RAC Rev. R. Acad. Cien. Serie A. Mat. VOL . 95 (1), 2001, pp. 121–143

Matem´atica Aplicada / Applied Mathematics

On some questions of topology for Sobolev spaces

 -valued fractional

H. Brezis and P. Mironescu

$ %            "&('*),+-    "/!#.0 1 )2 13&4  576 8!":9 ;C= D * E@? ?  F    8!BA 9      

Abstract. The purpose of this paper is to describe the homotopy classes (i.e., path-connected components) of the space . Here, , is a smooth, bounded, connected open set in and a.e. Our main results assert that classes as if maps in .

is path-connected if while it has the same homotopy . We also present some results and open problems about density of smooth

GIH      JK=KL=M!NO        "&('*),+-   /.0 1 )213&4 576 ;=YJ T [ V W@hlaeKXZ] YJ`‰[ G\z H^] ‚ H cŽ‚‹o"~ƒ ‚ X>YJ [ ‚G\‹ HE] HLY ™Œš› š ae}Uz ‚7‘€k v>’”“D• S H V W@XZYJ[ ¡{] H v>’^— ˜ “œ8ž “wŸ/  ¡ a¢k  ¡ S£T V W aNk  S{T V W aa}z S{T V W SˆT V W a€¤ I¥ WH S T V G@W H a}  a e¦}gz X h [ hª] ¨ § © o  ~ ƒ { S L T V = W … ¯ S V Z T W U S L T V W a}  –O«¬Œ­ §-® – H Here a simple consequence of the above results

Corollary 1 If

and

is simply connected, then

is path-connected.

Indeed, when

this is the content of Theorem 1. When , we use a) of Theorem 2 to connect to ; since is simply connected, we may write for and then we connect to via . When is a compact connected manifold, the study of the topology of was initiated in Brezis - Li [7] (see also White [26] for some related questions). In particular, these authors proved Theorems 1 and 2 in the special case . The analysis of homotopy classes for an arbitrary manifold and was subsequently tackled by Hang - Lin [15]. The passage to introduces two additional difficulties: a) when is not an integer, the norm is not “local”; b) when (or more generally ), gluing two maps in does not yield a map in . In our proofs, we exploit in an essential way the fact that the target manifold is . (The case of a general target is widely open.) In particular, we use the existence of a lifting of unimodular maps when and (see Bourgain - Brezis - Mironescu [4]). Another important tool is the following Composition Theorem (Brezis - Mironescu [10]) If , then is continuous from into

.

has bounded derivatives and

Remark 1 A very elegant and straightforward proof of this Composition Theorem has been given by V. Maz’ya and T. Shaposhnikova [18]. A related question is the description, when lifting. Here is a partial result

a}  Šc °±}² a ef}C°

Theorem 3 We have a) if , and

b) if

, then

a e}£z

, of the homotopy classes of

z³C´ a ef`° [ S T V W XjYJ[ ¯¦S V TjW XjYJ[ [ nŽµ T V W kUm^n v ’¶“ – o h·] H h\] x ´ nuµ TLV W k‰mQn ’”“ [ – o S T V W XZYJ[ h·] v xwt

S£T V W=XjYJ[ GIHE]

, then

aNk  7c ek¸z ´ nuµ T V W

_„`dh a¹ `  Šc °º}d² z,³da e"`d° ´ µ T V WB»½ ¼ m ’¶“ [ – o S TLV W XjYJ[ h·]  v xDt Y n ÁX À ]lk v HÂQ– à Äwà oÅNoSU´ T  V µW=°¾T VX W ¿ kR[²Šh·c ] kd¿ H cŠ_-nM`gk a`  c  Æ `Oef`bdcuz¹*³C™ W Tja W ef³Ç`Ʋ ` ¹*™ TZW jT W H v>’”“ _-`Ca`  c  `Oe…`gbdcŽ°±}Dz zB³a ef`Ç° ´ nuµ T V W kdn m [ – o S T V W XjYJ[ h·] v ’”“ xuÈÉ/Ê Ët ´ nuµ T V WB» n m [ – o S T V W XjYJ[ h·] v ’”“ xuÈÉ/Ê Ët S T V W XZYJ[ G H ]

Theorem 3 is due to Rubinstein - Sternberg [21] in the special case where solid torus in . When and , there is no such simple description of using the “non-lifting” results in Bourgain - Brezis - Mironescu [4], it is easy to see that

Here is an example: if a) b) there is no

in terms of

;

and

Y

is the

. For instance,

, then

such that

for

satisfying

.

However, we conjecture the following result

Conjecture 1 Assume that

and

. Then

We will prove below (see Corollary 2) that “half” of Conjecture 1 holds, namely

In a different but related direction, we establish some partial results concerning the density of into . 122

~-ƒ X YJ [ \G H^]

On some questions of topology for



-valued fractional Sobolev spaces

a ef³` a e… `gzc °Ì~ƒ }gX z YJ [ GI_-HE`C] ~aƒ `gX YJbd [ c G\ HE]S{`OefT V W|`XjYJb [ GIHE] S{T V W@XZYJ[ G\HQ] a a e…} }C° a e…}~ƒ z X YJ [ GI~HE] ƒ X YJ [ G\HE] S{T V W@XZYJ[ G\SUH^]TLV W@XjYJ[ G\HE] 

Theorem 4 We have, for : a) if , then is dense in ; b) if , then is not dense in c) if , then is dense in ; d) if and , then is dense in

;

.

There is only one missing case for which we make the following

SUT V W@XZYJ[ G\H^]

Conjecture 2 If .

_Í`(a‰`  c  `¾eÎ`ÏbdcŽ° }(²ŠcŽz£³(a eÎ`(° Y h¹

, then

~ ƒ 7X Y€ [ G H ]

is dense in

This problem is open even when is a ball in . We will prove below the equivalence of Conjectures 1 and 2. Parts of Theorem 4 were already known. Part a) is due to Escobedo [14]; so is part b), but in this case the idea goes back to Schoen - Uhlenbeck [24] (see also Bourgain - Brezis - Mironescu [5]). For , part c) is due to Schoen - Uhlenbeck [24]; their argument can be adapted to the general case (see, e.g., Brezis - Nirenberg [12] or Brezis - Li [7]). The only new result is part d). The proof relies heavily on the Composition Theorem and Theorems 2 and 3. We do not know any direct proof of d). We also mention and , Theorem 4 was established by Bethuel - Zheng [3]. For a general compact that for connected manifold and for , the question of density of into was settled by Bethuel [1] and Hang - Lin [15].

a„k 

Y 4k ¿ H ¡ Ja k 

a…k 

Y

~ ƒ X YJ [ {¡ ] S H V W XjYJ[ ¡{]

Remark 2 In Theorems 2 and 4, one may replace by a manifold with or without boundary. The statements are unchanged. However, the argument in the proof of Theorem 1 does not quite go through to the case of a manifold without boundary. Nevertheless, we make the following

Y

S T V W XjYJ[ ¡{]

Y Ï Š Ð Ó Ñ Ò } z _K`¸aÔ`4bgc  `‰ed`4b aeR`¨z S T VW X G H [ G H ] ЊÑÓÒ Y }Cz S£T V W@XZYJ[ G\H^] SUTLV W@XjYJ[ G\HE] _#`{aM`ˆbgc  `eÕ`ˆb

Conjecture 3 Let connected for every .

¡

be a manifold without boundary with . Then is pathwith , and for every compact connected manifold

Note that the condition

is necessary, since

Finally, we investigate the local path-connectedness of

is not path-connected when . Our main result is

a ef} 

.

. Then is locally path-connected. Consequently, Theorem 5 Let the homotophy classes coincide with the connected components and they are open and closed.

q¶q n ¬  q¶q `‰Ö _#`UaN`ˆbgS{c  TLV `CW eÕ`Ub Ö¦¤‰_ È É8Ê Ë  _B`ga`  c  `"ef`db ´ nuµ T V W » mQn [ – o S T V W XZYJ[ h·] dk n m [ – o S T V W XjYJ[ h\] v ’”“ x È É/Ê Ë v ’¶“ ˆS T V W|¯…× x È É/Ê ËŠt ƒ ´ nŽµ T V W¸» m^n ’¶“ [ – o S T V W XjYJ[ h·] v x

The heart of the matter in the proof is the following

Claim. Let . Then there is some may be connected to in . As a consequence of Theorem 5, we have Corollary 2 Let

such that, if

, then

n

. Then

Equality in Corollary 2 follows from the well-known fact that is a consequence of the fact that, clearly, we have

is an algebra. The inclusion

and of the closedness of the homotopy classes. Another consequence of Theorem 5 is Corollary 3 Conjecture 1

Ø

Conjecture 2. 123

H. Brezis and P. Mironescu

P ROOF.

By Corollary 2, we have

´ nuµ T V W » n m [ – o S T V W XjYJ[ h·] v ’”“ x È É/Ê Ët q¶q q¶q Ú=X ÜfYJ [­ ‚ SUT VÚ=W Ü Ù-¤gqÓ_ q Ú=Ü,¬ ‚ qÓq ‚ ¬`dÚ Ù È É/Ê Ë `ÇÙÛ Ú ~€ G\H^] k v>’¶“wß È É/Ê Ë ‚No m v ’¶“ [ – o S T V W XjYJ[ h·] x È É8Ê Ët SˆTLV W@XjYJ[ G\HE] à n o ´ Ú o nuµ T V W Ú o ´  µ T V W n Ú o m v ’”“ [ – o#~ ƒ X Y€ [ hl] x È É/Ê Ë n X YJ [ hl] x È-É8Ê Ë n S£T V W@XZYJ[ ¡{] ¡

n¦X Ú=k Ü ]¢ ÝÞ~ƒ ‚¦X o Y€ ´[  G\µ T H^V ]W Ú=Ü 

o

´ µ T VW  –lÜ

We prove that the reverse inclusion follows from Conjecture 1. By Proposition 1 a) below, we may take . Let . By Theorem 5, there is some such that . Let be such that in and . By Theorem 2 b), we obtain and are homotopic in . Thus for some globally defined smooth . Hence that

Ú {o ~ƒ X YJ [ G\H^] m v ’”“ [ – o S T V W XZYJ[[ h·] x È É/Ê Ë n#o m Ú v ’”“ – o#~ ƒ Y a cje a¢k  Y ÝCh ‹  `Ôef`gb á  ³aef`gz S T V W XZYJ[ [ G H] n  kUm^nfo

n ځ o

Conversely, assume that Conjecture 2 holds. Let . By Theorem 2 a), there is some such that . By Proposition 1 b), we have . Thus , so that clearly

Finally,

.

