;X) = BV ( ;X), and there
exists a constant C = C( ;N) 1 such that for every u2 BV ( ;X),
1 1K E (u) H(u) C( ;N)E (u):1;N
Remark 7 We emphasize that for p = 1, the result may be false if X is not a length space. Indeed
(see [16]), if X = f0;1g R, then u : ( 1;1) ! X de ned by u(x) = 1 if x > 0 and 0 if x 0
1;1 1;1belongs to BV (( 1;1);R) = H (( 1;1);R) (by Proposition 6) but not to H (( 1;1);X), since
X is not path-connected (see also Remark 6).
1;1Remark 8 Actually, the 1-energy in the space H ( ;X) is \close" to the BV energy in the sense
of L. Ambrosio (see Lemma 8 in section 2.7).
1.5 Other characterizations of Sobolev spaces
In [2], J. Bourgain, H. Brezis and P. Mironescu have introduced a new characterization of
1;p s;pthe usual Sobolev spaces W ( ;R) of real-valued maps, viewed as the limit of the spaces W ,
0 < s < 1, as s! 1. This does not require a notion of gradient, and is very close to the two previous
de nitions ([12] and [11]). We generalize in the following Theorem the results of [2] and [8] in the
case of a metric space target.
We consider a family of radial molli ers ( ) (or a sequence ( )). We recall that this means" ">0 nR
1 Nthat is in L (R ), radial, 0, = 1 and for all > 0," " N "R
Z
(x) dx! 0 as "! 0:"
jxj>
pThen, given 1 p <1, we de ne, for u2 L ( ;X),
Z Z
pd (u(x);u(y))
F (u) (x y) dxdy2 R :" " +pjx yj
Theorem 4 Let 1 p <1 and (" ) be any positive sequence, " ! 0 as n! +1. Then, for everyn n
pu2 L ( ;X), lim F (u) and lim F (u) exist in R [f+1g and"!0 " n!+1 " +n
Z Z pd (u(x);u(y)) plim F (u) = lim (jx yj) dxdy = lim F (u) = E (u):" " "np n!+1"!0 "!0 jx yj
pMoreover, the functionals (F ) (and (F )) -converge as "! 0 (and as n! +1), in the L ( ;X)" "n
ptopology, to E .
8Theorem 4 combined with Theorem 1 extend the results of [2] and [8] which correspond to the
pcase of real-valued maps. Indeed, from [2] and [8], given 1 p <1 and f 2 L ( ;R), then
Z Z Zpjf(x) f(y)j pliminf (jx yj) dxdy = K jrfj dx2 R [f1g: (4)n p;N +pn!+1 jx yj
1;pThe quantity (4) is 1 if f 62 W ( ;R) (if 1 < p < 1) or f 62 BV ( ;R) (if p = 1). Moreover,
the right-hand of (4) has to be understood as the mass of the measure jruj in the BV case. This
approach de nes the same p-energy as [12] (thus extends the classical Dirichlet energy), but does not
1;1allow to de ne W ( ;X).
Remark 9 Theorem 4 emphasizes that the result of J. Bourgain, H. Brezis and P. Mironescu (and
also J. D avila) does not require a linear structure, neither usual Sobolev tools, such that density of
smooth functions or integration by parts.
Remark 10 It is a natural question to wonder what may occur if one chooses non-radial functions
. The problem is that in the non-radial case, there can be privileged directions for , for instance,n nh i h i
2 1 1 1 1inR , can be the characteristic function of the thin rectangle in the x direction ; ;n 1 2 2n n n n
suitably normalized, and then we have
Z Z Z
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