On the definitions of Sobolev and BV spaces into singular spaces and the trace problem

David Chiron

Description

Niveau: Supérieur, Doctorat, Bac+8
On the definitions 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 : 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 define the same p-energies. We review also other definitions, due to L. Ambrosio (for BV maps into metric spaces), Y.G. Reshetnyak and finally to the notion of Newtonian-Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different 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 terms of “limit” of the space W s,p as s ? 1, 0 < s < 1, and finally following the approach proposed by H.M. Nguyen. We also establish the W s? 1p ,p regularity of traces of maps in W s,p (0 < s ≤ 1 < sp).

• into metric


• sobolev spaces


• sequence u? ?


• existence results


• bv maps


• sequence ?n


• standard quantities

Sujets

• Antipolis
• Chiron
• Gromov
• Nguy?n
• Sobolev space

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 23
• Langue: English

Extrait

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 : [e-mail]
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