, i.e.

may be approximated by smooth maps.

In the same vein, we raise the following

locally path-connected Open Problem 1. Let be a manifold with or without boundary. Is for every and every compact manifold ? The case can be settled using the methods of Hang - Lin [15]. We will return to this question in a subsequent work; see Brezis - Mironescu [11]. The reader who is looking for more open problems may also consider the following Open Problem 2. Let and

á  e Ô³ e…`gz`"z ef`² >â eMk:z

S{T V W@XZYJ[ G\H^]

is known to be dense in Comment. a) and ; see Bethuel-Zheng [3] b) and ; see Bethuel [2] c) and ; see Rivi`ere [20].

¬  >â z

The paper is organized as follows 1. Introduction 2. Proof of Theorem 1 3. Proof of Theorems 2 and 3 4. Proof of Theorem 4 5. Proof of Theorem 5 Appendix A. An extension lemma Appendix B. Good restrictions Appendix C. Global lifting Appendix D. Filling a hole - the fractional case Appendix E. Slicing with norm control 124

S T V W XZYJ[ G xwH t]

is smooth except at a finite number of points

(Here, the number and location of singular points is left free). Is

a¢k aJk a¢k

_`ga`S{bdT V W|c XjYJ[ G\HE]

~ƒ X YJ [ \G H^]

be a smooth bounded domain. Assume (this is the range where is not dense in

á

dense in

in many cases, e.g.:

?

). Set

On some questions of topology for



-valued fractional Sobolev spaces

2. Proof of Theorem 1

a e…aef` `  

Case 1: When

SˆT V W@XZYJ[ ¡{] a"¤¸_Šc  Þ` e:`4bgcua ed`  ¡ r of¡ #n o S{T V W|XjYJ[ ¡{] Y nMã k¨ä nFr cc hli,å Y t n£ã oé¸S oOæy~T ç V èW X X´ _Šhlc i µ [ [ ¡{S{] T V WXjYJ[ é¡{Xj]ê ] c À ]Rk é n ã XÀ â X  ¬n ê ]]ëcŽ_d³ ê `  c À o Y Xé  c À a ]¦e£ì ` r   r a ef`Ç°  `ga ef`CzŠc °±}Cz Y XÀ Y À o ¸ Ù í ¤ _ î m [  X À Y À J Y [ C Y Ô ð ñ  hlñ i k±m À o…hlc îi,å ]YJ[ `4z7Ù x XÁÀ c î Y ]-ïÍkgkàÙ m odh·i-å c î ]N`©Ù x îuï(kíî x`ˆa„` =¥Þ>â eSUT W7V W|X nÇo [ SˆT V W SUT ™òH W7V W ˆ `  a Ç e { ` z Õ e C n o U S T V @ W Z X J Y [  >  â nfo G\H^] ‚„o ™2H  îuï GIH*] r o…GIH ‚Çk©ä r c nsc îuñ ï t ófk  ¬ >â e3c>ef`z ‚£o S‰÷V W=X îuï [ GIH*] Ú o SÞ÷ ž H W7V W@X _Þï [ `G\H^óô] `  c  `ôe¨`õbgcŠóöe¸`  Ú ãn¦kùú ûø Ún ïY Ü XZYÔð h å ï] r L T V W nfnã ã o S æyç è X hli n [ GIHE] n ã a ef`° r aeMk  c=°±}z S H  W7V W{k ü ×W aBk >â e eÔk zc> e…kÞz  a k >â ý H  ‹ ý H  ‹ _   > W V W S j X J Y [ nfoH W7V W@H XZ YJ[ G\H^] G H ] n o S H  W>V W=XjYJ[ G\HE] r ofG H S © n o H îY , we have the following more general result

Theorem 6 If connected. P ROOF.

and

Fix some

. For

is a compact manifold, then

is path-

, let

in in

Since only

, we have . Then clearly ).

. Let

and

connects

to the constant

and (here we use

Case 2: In this case one could adapt the tools developed in Brezis - Li [7], but we prefer a more direct approach. Let be such that the projection onto be well-defined and smooth in the region dist . Let dist . We have , where dist . Since , we have ; thus, for we have tr . Let . Fix some and define by tr

on on

We use the following extension result. (The first result of this kind is due to Hardt - Kinderlehrer - Lin [16]; it corresponds to our lemma when .) Lemma 1 Let

. Then any

has an extension

.

The proof is given in Appendix A; see Lemma A.1. It relies heavily on the lifting results in Bourgain Brezis - Mironescu [4]. Returning to the proof of Case 2, with given by Lemma 1, set in in in

Clearly, we may use

to connect

and is constant outside some compact set. As in the proof of Theorem 6, to , since once more we have .

Case 3: The idea is the same as in the previous case; however, there is an additional difficulty, since in the limiting case the trace theory is delicate - in particular, tr (unless ). Instead of trace, we work with a notion of “good restriction” developed in Appendix B; when , the space of functions in having as good restriction on the boundary coincides with the space of Lions - Magenes [17] (see Theorem 11.7, p. 72). Our aim is to prove that any can be connected to a constant . Step 1: we connect to some having a good restriction on 125

H. Brezis and P. Mironescu

Y YJ[ XÁÀ Y À oUhIi [ î m _B`ÖB`ÕÙ o c î ]\kgÖ x _-`CÖB`Ù X ] q W    q XÀ n q  o S H Â W7V W X
]  n q À ¬] ¬ òqn i a `b žH   Ä t q   kôn q   n n
q   ­ î
Y  ` "`dÖ N _ d ` , Ö  ` Ù N _    ÿ Y š
7c _³ Y ê ³ kº m À o YJ[   k XÀ c î Y q  ]¦  ¤š q  x q k   š
q   ê «­ n®  š S H  W7V W Y  š  Y  ­ n š kôn® š n  kgn n H kdn ® 
nH î n hi ïkUm À o…hli,å YJ [ H XÁÀ3[ î Y ]„`ÇÙ x o#GH r Y ‚k¨ä nr cH c îñ t ‚Mo S HÚ Â W7Vo W@X S îuï÷ ] H  W7V W X ï ‚¦[ G o H ] S‰÷V W@X îuï] Ú q  _,kd`d‚ ó` >â e _N`dó` 7â e ž ãn H k ú ûø nÚ H c c ïY XjYOð r c hli-å ï] t W7V W n:ã o S æyH ç  è W>V W X h·i,å Y ] nã H o S H W7V W@XZYÔð ïn]ã H o S æyHç  è X hi [ G\HE]    U S L T V  W Z X Y S EV WŽX ï]  Ú , _ d `  a ` c O ` # e R ` d b c

a # e } M¤ga n o ] o H q  Y q  q     Ú k nH nH nH î Y ä nÚ H c c ï S{T V W@XZYOð ï] Y k X ¬  c  ]iª™2!H X _Šc  ] ï¨k X ¬  c  ]i|™ò"H X ¬  c _ö] a:k >â e3S #c ©W7k V W ó ¥¨7â e nã H o S æÓHç  è W7V W X hli-] n H H  ³ga Y ef`CzŠcŽ°Îk  Y Sˆ[ T V WŠXjYJ[ G\HE] – o SUTLV W@XjYJ[ h\]

a f e } f n o X  nÔS{k T v7V W ’¶“ § –Õ«­ §,® a –eK}Þ° ~ƒ h Sˆhª] T V W a a eÕeNkR¤{° ° ê «­ v7’Á˜ Hë™uš› “ nMk v7’¶“  XÀ cÙÇî Y¤þ]N_ `gz>ÙD] x

Let dist we have

ªÿ kÞm À

be such that the projection . For , set 


onto

be well-defined and smooth in the set dist . By Fubini, for a.e.

and

,

(1)

By Lemma B.5, this implies that has a good restriction on , and that Rest a.e. on . Let any satisfying (1). For , let be the smooth inverse of . dist . Consider a continuous family of diffeomorphisms Let also , such that id and . Then is a homotopy in . Moreover, if , then and . By (1), has a good restriction on . Step 2: we extend to Let dist . As in Case 2, we fix some and set on on

Clearly, 1, there is some

, so that

such that

for

. We fix any . We define

. By Lemma

in in in

We claim that

. Obviously, . This is a consequence of

Lemma 2 Let that has a good restriction Rest

on

. It remains to check that

and . Let and that

tr

and Rest

. Assume . Then the map

in in

belongs to

.

Clearly, in the proof of Lemma 2 it suffices to consider the case of a flat boundary. When and , the proof of Lemma 2 is presented in Appendix B; see Lemma B.4. , we obtain that Returning to Case 3 and applying Lemma 2 with . As in the two previous cases, this means that is -homotopic to a constant. Case 4: In this case, is an interval. Recall the following result proved in Bourgain - Brezis - Mironescu [4] (Theorem 1): if is an interval and , then for each there is some such that . Recall also that, when , then functions with bounded derivatives operate on ; that is, the map is continuous from into itself (see, e.g., Peetre [19] for , Runst - Sickel [23], Corollary 2 and Remark 5 in Section 5.3.7 or Brezis - Mironescu [9] when ; this is also a consequence of the Composition Theorem). By combining these two results, we find that the homotopy connects to . The proof of Theorem 1 is complete. 126

On some questions of topology for



-valued fractional Sobolev spaces

3. Proof of Theorems 2 and 3

S T VW #n o S LT V W XZYJ[ G H ] ´ nuµ T V W ´n´ ‚ µ T V W k ´ ´ nŒ‚ µ T V W _´ `a`gbdc ´  `Ôe…`db sn c ‚No SUT V W=XjYJ[ G\HE] ´ nunuµµ TTLVV WW ´ k ‚ µ T V ‚ W µ T k V W ´ Ø nŒ‚ µ T n V W ‚D µ TLV W k  µ T V W n Ü ­ nFc‚ Ü ­ ‚ SUT V W qÓq n Ü qÓq $&% ³ô~Jc ´ q¶q ‚ Ü qÓq $&% ³Î´ ~ S{n Ü TLV ‚W=Ü ¯…­ × ƒ nŒ‚ SUT V W TLV W T VW Ú o ´ nŒ‚ µ T V W n H Ú n o¦n ´ ‚ ‚ µH T V W ‚ é' Ú n" o ´ n ‚DHµ T ‚ V W H n H nŒo ‚ nuµ Ú´cnŽ‚ µ kTH V W on ´ ‚DX Ú µ‚DT µn2V W ]Ý o ´ ´ nunŽ‚Dé@µ TLµ (cVTLW ' V W ´ ‚ µ T V W S{TLV W XjYJñª[ G[ H ] Y ñ H M a k ý { S L T V @ W X  h¹ _-`ga`dh+bd* c  `Ôef`db fo ñ cn -Xa ,efc( Œ}g]Ioz S{T V W=n"X oG\H [ G\H^] GIH) G\HE] òc n X., c/ 0] 0jn V W X., /c 0.] 1 £ S T a e¦}gz a}  X n -X , c( Œ] ],k  4 n XÀ c( Œ]65 îŒn XÀ c/ 0] a Ä c 3z 2 8î 7  œ n95M‚Çkôn H ‚ ‹ ¬ n ‹ ‚ H 4 a}  a e…}z  4  X n q ]>k = = n XÀ /c 0] 5 îun XÀ (c Œ] a Ä t î87   œ:­° ’¶“ n v>’Á˜ HL™Œš› “ –C«­ x n §Nv7’¶® “ – n S{T V W S£´ TLV W §"of~ƒ X h [ [ hª] µ T V W Ý m v ’”“ – o SUT V W@XZYJ» [ h·] x ´  À o Y ¿ÄÝ Y À ‚Ço  µ T V W é [ h·] hi a eO}R° { S T V @ W X [ { S T V @ W X Ú Ú o é I G E H ]  S o é k À o Y SUTLV W@X ¯ – Ä o [ SUT V W@X ¿ Ä [ h·] q k v>’”X“ V ¯ [ v>¿’UT Ä ¯ ¿W ‚ – Ä ¬K– W o – ¬– ¿ Ä ¿ W 3z 2RX¢] – Ä ¬– W Yo IV '¡ Z ¿ Ä ¿ W z32RX¢] a eO}R° ¯ Ä W À Y ¿ Ä À¿ W Y – À‚ À o o [ ]– \ o SUTLV W@X [ @ [ hl]  q \ G  k v>’¶“_^ Ç Ù Î ¤ _ ‚ q–`\ G bÄ a› k – Ä a q G bÄ a› [c@ Ù [ I ˜ I ˜ q– G ÄQ› k – \ q G ÄQ› t I ˜ I ˜ – 4 X Ú q ]lkR_ _…`‰a,`‰bdc  `CeOSg`{o bdS{cuT a VeOW=X }ÞG\H°f…¿cŽ° H ]}Þz Ú o Ú S{k T v>V W@’UT X G\H #¿ H [ G\HE] ¿ H œ :&I œ X ‚ q FHG ]-k _ –h·iª™2H ? ‚„o ´  µ TLV W – aef}C° a}  c  `Ôef´`gnuµ bgT V W cukU°Ìm^}n ²ŠcŽ[ zB– ³go a SUef`ÇT V W|° XjYJ[ hl] ¯,S H V TZW@XjYJ[ h·] » v>’¶“ x› ê SUTLV W ­ « n L H Œ ™  š n n ´ [ U S L T V @ W j X J Y [  ¯ S V j T | W j X J Y [

v Á ’ ˜ “ > v ” ’ “ – o L T V W Ý µ Ý m h·] H h·] x ´  Q v ” ’ “ ‚o  µ T V W aÞo } SU TLcŽV W=a eôX é }([ h·]z ¯ÕS é H V TjW@X é [ h·] hIi Ú o SUT V W|XÀ é [ G\H*] Ú S k SUT V W@X ¿ Ä [ h·] ¯KS H V TjW@X ¿ Ä [ h·] q k v>v>’¶U’“ T V – o ‚ V Ä S X ¯ [ ¯ – Ä ¬¸– Wfo H H ¿ Ä ¿ W 3z 2RX¢] – Ä ¬Õ– W ¿ o Ä SÍ¿T V W¢WI¯#V S H V TjW ‚ – –4 X Ú q ]¦k±aN_ }  c  `ÇeK`{bdcŽ° S‰o}ÞS{²ŽcŽT z¦V W@³‰X G\a eH eKf¿`‰° H [ h·] Ú ¯fo S S{H V TjT W=V W@X G\X G\YH YH ‰¿U¿ H [H h·[ G\] HE] ‚Çk v ’fT œ :dI œ

Thus the closed form K

has the property that

for any simple closed smooth curve

. By the general form of the Poincar´e lemma, there is some easily check that for some constant . Then

such that . One may connects to in .

We now turn to the proof of the remaining assertions in Theorems 2 and 3.

Case 1: Step 1: each can be connected to a smooth map This is proved in Brezis - Li [7], Proposition A.2, for and ; their arguments apply to any and any such that . The main idea originates in the paper Schoen - Uhlenbeck [23]; see also Brezis - Nirenberg [12], [13]. Step 2: we have Let . Then connects to in . (Recall that, if has bounded derivatives and , then the map is continuous from into itself.) This proves “ ”. To prove the reverse inclusion, by Proposition 1, it suffices to show that . Let . For each , let be a ball containing . We recall the following lifting result from Bourgain - Brezis - Mironescu [4] (Theorem 2): if is simply connected in and , then for each there is some such that . Thus, for each there is some such that . Note that , in , we have . Therefore, , since . It then follows that is constant a.e. on ; see Brezis - Nirenberg [12], Section I.5. By a standard continuation argument, we may thus define a (multi-valued) argument for in the following way: fix some . For any , let be a simple smooth path from to . Then, for sufficiently small, there is a unique function such that and ; here, is the -tubular neighborhood of . We then set

We actually claim that

is single-valued. This follows from

Lemma 3 Assume that that deg , then there is some

. If such that

.

is such

Here, is the unit ball in . The proof of Lemma 3 is presented in Appendix C; see Lemma C.1. Returning to the claim that is single-valued, we have that deg for each , since . By Lemma 3, a standard argument implies that is single-valued. The proof of Theorems 2 and 3 when is complete. Case 2: Step 1: we have

For “ ”, we use the Composition

Theorem mentioned in the Introduction, which implies that connects to in . For “ ” it suffices to prove that . We proceed as in Case 1, Step 2. Let . The corresponding lifting result we use is the following (see Bourgain - Brezis - Mironescu [4], Lemma 4): if and is simply connected in , then for each there is some such that . As in Case 1, for each there is some such that . Since , we find that is constant ae. on (see [4], Theorem B.1.). These two ingredients allow the construction of a multi-valued phase for . To prove that is actually single-valued, we rely on Lemma 4 Assume that that deg . 128

. If

, then there is some

is such such that

On some questions of topology for



-valued fractional Sobolev spaces

The proof of Lemma 4 is given in Appendix C; see Lemma C.2. The proof of Step 1 is complete. Step 2: assume ; then, for each , there is some such that . Consider the form K (see Bourgain - Brezis - Mironescu g5M . Then K [4], Lemmas D.1 and D.2). Let be any solution of h div K in . By the Composition Theorem, we then have , and thus . We claim that . Indeed, let be any ball in . Since and , there is some such that . It then follows that K S iMNS . Thus h jh9S I haveUT in , i.e., S is harmonic in . Since in we ,I we obtain that , fT so that the claim follows. Using Step 1 and the equality , we obtain that . , there is some such that . Step 3: for each In view of Step 2, it suffices to consider the case where . We use the 6   same homotopy as in Step 1, Case 3, in the proof of Theorem 1: , where is a continuous  family of diffeomorphisms 
such that  Pk . Clearly,  .  The conclusions of Theorems 2 and 3 when follow from Proposition 2 and Steps 1 and 3. We now complete the proof of Theorem 2 with

S{TLV W=XZYJ[ G\H^] a d } c þ ` ¨ e º ` g b Œ c  a ¨ e º } z ¸ n o ´ U S T V @ W Z X J Y [ ¯ X Z J Y [   N‚ o G\HQ] ~ƒ kôn \G H^] ,n ±‚No o S{nŽµ TLT V W™2H V W|XjY ] ¯×TjW@XZY ] – o ˆS T V W|XjYJ[ h·] ¯#SUS TH V V W=TjW@XjYJXZYJ[ [ h·] – k SUT V W@XZYJ[ Y ™ ’”“ o G\HE] af} ‚dkÌn a v eC™ ’¶}͓ o z G\H^] j X J Y [ Y v K ‚ g o  ~ ƒ \ G E H ] ¿ – kX g¿o SUTLV W@X ¿ ¬,[ – h\] ¯S H V TjW=X ¿ [ ¿ h\] n¿ q k v>’ ‚kôn ™ k ™ › q k ‚No#~ ƒ ¿-] v ’¶“ ´ v ’Á˜ “ Ú o#~‚Nƒ o SˆX YJnŽ µ T [ T V GIW=V W HEXjYJ] [ ¯ Ú o XjYJ´ nŽ[ µ TLV W nfo SUT V W=Xj‚-YJ[kgG\nHE] v ™ ’”“ no ê «­ n¦GI® HE] ~ƒ G\HQ] š H o…~ƒ š X Y€ [ G\H^] k Ç ‚ ô k  n ® š Y ­ Y š  a…}  c  `ReC`©bdcŠ° }ˆ²Žcuz"³£a eC`£° _B`ga`  c  `Oef`gbdcŠ°±}C²ŽcŽz³Ca ef`° ngo SˆT V W=XjYJ[ G\HE] X Y€ [ GIHE]  ‚ g o  ~ ƒ ´ ‚„o nŽµ TLV W l n m SˆT V W Y  n Y  _ ` K a ` Y Ù¤Ç_ À o¦h\i  n Ä j k o©m À ¥ 3Ù p ¥ X _Šc/ÙD] i [ ps"o X i À ¥ JÙ p ¥ X _Žc/ÙD] i PÝ m xDt i n Ä rc qk c c ° ¬ n n Ä ‘  t¶tÓÀ t oCh·i  ‘ž H qn s V o SUT V W _c q"‘ Ä k  c t¶tÓt cŠ° ¬  À n>â q s e#tvV `Ru aB`  k n q s n tvV q u s tvV u œ o SUo T ™2H  W7V W SU• TLV W À À À w^n q s œ tvV u s tvV u o SUT ™òH W7œ V W w^n q s tvV u s tvV u œ/xzyy Ÿ kn q s tvV u Ÿ œ/oxzyyS À T V W Ÿ À nï H n {Þn k ´ a eŽµ zR³|n"ã {{o ³±S{T ° H¬Â W7V W| X ï [ GIHE] qn s} q so } SUT V W aef~} { n ž n ~ÍÝ * n * ž H* ž H n q Q € n ã [ n ã nã oÇ~S{ƒ T ž X ïH  å W>n V W ] 7â e`Ua ¥:7nâ ã e` n @¥:>â e n Î ~ Ý * *ž H ¯~ ï * ž H { q ã {Õ}£zŠX c n Q Q GIH 2 * GIHE]-k _ Q nã Q 㠂k ä nFn ã c c Q Q Q ‚Uo SˆT ž H  W7V W=X n * ž H [ G\HE] q‚ s} kgn q s} S{T ž H  W7V W=X n * ž H ]

Case 3: In this case, all we have to prove is that, for each , there is some such that . The ideas we use in the proof are essentially due to Brezis - Li [7] (see 1.3, “Filling” a hole). We may assume that is defined in a neighborhood of ; this is done by extending by reflections across the boundary of - the extended map is still in since . We next define a good be small enough; for , we set covering of : let and

Define also By Fubini, for a.e. , we have tr tr

, by backward induction : is the union of faces of cubes in . , we have , in the following sense: since for all . However, for a.e. , we have the better property tr . For any such , we have

, but once more for a.e. such we have the better property tr

, and so on. (See Appendix E for a detailed discussion). We fix any having the above property and we drop from now on the superscript . Step 1: we connect to some smoother map Let , so that . Since and , there is a neighborhood of in and an extension of . This extension is first obtained in each cube starting from (see Brezis - Nirenberg [12], Appendix 3, for the existence of such an extension). We next glue together all these extensions to obtain belongs to since . Moreover, the explicit construction in [12] yields some . We next extend to in the following way: for each , let be a convex smooth hypersurface in . Since is -dimensional and may be extended smoothly in the interior of as an -valued map (here, we use the fact that ). Let be such an extension. Then the map outside the inside

belongs to

.

’s

. To summarize, we have found some

such that

129

H. Brezis and P. Mironescu

aC`Ìa H ` m>a ¥:>â e3c  x e H a H e H k{a e ¥: `Õe H `Þb  SUT ž H W7V W=¯¦× ƒ4Ý SUT œ V W œ ‚No S{T œ V W œ X H ] ¬  ‚ ž n " Í k ° H ¬ n H kg‚ M`C°  ‚ i i ´ S T œ V W œ X n ‘ [ G H ] _Í` a H `Ïbdc  n H `ào eS H T œ `ÏV W œ bdX n c i  [ G ` H ] a H e H ` °fn H c q sa ‚ H oe H S µ"³T œ V W q¨œ `(pò° kƒq c tÓt¶t c ° ¬  ‚¸o aH k  ´ {Ík SUa eŽT µ V W X n [ a{0H c ` c °  c ¬  `ge H n `£q sbg} kcŒa n H e q sH } k©ae ¥Þ n H o œ œ i GIH*] n a HH cjq s e • H o SUT œ V W œ cca:q…k` n H t¶q st¶t • o SU T V W _c qB„k H {0c t¶t¶t c ° ¬  n n H SUT V W ´ a eŠµl³ƒq…`:° n q s‚ o SUT V W cŽn q s‚ o SUT V W c…pFk „ _ : `  a ` c  ` Ô e ‰ ` g b c : `

a O e R ` f ° c T V W S q s q s H    qDc t¶tÓt c ° n • kgn H • n n H a-k  ´ a eŠµ ´ n [ G\H^] U S T V W X {Rk a eŽµk±° ¬  n n o a`ga H `  c  `ÔS{e HT `gV W b a H e H k:S{aT e V W ¥g }° a ef}H ° œ œ i n H ‚Nof~ƒ X YJ ´ [ GIHE] œ ¬ œ ´ S{T œ V W œ {NS{ k T V Wa eŽµ2k:° ‚Nz o#~ƒ aX Y€H e [ H G\µòHEk:] ° ¬  nH

Pick any min and let be such that (note that ). By Gagliardo - Nirenberg (see, e.g., Runst [22], Lemman 1, p.329 or Brezis - Mironescu [10], Corollary 3), we have . Thus * . We complete the construction of the smoother map in the , then is n n following way: if { defined in and we set ; if { , we extend to with the help of

Lemma 5 Let

. Then any

has an extension

such that

for

.

When , Lemma 5 is due to Brezis - Li [7], Section 1.3, “Filling” a hole; for the general case, see Lemma D.3 in Appendix D. We summarize what we have done so far: if , then there are some such that and a map such that and . By Gagliardo - Nirenberg and the Sobolev embeddings, we have in particular . Finally, and are - homotopic by Lemma 6 Let , and

. If

, then

and

are

-homotopic.

is due to Brezis - Li [7]; the proof of Lemma 6 in the general case is presented in the The case Appendix D- see Lemma D.4. Step 2: induction on . If , we have connected in the previous step to , where and . Using Case 1 (i.e., ) from this section, may (and thus in , by Gagliardo - Nirenberg and the Sobolev embeddings) to some be connected in . This case is complete. If , then . By the previous case, can be connected in (and thus in ) to some . Clearly, the general case follows by induction. The proof of Theorems 2 and 3 is complete. We end this section with two simple consequences of the above proofs; these results supplement the description of the homotopy classes.

´ nuµ T V W k ´ D‚ µ T V W Ø _‰`õX n agq F ` G ]·kbgc  `þX ‚ e¸q F G `õ] bdcua e¨}º? zŠcŠ° }z nFc‚ˆo S{T V W@XZYJ[ G\H^] ´ bd‚Dµ T cŽaV W H e ØH } ´ nuµ zT cŽV W a ‹ e k ‹ ´ ‚ } µ T zŠV W cŠ° }Ïz nFc‚¸o S T œ V W œ XZYJ[ G H ] ¯¦_ÍS `T Ÿ V W a Ÿ H Xjc YJa ‹[ G ` H ] bgc  `à´ enuHµ T cje V W‹ `Ï k œ œ œ œ Ÿ Ÿ Ÿ Ÿ X>YJ [ n H c‚ H o#~ƒ G\HE] ´ n µ T V W k ´ nŽµ T V W ´ ‚ µ T V W k ´ ‚Dµ TLV W H ´ ´ H ´ nuµ T V W k ‚Dµ TLV W Ø n H µ TLV W k ´ ‚ H µ T V W Ø ´ n H µ Q a k ´ ‚ H µ Q a Ø X n H q F ]-k X ‚ H q F ]Lc‡†ˆ? t

Corollary 4 Let

. For

deg

deg

Corollary 5 Let

, we have

for every .

. For

, we have

.

Clearly, Corollary 5 follows from Corollary 4. As for Corollary 4, let and . Then, by Theorem 2 b),

be such that

deg

Moreover, we have deg

X n H q F ]·k

deg

X ‚ H q F ]FØ

deg

X n H q FHG ]lk

by standard properties of the degree. We obtain Corollary 4 by combining (2) and (3). 130

deg

X ‚ H q FHG ]Ø

deg

deg

X n q F_G ]·k

deg

(2)

X ‚ q FHG ]ëc#†ˆ?\c

(3)

On some questions of topology for



-valued fractional Sobolev spaces

4. Proof of Theorem 4

nKo SUTLV W=XZYJ[ G\H^– ] o S{T V W=XjYJ[ h\] ¯fS H V TjW@XZYJ[ h·] ‚#oKaC~} ƒ ‚NX k:YJc [ n G\`¸H^] e£` v>’”“ –Ü,­– S{SUT V W=T XjV YJW@¯M[ S H V TjW G\HE] n

bgcŠ° ‚…o }‰´ nu²Šµ cŽT zMV W ³UaeK`Þ° X –Ü ]Ý~ƒ X YJX [ h·] ‚ v ™ ’”“wß ]

According to the discussion in the Introduction, we only have to prove part d). Let . Let . By Theorem 2 a), there is some such that . By Theorem 3 b), there is some such that . Let be such that in . By the Composition Theorem, the sequence of smooth maps converges to in . The proof of Theorem 4 is complete.

5. Proof of Theorem 5

a aefeÕ}C}‰z z S T VW S T œ VW œ a He H }  nfo SUT V W@XZYJ[ G\H^] _#`ˆaM`£bdc  Ö`g¤CeÇ_ `£bgcu_#`ˆa H `ˆaDc  `ge H `£bgc  ³ˆa H e H ³ˆa e m^‚No S T V W XjYJ[ G H ] [>qÓq ‚ ¬ n q¶q œ œ `Ö x Ý ´ nŽµ T V W t È É ÊË SUT V W|XjYJ[ G\HE]MÝ SUT œ V W œ XZYJ[ G\H^] a ef}Cz ae…`gz S{T V W

X n qF G ]

We start this section with a discussion on the stability of the degree: recall that if , then deg is well-defined and stable under convergence. However, while the condition is optimal for the existence of the degree (see Brezis - Li - Mironescu - Nirenberg [8], Remark 1), the stability of the degree of maps holds under (the weaker assumption of) convergence, where . This property and Corollary 4 suggest the following generalization of Theorem 5 Theorem 7 Let

there is some

. Then for each

such that

Note that , by Gagliardo - Nirenberg and the Sobolev embeddings, so that Theorem 5 follows from Theorem 7 when (when , there is nothing to prove, by Theorem 1).

aDc a H cÁescÁe H ¬  ] â esc  `"ef`dbgc  `Oe H `dbgcŽa eMk:zcŽa H e H k  c °Ì}z t _ÔÍ` a  `©bdc  ` a  e  }àz ° k aDc a H cÁesa cÁ e e H  `ÌaMz `{a  Ö  ¤_ m^m^‚„‚„oo SUSUTTLaV W@V W Xja YJXj[YJG\[ HEG\] HE[Q] q¶q[Q‚ q¶q ‚¬ ¬ n qÓn q qÓq a a`C`CÖ Ö  x Ý È œ É œ Ê Ë xwt È ‚Mo#¡ Ûº‚„o ´ nuµ T V W É Ê Ë ¡ Ý ´ Žn µ T a V W a

Proof of Theorem 7 Step 1: reduction to special values of . We claim that it suffices to prove Theorem 7 when

_-`Ca H `a`  ¬ X ° enÕ o `ÌSUbgT a c V W ° a }àz aDc a cÁescÁe H H

(4)

Indeed, assume Theorem 7 proved for all the values of satisfying (4). Let be such that (when or , there is nothing to prove). Let and let satisfy (4) and the additional condition . By Gagliardo - Nirenberg and the Sobolev embeddings, there is some such that

¡ k

´ nŽµ T V W nMk  nfo SUT V W@XZYJ[ G\H^]

(5)

By the special case of Theorem 7, we have . By Corollary 5, we obtain , i.e., is open. a a In conclusion, it suffices to prove Theorem 7 under assumption (4). Moreover, by Proposition 1 we may assume . Step 2: construction of a good covering. We fix a small neighborhood m of . By reflections across the boundary of , we may associate to each  an extension satisfying m

and

Y ãnfo SUY  T V WŠX [ GIHE] qÓq n ã ¬ ‚ ã q¶q › ³~ H q¶q n ¬ ‚ qÓq › È É8Ê Ë ˜Š‰ È É/Ê Ë ˜  q¶q n ã ¬ ‚ ã qÓq q¶q n ¬ ‚ qÓq › t  ³ ~ H › È É œ Ê Ë œ ˜Š‰ È É œ ÊË œ ˜

(6) (7) 131

H. Brezis and P. Mironescu

In this section,

À o¦h i

~ H cL~ ‹ c tÓt¶t ÙO¤£_ ‚

nFc‚Œc t¶t¶t ‚o S£T V W=XjYJ[ GIHE] n Ä q‚ s V o S T V W c+qBk  c itÓt¶t c ° ¬   • qÓq ‚ q s V ¬  q¶q s V › ³g~I‹ qÓq ‚ ¬  q¶q q¶q ‚ ¬  qÓq ›  ³ I ~ > ‹ ~ H › œ È É œ Ê Ë œ ˜ œ n È É œ Ê Ë œ ˜‹‰ È É œ ÊË œ ˜ ‚ ‚ iÄ n Œ [ ‹Ä n n ~H7c ~ ‹ ‹Ä ~H ‹Ä ~ ‹ Ö ~{k X _Šc/ÙD] ‹ ‚„o S{_,`RT V Wa H X ` î2~  [c G\ H^`K] e H `:bdcua H e H k  t Ö H ¤d_ œ œ qÓq ¬ q¶q  ‚  œ œ ˜ Q › `CÖ^‹ È É ÊË So UT œ V W œ X î2~ [ hl] qÓq – qÓq  › `CÖ H t È É œ ÊË œ ˜ Q ~ qÓq – qÓq $  › `C~ ¹ Ö H œ˜ Q 3Ö  ÖH Ö H Ž` 20Ù â ~ ¹ t denote constants independent of

.

We fix some small . By Lemma E.2 in Appendix E, for each (depending possibly on ) such that the covering has the properties

and

À

(the last inequality follows from (7)). While may depend on , the covering

there is some (8) (9)

has two features independent of :

the number of squares in

has a uniform upper bound

if are two squares in , there is a path of squares in having an edge in common with its neighbours, connecting

each one to .

(10) (11)

Step 3: choice of . We rely on

ÖE‹€¤C_

Lemma 7 Let and such that every map

has a lifting

–

satisfying

Then for each

there is some (12)

such that

(13)

Clearly, in Lemma 7, may be replaced by the unit disc. For the unit disc, the proof of Lemma 7 is given in Appendix C; see Lemma C.3. In particular, if (12) holds, then we have

for some

With

֋

~¹

independent of the

s. We now take

(14)

such that

(15)

provided by Lemma 7, we choose

րk

m>ÖE‹ â ~  c ÖE‹ â ~ H ~‹ xwt qs V ‚ ¶ q q Ó q q U S T V = W j X J Y [ ¬ ‚„o G\‚HE] q  ‚  œ ~ œ `֜ n Ä ÖB³gÖE‹ â ~ H ~‹ ~õo n Ä ‚ q  – ‹ Q –– o SUSUQ T T V œW=V W X œ X î2~] È É Ê Ë ‹ Q U S L T V @ W Á X × – Q o î2~] î2~ × o ] × Q Q S{TLV W@XÁ× × _-`Ca`×  a eMk:z}  × ¬S f– o XZSUSgT o V W ¥ SUT œ V W ]œ ] XÁ×|[ zJ2RX¢‚] k v>U’ T S ¬#– Q Q a ef¤  ‚ q s œV o SUT V W c – Q o SUTLV W – X ]Ik – X ] ‚ q s œV oO~ – Q n Ä H Q min

(16)

Step 4: construction of a global lifting for . Let satisfy . Since , (9) implies that the conclusion of Lemma 7 holds for and every square in . Thus, for every , has a lifting satisfying (14) and . We claim that . The statement being local, it suffices to prove that , where is the union of three edges in . Since is Lipschitz homeomorphic with an interval, by Theorem 1 in [4] there is some such that in (here we use and ). In , we have ; thus is constant a.e. in (see [4], Remark B.3), so that the claim follows. Since and , we may redefine and on null sets in order to have continuous functions. We claim that the function , if is well-defined on (and thus 132

On some questions of topology for



-valued fractional Sobolev spaces

n S{T V W ‹ B ~ > H L c ~  ‹Ä pFo"X – œk – Ÿ  – –  k 3 z

p 2 Q Q ¶q q – ¥ zHp 2 q¶q $ › k qÓq – Q Ÿ q¶q $ Q› œ `g¥ ~ ¹ Ö H c œ ˜f Q Ÿ œ ˜U Q œ z q p q 20ـk qÓq z3p‘2 qÓq $ œ ˜f › ³ q¶q – Q œ q¶q $ œ ˜f › ¥ ~ ¹ Ö H `gz ~ ¹ Ö H c p3kd_ q s – o Ú S T V W qX s n HÄ [ h·] ‚ œV ‚ – œV – SUT ‹ W7V W@X n [ – ‹ – o SUT ž SUH T W7V W=X n ‹Ä [ h·W7] V W@X n [ –‹ o Â Ä · h ] ¹ ž ¹ Ú k v7Ú ’”“ t ´ o SUT žl˜ iª™2H›Á W>V W=X n iÄ [ G\H^] aÔX a ` X¢° ¥ ¬ X ° i] ¬ e2o ]  ], âeMe kdž° ˜ iª™2H›Á ¤° iÄ h·] o  µ T ž˜ i|™òH/›j W7V W ¥ Ú  o â ´  µ T V W ¥ Ú o ´ ‚Dµ T V W q‚ s‚ o SUSUT TV W V W _„c Ú`Rq sa-‚ ` o SUc T V W `c€pseÔkŽ`‰q bgc tÓt¶ct c  ° `:¬ a eO `R°fc ´ aeŽµ·ƒ³ ‚ q¦q s `R° Ú q s ‚Œc Ú o SUS{T V W T V W@X n i [ GIHE] ‚ Ú • • qs Ú qs qÔk¸z ‚ n SUT V W °í}£² q q Ú Ÿ Ÿ ( ~ o é ‚ ‹ Q Q _Q `þaO`  éSgo Q S{T V W@X ~ [ h·î2] ~ a‚ e„k kRzk ~ ~ ~ q  k v>’ T k v>’”“ ‚ j X U S T 7 W V W U S T V W X [ 7 v ’ f T Q S ¬– o 2 ™

H  ] 2 î ~ 3z 2RXJ] ê S ¬– ¥ « u ¬ ­ – – Q Q L H Œ ™  š › š › SÇk k é Ú Q o ´ ‚ µ T V W ‹ v ’˜˜ TŠž “ Ÿ Q °¾k:z

continuous and common, then

). By (11), it suffices to prove that, if on . Clearly, on we have

are squares in having the edge for some . Thus

in

by (14). It follows that

(17)

by (15) and (16). which implies In conclusion, has a global lifting . Step 5: construction of a good extension of . Let be an extension of , an extension of , and so on; let be the final extension. Note that these extensions exist since , so that trace theory applies. We set . Since , we obtain by Theorem 3 that . By Corollary 5, we also have . We complete the proof of Theorem 7 by proving Step 6: . We rely on the following variant of Lemma 6 Lemma 8 Let that and are -homotopic.

. Assume that

. Let and are

be such -homotopic. Then

The proof of Lemma 8 is given Appendix D; see Lemma D.5. , we are going to apply Lemma 8 with . In order to prove that and are When -homotopic, it suffices to find, for each , a homotopy from to preserving the boundary condition on ; we next glue together these homotopies (this works since ). We construct using the lifting: since dim and is simply connected, by Theorem 2 in [4] there tr is some such that in . By taking traces, we find that ; thus tr . Therefore, tr is constant a.e., by Remark B.3 in [4]. We may assume that tr tr . Then is the desired homotopy . When , the above argument proves directly (i.e., without the help of Lemma 8) that . The proof of Theorem 7 is complete.

Appendix A An extension lemma

ÞS ÷ ž H W7V W@X ï [ G\H^] ‚„o S‰÷_NV W=`dX îuïóK[ ` G\H^ ] c  `KeÔ`:bgcŽóöeOÚ ` o  cŠS‰°õ÷ ž }RH z W7V W=X ï ï[ G\H^] ¬ 7â e óÔS‰¤ ÷V W ¬ 7â e " ó ³  ¬ ód³  SR>o â e S‰÷V W@X îuïóöeC[ h·` ]  c‚ ‚Nk SÞ÷ W7V W=X [ Ú k v>’”“ o SÞ÷ ž H W7V W=X ï [ G\H^] ó ¥g7â e…³ v> ’UT À – «­ o v>’ Ä ž H  ï h·] óK¤  ¬ 7â e S‰÷ V W S S ¬ ó"¤  >â e

In this appendix, we investigate, in a special case, the question whether a map in extension in . Lemma A.1 Let Then every

. Let

has an extension

Sˆ÷ V W@X îŽï [ G\H*]

admits an

be a smooth bounded domain in .

h\i

.

P ROOF. We distinguish two cases: and . Case : since may be lifted in (see Bourgain - Brezis - Mironescu [4]), i.e. there is some such that . Let be an extension of S . Then (since and is Lipchitz). Clearly, has all the required properties. Case : the argument is similar, but somewhat more involved. The proof in [4] actually yields a lifting which is better than ; more specifically, this lifting belongs to for , see Remark 2, p.41, in the above reference. On the other hand, since , we have .

Ú

š ÷V W Â êš ke _¦X `óöe ê ³ ]I ` â ¥# 

133

H. Brezis and P. Mironescu

ê go SU÷V W¯MS LH ™òH ˜ ÷QW ž H› V ÷QW ž H – o ÞS ÷ ž H‰S  W7÷ V W¯W7S V W|H X V ÷QW[ ž H ‚ Ú k v ’”“ Ú kd‚ ž H  ï GIH*] ¬ Ôe `Rz ó k  >â e fe `dz óöe#`  ef}gz

For this choice of , we obtain that has a lifting S . This S has an extension . By the Composition Theorem stated in the Introduction, the map belongs to . Clearly, we have tr . Remark A.1. The special case and was originally treated by Hardt - Kinderlehrer Lin [16] via a totally different method. Their argument extends to the case and , but does not seem to apply when .

Appendix B Good restrictions

aNk 7â e X _Šc  ]iª™2HQc Y k ã XÀ ê X _Šc  ]ëc XÁÀY ™ k XÁÀ ê X ¬  Y c _ö]Lc Y k Y ðMY ™ k X ¬  c  ] U k ž ‚ Pc ]-kô‚ ] Zc ]Io ž ‚ _`ga`  c  `Oef`gb fn o {S T V W=XjY ž ] ‚ ’ ‚Mo S{T V W@X ’] – € — q XÀ ¬ ã XÀ q W X‘˜ t¶ ] k n ] À TjW ‚ ]  À `db [ i Y S{T V W=XjY ] Ú H k ä nF‚Œã cc Y ž c S{T V W@XZY ] Ú ‹ªk©ä n_Žc ¬ ‚Œã c ™ YY ž ™ ©S T V W=X é] é W

H   X À ê  X À q W   ‘X ˜ Dz ] § ¥ ê v Tj‘QW ] ¬ § ] À ꢡE¤ £ § «¬Œ­ ™š› ‘œ i½ ƒ t H   ž H  C HÄ cžRŸZÄ ž š   • cžAD ‚Ôo S{T V W=X ’] ‚"o S{T V W=X ’] SUTLV W Y Y ™ Ú HcÚ ‹ ž  €—   q n XÀ ] ¬ ‚ ã XÁÀ ] q W ê À X‘˜ t ²w] Ú H o S T V W XZY ]¦ Ø ¥fk XÀ i ¬ ê ] TjW H§  `db ™òH ž X‘˜ t ¨ ] Ú ‹ o S TLV W XjY ]Bئ¥O`gb t ¬ zö™ TZW – ³©¥ˆ³  – ae a e t a ef}  _-`Ca`  c  `Ôef`gb a e…}  n#o S TLV W XZY ž ]

; it is known that the In this appendix, we describe a natural substitute for the trace theory when standard trace theory is not defined in this limiting case. For simplicity, we consider mainly the case of a flat boundary. However, we state Lemma B.5 (used in the proof of Theorem 1) for a general domain. We start by introducing some Notations: let ’ „’“ „’” ’• . If is a function defined on ’ , we set for  .  Lemma B.1 Let the following assertions are equivalent: a) and

b) the map c) the map

P ROOF. Recall that, if is given by

in in

. Then for

belongs to

and for any function defined on

,

;

in belongs to . in is a smooth or cube-like domain, then an equivalent (semi-) norm on

(see, e.g., Triebel [25]). . Conversely, for we have to prove the Clearly, both b) and c) imply that equivalence of (B.1), b) and c). We consider the norm given by (B.2). Taking into account the fact that belong to in and , we see that

and

The lemma follows from the obvious inequality

We now assume in addition that

Corollary B.1 Let 134

and derive the following be such that

. Then, for every

we have



On some questions of topology for

a) for each

_-³ ê  ` 

S T V W ê XjY ] _-³  ` 

‚

, there is at most one function defined on ’ such that the maps

Ú šH a k¨ä sn‚uã cc Ú ‹š a k¨ä n_Žc ¬ ‚0ã c ‚kdn -X , c ê  ]

and

belong to b) for a.e.

-valued fractional Sobolev spaces

;

, the function

‚

XÁX ê  c  ê] ¬ c ] ’« XjX ê  c  ê ] ’« ¬  c  ] Ú š H a c Ú ‹š a o S T V W XZY ]

in ’ª in ’ª in in

has the property that

(As usual, the uniqueness of is understood a.e.) The above corollary suggests the following

_-`C‚ a`  c  `Oe…`gbdcua ef}  cŠ_-³ ê  ` n  m À nfo k SUê T V W=XjY Ú ž š ] c Ú š q™ ½ ê `  i x H ‹ " ‚ k n " _ ` Ä š ‚ n q Ğ S ½ š kg‚ ã XÁX ê  c ê  ] S TLV W XjY šH k©ä nF‚Œcc ] _Žc  ] o ž XjX ê  c ê  ] S T V W XjY S ‹ š k©ä _Šc ¬ ã o ]t n Œ ‚ c Š _ c ] ž  ‚ nq ½ Ä š a qn Ä t ½ š a kd_,n `d-X , c aBê  ` ]  c  `e#`RbgcŒae"}  n"o S{T V W@XZY ž ] a eͤ  n©o SˆT V W@XZY ž ] °Ïk¸z _d³ ê  À «­ XÀ ¬ 7â z v H ] â q À ¬ >â z v H q SUT V W=XjY ] _-`a`  c aef¤   `Ôef`gbgcLaef`gz À n q Ä Ÿ ½  k¨ä ¬  cc À HH ¤` >>ââ zz S{T V W|X _Žc  ] SˆT V W a€k 7â e SUT V W=XjY ¬\]  _  `  a ` c Ô ` f e d ` g b Ž c

a f e }  n  ¬ o    SUT V W Ú k©ä nn ž cc YY ž n ™ q žÄ t ½  ™ ™ qn ™ t ½ _f‚`ˆk aM` n q Ä t c ½   `eÕ`ˆbdcua e¤  nÇo S{T V W@XZY ž ] Ä  Ú k ä n_Šc ¬ ‚0ã c YY ž Ú o S{T V W=XjY ] _k ™ Út q Ä t ½  n q Ä t ½  kg‚

.

o ‚ SUT V W|XjY ]

Definition: let . Let and let be a function a if a ; we defined on ’ . Then is the downward good restriction of to t a . Similarly, for then write Rest we may define an upward good restriction Rest as the unique function defined on ’ satisfying the two equivalent conditions t a in ’ª a) a in ’ª and in ’« b) a in ’« If is both an upward and a downward good restriction, we call it a good restriction and we write t . Rest Corollary B.2 Let have Rest

. Let

. Then, for a.e.

.

_,` ê  `  ³ 

‚Ok

, we

Remark B.1 If , then functions have traces for all . However, these traces need not be good restrictions. Here is an example: For , one may prove that the map belongs to if . However, if , its trace if if

tr

does not belong to

, so that it is not a good restriction.

Remark B.2 In the limiting case , functions in good restrictions a.e. Here is yet another simple consequence of Lemma B.1 Corollary B.3 Rest

Let

do not have traces. However, they do have

. Let

in in

. Then the map

belongs to

be such that Rest

.

n ž q ę t ½  k

The following results explain the connections between good restrictions and traces.

Lemma B.2 Let Rest . Then P ROOF.

. Let

tr

Let

continuity of the trace, we have

. Assume that there exists

.

in in

By Lemma B.1, we have

tr

, so that tr

‚fk

. By trace theory and

.

135

H. Brezis and P. Mironescu

S{T ž H  W7V W|XjY ] q _  `  a ` c ` O f e g ` d b Ž c

a f e } " n o À    ž n Ät ½  n m i kg_ x q U S T V = W X ,‚ k n Ä t ½  ¦‚ o €— ’] q n XÁÀ ] ¬ TjW ‚ ã XÁÀ ] q W À  À `gb t  i a ¥d>â eNk  #¯ H q n XÁÀ jc ê ] ¬ n XÀ jc _ö] q W ê À W VW êW  ³ N°±M,n]° $ cc†Œnfo S H XjY ] t ~   ž  Ë ü a ¥d >€â — e k q  XÀ ó#ã k:XÁÀ a q ¥dW >â e n ] À ¬ TjW ‚ ] À ³gN~ °ën]° W i SU÷V W  È ³ ÊË  W  H  q  X À ê Á X À ê  X À q W  ƒ  ¬ œ i § ¥ z v ‘ ] z ê ÷Q§ W ¥ v ‘ ] ¥ § ] À ꢡE¤ £ § «­¶™š› ‘ ½   žH H  C 3Ä cžRŸZÄ ž š   • cž V Ä ž ‹ š   • cžAD À ò¸o ’ n Ä ¹ kgn XÀ Zc , ]Io SÞ÷ V W@X _Žc  ] XÀÀ jc ê ]Lc êê ¤_ j X ê n  X § Ä ¹ ]·k¨ä ‚  ]Lc `_ S‰÷V W@X ¬  c  ] À W V ³ÎN~ °ën ° W V c °E§ bÄ ¹ ° Ä ¹ ™òH H›  H› ˜  È ³ Ê Ë q § Ä ¹ X¼½ ¥ z ê ] Ȭ ³ Ê z Ë § ˜ Ä ¹ Xr½ ¥ ê ] ¥ § Ĺ Xr½ ] q W ½ ê  ƒ  ê ³   C/»  ˜ ™òH V H›¼Ÿ » ž š¢ ˜ ™2H V H› V » ž ‹ š¢ ˜ ™2H V H/ ›rD q Xr½ ê ¬ ÷QW ž XrH ½ ê X¼½ q W   ~ ƒ   V H¼› Ÿ ¢š  ­  V H› V ‹ ¢š   V H/r› D n bÄ ¹ ¥ z ] z7ên ÷QÄ W ¹ H ¥ ] ¥ n Ĺ ]  ½  ê t  C…» ˜ » ž ˜ » ž ˜ ž –  ‹  q ¼X ½ ê ¼X ½ ê ¼X ½ q W k H  ™u‹ š § bÄ ¹ ¥ z ] ¬ zDê § ÷QbÄ W ¹ H ¥ ] ¥ § bÄ ¹ ]  ½  ê ³gN~ °ën Ĺ ° W ÷V W t  ™š ž È –  q XÀ ê XÀ q W  q XÁÀ ê XÀ q W }g~ H  ¹ n Ác ] ê ÷Q¬ W ‚ ]  ê kR~ H ¹ n Zc ê ]TjW ¬ H ‚ Á]  ê c   ž  H  ¹ q n XÀ Zc ê ] ¬ ‚ XÁÀ Á] q W ê ê TjW H  ³CN~ °*n bÄ ¹ ° W t  ž È ³ ÊË  H q n XÁÀ jc ê ] ¬ ‚ XÁÀ ] q W ê ê TjW H  ³gN~ °ën Ä ¹ ° W$ ¥ ~ q ‚ XÀ  ] q W t H  ¹ ž Ë

Lemma B.3 Let function, has a good downward restriction to P ROOF. that

Let

tr

. Then

Assume first that

. Let

. Then, considered as a which coincides with tr .

, by the trace theory. By Lemma B.1, it remains to prove

. Then (B.5) follows from the well-known Hardy inequality

Consider now the case where

. Let

for some convenient equivalent (semi-) norm on

(see, e.g., Triebel [24]). For any such that

S{TLV W

. We are going to prove that

X‘˜ t®­ ]

X‘˜ t ² ] X‘˜ tµ´ ]

. It is useful to consider the norm

X‘˜ t · ]

, the map if if

belongs to

i.e.

, by standard trace theory. Moreover, for any such

In particular,

Since

we find that

On the other hand, we clearly have

136

we have

X‘˜ t º ]

X¼˜ tÓ w_ ]

X¼˜ tÓD ] X¼˜ tÓ zD] X¼˜ tÓ w² ]

On some questions of topology for



-valued fractional Sobolev spaces

By combining (B.12), (B.13) and integrating with respect to is complete.

À

, we obtain (B.7). The proof of Lemma B.3

S¾^V W=XZY ™ ]  _ g `  a ` c " ` f e d ` d b Ž c  a f e } N¤ga q n H o SUT V W=XZY ] n o ‹ q    nH ‚…k n H ę t ½  ‚fž k n ‹ Ä t ½  Ú k ä nn0‹H cc YY ž ™ S T V W XZY ] n ¹ o SUT ž H  W>V W|XjY ™ ] ‚ Ú k ÚH¥ ڋ Ú H k¨ä nn H cc YY ž ¹ ™ Ú ‹ k¨ä nŒ_Šc‹ ¬ n c YY ž ¹ ™t q q ™ t ½ SUT kV W=XjY n ¹ q ž t ½ ™ # ‚ k n n tÄ ½  H H U S T V @ W Z X Y Ú H ÷V W=o XjY ] Ú ‹SUÄ o T V W= XjY ] Ä ó¸k  Þ S Ú Ú mŒc a ¥d>â e3c  x ‹€o ] ‹€o ] S H W7V W@XZY ] f n o _-`ÖB`Ù  qn  o S H W7V W X
]      q n q XÀ À ¬] ¬ 2qn ‘X ] q W a Ä `gb [  X¼˜ tÓ¨ ] i ž H   Ön 

nq A simple consequence of Lemma B.3 is the following

Lemma B.4 Let Assume that has a good downward restriction map

belongs to P ROOF.

and Rest

. Let

and tr

and that

. . Then the

in in

.

Let

be an extension of . Then

, where

in in

and

in in

By Lemma B.3 and the assumption Rest , we have Rest Corollary B.3, we find that . It remains to prove that . Then , by standard trace theory. Thus

Rest . Let

.

. By min

We conclude this section by stating the following precised form of Corollary B.1, b) in the case of a general boundary. We use the same notations as in the proof of Theorem 1, Case 4. Lemma B.5 em Let we have a) for a.e.

. Then

and

b) for any such ,

has a good restriction to

which coincides (a.e. on

) with

.

Appendix C Global lifting

X n q ]lkR_ _„`Þa`Þbgc  `eO–`Þo bdS cua T eÔV W }RX G °#H c …¿°õ}ÞH [ G z H ] nÔo SUnMT V W=k X G\v ’”H“ #¿ H [ G\H^] ¿ H œ :&I œ ‚  h!¿ ­ hliªGIH>™2cH ‚ XÁê c À ]\kgn X šLc À ] ‚No S æyT ç V è W X h!¿ [ GIHE] ê H H Sgo S À æyT ç V èW 4 X hPN¿ H [ h·] vQ’ 4 ê S X q zJ2 ÀX-, À S TLV W odn X q ¿ H ÄJÀ D [ no SUT V W|‚KX G\k HLv>m’UT ¿ À x [ GIH*] n œ :€C ÄJD ]ªkˆ_ À oC¿ :&C S Ä kÀS c ]Bo æyç è h“œ Km X Ä Áê ]Ák Ä ¿ Ä X v>’ š ] À od¿ H H x zJh·2R]X ¬ S Ä À Ä

Ä z3 2 S ÄX À oXÁê ¿ c À H ] S ê – ­ – zJ2  G 9 H O¿  h c ë š c  ]  k S H nMk v’¶“ vQ’ S{T V W@X G\H …¿ H [ h·]

In this appendix, we investigate the existence of a global lifting in some domains with non-trival topology. 4 Lemma C.1 Let deg . Then there is some

. Let such that

be such that

.

Here, is the unit ball in . P ROOF. Let . Then , where “loc” refers only to the variable . By Theorem 2 in Bourgain - Brezis - Mironescu [4], there is some such that . We claim that is -periodic in the variable . Indeed, for a.e. , we have and deg . In particular, for any such the map has a continuous lifting . On the other hand, for a.e. we have . Thus, with , we find that for a.e. the function is continuous and -valued; therefore it is a constant. Since is -periodic, so is for a.e. . We obtain that is -periodic in the variable . Thus the map is well-defined and belongs to . Moreover, we clearly have . In the same vein, we have

137

H. Brezis and P. Mironescu

S{X T V W@X G\H[ #¿ H [ G\HE] , a } c Õ `  e U ` g b Ž c ( ° : } Ž ² u c M z Þ ³

a  e : ` ° K n o { S L T V @ W X [  ¯ S V j T  W   – o GIHÄ¿ H h·] H G\Hd-¿ H h·] nMk v>’”“ œ :ÃI œ ]lkd_ 4 4 S > W V | W X GIH [ GIHE] " ` f e d ` b Ö C ¤ _ Ö C ¤ _ N ‚ o

H  ‹ H S > W V = W X [  – o H  – G\H h·] °*‚ ¬  ° °– ° `CX – [Ö H › œÈ Å Ë*Ê Ë – ˜ œ › `Ö ‹ ¼ œ r Å œ ˜ S 7 W V @ W X [ S 7 W V @ W Ú o – H G\H^] S4È o Ë*Ê Ë H h\] Ú Ú  o S H  W>V WŽX [ 4 G\HE] Ö  ¤Ç_ Úm [ ° ÚC¬OÚ  ° Ÿ œ › `Ö  x œ r Å ˜ U Æ S H W7V W È Ë*Ê Ë ÚR«­ S S – – 4 „ e : k z ´k ¬ z32lc zJ20µ S 7 W V @ W X [ S 7 W V @ W X [ j X ê X v>’ š ] Ú Ú ‚No HÂ Ö G\¤H _ G\HE] o

H  I G ^ H ] „ ] à k ‚ ±Ö Ǥg_ °*‚ ¬  ° `õ±Ö Ç ¹ Ú rœ Å ˜ œ › S XÁê °Xjê ES° : ` Ö S z 3 2 Ö È › ¹ S H  W7V WŠX ´ _ŠcLJz 20µ [ Jz 2RXJ] ¹ È Ë*Ê Ë È ]JÉ k S ¬ $ 3z È 2s] œ¼Å ¬ Ë*Ê Ë S ˜ŠÆ $ Xjê ] 4 °EÈ&°  ³ °ÊS6– ° X œ š ]„`gkj~€Ö S ¹ Xjê ] ȐkR_ S Jz 2 S H  W>V W ~€Ö ¹ `gJz 2 œ Ö ¹`Ö H n¦k v>’”“ v’ °– ° È œ¼Å ËEÊ Ë ˜ œ › Xn q

4

Lemma C.2 Let deg . Then there is some

. Let

be such that such that .

The proof is similar to that of Lemma C.1; one has to use Lemma 4 in [4] instead of Theorem 2 in [4].

Lemma C.3 Let satisfying

and . Then there is some has a global lifting

such that every such that

.

P ROOF. Recall that if is an interval, then every has a lifting (see Bourgain - Brezis - Mironescu [4], Theorem 1). Moreover, this lifting may be chosen to be (locally) continuous with respect to , i.e. for every there is some such that in the set

there is a lifting continuous for the norm. (This assertion can be established using the same argument as in Step 7 of the proof of Theorem 4 in Brezis, Nirenberg [12]; it can also be derived from the explicit construction of in the proof of Theorem 1 in [4]; see also Boutet de Monvel, Berthier, Georgescu, Purice [6] when ). Let . To each we associate the map , . By the above considerations, for every there is some such that, if , then has a lifting such that . We claim that is -periodic if is small enough. Indeed, the function belongs to , so that is constant a.e. (see [4], Theorem B.1). Since , we have (i.e. is -periodic) if . Thus, for small enough, the map is well-defined, belongs to and satisfies and .

Appendix D Filling a hole - the fractional case

We adapt to fractional Sobolev spaces the technique of Brezis - Li [7], Section 1.3. The first two results are preparations for the proofs of Lemmas 5,6 and 8 (see Lemmas D.3, D.4 and D.5 below).

ãKn o US T V W=X S{~] T _ÕV W@`þX aOn ã` XÀ  ]=c k‰SU nT`{V W=XÀ e‰X â q `ôÀFq ] bgc  q€`þq a eÞ`4× ° ƒ ~±k h·i X ¬  c  ]i nÞo US n T V W|­« X î2n ã~] î0~] ã $ ~] $  ° n`° SUTLV W ˜Ë fQ › ³£~  °*nË° Ë ˜ Q › W ³~ X °ën]° W  °ën]° W$  ] ‘Ì ] X ° `n ã ° H tÓ È É8Ê Ë ˜fQ › È É/Ê Ë ˜ Q › ¥ Ë ˜ Q › t   q n ã XÁÀ ] ¬ n ã X ] q W À  H  H   q n XÁÀ ] ¬ n X‘ ] q W X‘Ì t zw] | i ò ™ H ª i ò ™ H q À ¬ òq i TZW   k q À ò q 7 ó a b a W 7 ó j T W ¬ Ä t   7 ó iž       ž Q Q Q Q –   X‘Ì t ²ö] k H H q 7 7 À iª¬ ™2óH ó òiªq i ™òH TZW  7  ó³C~‹ â q À ¬ òq i ž TZW t   ž –   X k H H /Â Í q 77ŽÀ iª™2¬ H à &7 7u ò]/q iiª™2TjH W   7 k – – X‘Ì t ¨ ] ž   H   H /Â Í TjW 7 iª™ ™2H q À ¬ Œ i| ò™òq H TZW 7f³ H ¥ ‹ c iž    

Lemma D.1 Let Then ; here, continuous from into P ROOF. Clearly, we have norms in , the inequality

We have

We claim that

Indeed,

138

and

is the

. Let norm in

and . Moreover, the map

. is

.

. Thus it suffices to prove, for the Gagliardo semi-

On some questions of topology for

–

H kO  H O  ‹

–



-valued fractional Sobolev spaces

‹ k H O ‹ƒ –   kH H ‹ 7 iª™ TZW ™òH q À ¬ u iª ò™2q i H TjW   7 X‘Ì t ­ ]    H  ‹ TjW iª™2ž H TjW ³~ ¹ 7 iª™ ™2H q À ¬ òq TjW 7…³~dÇ â q À ¬ òq i ž t iž     –   ‹ k H ‹ ƒ 7 iª™ TjW ™òH q À ¬ Œ iª ò™2q i H TjW   7 X‘Ì t ² ]   H  ƒ TZW Œiª™2H ž  H  ƒ Z T W Z T W ³g~ÄÎ ‹ 7 iª™ ™òH i TjW   7Mk:Ä~ Î ‹ 7 iª™ ™òH ™ ™òH   7f³g~dÏ t ž   q‚  X k ,Ú q  o SUT _RV W@XÁ`±êX î0,~aq ] `  c  q  `4eÍê `±´ bdc  `Ìa e£éÍ`±o~€°  X ´ _Žc  µ [L‚ŒSUc Ú T V W@o X ~ S [ G\T V H^W ]X] ~ [ G H ] é X _Žc , ]\k ‚ucué Q  c ]lk Q Ú é q  c ] Q kg‚ Q Ðc † o _Šc  µ Ú k nã f n : k ‚ ê Q _-³ `  Xé Áê c À ]·k ä ‚ ã XXÀÀ â X  ¬ ê ] ]Lc q À¬ q ³ ê  ¬ q Àê q [ n T V W@]LXc [  ` ³  Xjê , ­ ã ê ­ ´ Xé  c , ]k n ã X L [ U S é{o#~J _Šc  ] ~ G\H^]] é c] n  q  À q  X À § XÀ ]lk ä ‚n ã XÀ ]L]ëcc q Àq ³¤  T ç V è W X hli,] фo S æÓT ç V è W X h·i,] Ñ-kg_ S ã ã ¬ Ó æ Ñ R k § n Œ § c " n o ~ ÑNo SUT V W@X hi,] W ”k °(Ñ .X , X ¬ ê ]±] ° W ³ °^é Xjê c , ] ¬ ]n ã ° W › â  TjW W › t ­ _ °(Ñ .X , â X  ¬ ê ]]±È° É8Ê Ë f˜ Q t k X  ¬ ê ] iª™ È °(ÑRÉ/°Ê Ë f˜ Q ê­  È É/Ê Ë µ˜ Ò › È É/Ê Ë µ˜ Ò › SˆT œ V W œ X n * ] _-`ã a H `  c  `Ôe H `gbdc  `ga H e H `° # n o n q  ~ o n* 㠞 H c n n Q ‹ ~ n n X * i ž ý * n2] ´ R _ ¾ ` a ` c ¨ ` e Ì ` g b c à ` a e Î ` # ° c SUT œ V W œ X n ‘ [ G\H*] H  n H o SUHT œ V W œ X n i [ G\HE] H H n H q sa ‚ H oe H SUµÓ³ T œ V Wq:œ `¾° pòƒk q c tÓt¶t c ° ¬  ‚‰o X ‚Š] n £ k  ý ‘ pI¾ k q ¥Þ c tÓt¶t c ° H ` a H e H Ž` p SUT V W …c pòÔk qDc tÓt¶t c ° ¬ _Õ `Îa` n q s  c k n `£H q e:s ` bgc n  `În ae‰H `4°#SUc T ´ V a W eŽµ”³ q`þ° n q s ‚ o SUT V W cn H q s ‚ o • q • q q qRk ° ¬  n ~ o S T VW i n Q nHQ

where and ~O On the one hand, we have

.

On the other hand, we have

We obtain (D.3) by combining (D.4), (D.5) and (D.6). Finally, (D.1) follows from (D.2) and (D.3). The proof of Lemma D.1 is complete. Lemma D.2 Let

and

P ROOF. for

set

Let ,

. Let

. Then, there is a homotopy .

be such that such that

. It clearly suffices to prove the lemma in the special case

. In this case, let,

if if

. Clearly,

. It remains to prove that

as

. Let

if if

and

as

. Then . Thus

, so that

. Since

outside

, we actually have

. The proof of Lemma D.2 is complete.

We introduce a useful notation: let , where . We extend, for each to as in Lemma D.1. Let be the map obtained by gluing these extensions. We next extend to in the same manner, and so on, until we obtain a map defined in ; call it . Lemma D.3 Let has an extension P ROOF.

for

. We may use repeatedly Lemma D.1, since for

We take .

Lemma D.4 Let

such that

, and

, then

and

are

-homotopic.

. Then every

.

we have

. If

P ROOF. We argue by backward induction on . If , then for each Lemma D.2 provides a -homotopy of and preserving the boundary condition. By gluing together these 139

H. Brezis and P. Mironescu

n n H S{T V W eR`4aK` }  ´ aeŽµ >  â n ‚:ýk±‘ ž ýH X‘ n q X• n œ q ] ] US T V W ‚ kºý‘ ¥UX n  q ] n H S{T V W ý‘ ž H X n H q • œ ] ~Ïo ‘ ž H ‚ q kU‚ H q î2ž ~ H kˆn q • œ kUn H q H ž H H • œ ‚ q ‚ H q ‚ q q SUX T V W Xjê ‚ H ê q q «­• œ ý‘ H é ]] éSUT V W@X [ G\‚ HE] • œ ‚ H ý• ‘ œ X ‚ q ] ý‘ X ‚ q ] ‚ • ‚ œ ž i H žH • œ žH H • œ ´ q‚ s‚ o SUS _…T T V W `{V W c Ú a,q s` ‚ o c SU `CT V W eK…c p|`ˆ“k bgqDc c tÓt¶ t c `U° a ¬ eK `Þ°#c a eŽµI³Pq#‚ `‰q s ° Ú ‚uq sc Ú o SUSUT T V W|V W X n i [ G\H*] • SUT V W • ‚ Ú s ] s ] X q X q ‘ Ú Ú ‚ B ý ‘ ‚ ý é s s q q U S T V W ê X j X ê Ú ­ « • • ‚  ý ‘ é  ] ] ý ‘ X ‚ q s • ] • ý ‘ X Ú • q s • ] SUT V W ‚ Ú S{T V W

— that and are homotopies we find -homotopic (here we use — ). Suppose now that — — prove it for q , assuming that q the conclusion of the lemma holds for q ; we . By assumption, s s . It suffices therefore and are -homotopic, and so are and to prove n s s — — that and are -homotopic. For each , we have   —  —  . By Lemma D.2, and are connected by a homotopy preserving s and s are — homotopies, we — find that Q Q Q Q the Q trace on Q . Gluing together these -homotopic. s s If connects to , then Lemma D.1 used repeatedly implies that connects n s s , i.e., to . in the map to The proof of Lemma D.4 is complete.

Lemma D.5 Let . Let such that . Assume that and Then and are -homotopic. P ROOF. By Lemma D.4, and (respectively and ) are nects to in , then as in the proof of Lemma D.4, we obtain that to in . Thus and are -homotopic.

Appendix E Slicing with norm control

be -homotopic.

are

-homotopic. If conconnects

S£T V W

h\i ÙÀ k o…~  i k X Š_ c ] i k  c tÓt¶t c ° ¬   œ ‘ ê } iªœ ™ ‘ æ ‚ [Žê ~·‘Jk o ä ½ * v ’ ¥ æ ½ v ‘ * o…hc( æ o"Xc*m v ’ } x ð m v ‘ ‚ x k‰m v H c t¶tÓt v i €x Õ H X~ ‘ À ]·k À ñ ¥ ~ ‘ * H XÀ ]·k n ‘ Ä Y Rk hli ~ ‘ q € ñ q Ý:m  c t¶t c ° x kÔq ~ ‘ ; k ä œ ê ’ v ’ ¥ ‘œ ö‘ v ‘ [Žê ’ o…hc/ Š‘No!X Õ c Â ; ’ ; ~·‘Jk ð m>~ ‘ ; [ ñ ÝRm  c tÓt¶t c ° x c q ñ€q kÔq x c ~ ‘ XÁÀ ]Ik ð m>~ ‘ ; XÁÀ ] [ ñ Ý:m  c t¶tÓt c ° x c q ñJq kƒq xDt S£T V W c _`¾aK`  c  `Íe{`¾b ê v ] ¬ § XÀ ] q W ê À   q § XÁÀ œ q § q W‘ k ¥ Ž Ö  q * ê q ’ ; ‘ ž ’ TjW ’   ñq ñ€ÝRq m  c t¶t¶t c ° x Ò t Ò • kq § ~ ‘ ; XÀ ]

In this section, we prove the existence of good coverings for maps. The arguments are rather standard. Without loss of generality, we may consider maps defined in . Throughout this section, we assume , i.e. we consider a covering with cubes of size 1. We start by introducing some useful notations: for and for q , let

and For a fixed set

. (With the notations introduced in Section 3, we have such that , let also

when

).

so that

and with obvious notations

Instead of considering a fixed (semi-) norm on consider a family of equivalent norms

, it is convenient to

(see, e.g., Triebel [24]). An obvious computation yields, for the usual Gagliardo (semi-) norm on 140

,

 _`ga`  c  `Ôef`db  fn o S{T V W œ W À ³ q n q W‘ X¼Ø t¶ ] °*Ën ° t ^ Ä  › ¢ ›  ñq ñ€ÝRq m  c t¶t¶t c ° x Q È É/Ê Ë ˜UQA•× ˜ kq Ô ~ n °*Ën ° È°*Ën ° WÉ/Ê Ë ˜UQ • ˜ ÄQ›› k œ — °ë]n ° W  È É/Ê Ë U˜ Q • ˜ Ä^›› Q  Q • œ ˜ ÄQ› È É/Ê Ë ˜ Q › t À o…~iBc_n `gq aÄQ` › o  c S  æÓT ç `ÔV è W efrc qB`dk b  c tÓt¶t c ° ¬  n#o S{T V W Q •˜ ÙUÝC~Bi W q n q W‘ ‡ À "o Ù t X¼Ø t zD] °*Ën °  ³ ~ c † È É8Ê Ë ˜fQ • ˜ ÄQ›› n q Q • ˜ Ä^› a ef¤  qnSUT tvV W u Ä^› o SUTLV W _R` aÕ`  c  `ÍÀ eˆoÕ`¾~i bdcua e{n q ¤ tvu ÄQ› nˆo S T V W ~ iª™ ‹ XÁÀ ] À oˆ~ i Q œ˜ Q œ˜ q¦k  c °õk£z  X À ~  o ~ ] $ ‚ À o"~ ‹ À ~ °ë]n æ ° W ~6‚ ڌcu³ ~ cŽ~Û q n q W n q Q o H SUT V W Ö Q  Q2œ ˜ ÄQ› È – É/Ê Ë f˜ Q  › œ  H  q X‘ ¬ X‘Ý q W W X¼Ø t ²w] n q ¬] ݌q n ‹ TjW] Ý ³ ‚ k °*Ën ° ž   È É/Ê Ë Q Ÿ Q  Q0œ ˜ Ä^› Q QvSˆÜ T V W@X ‹h ] –  XÁÀ ᥠà H v H §¥ à ‹ v ‹ ¥ ó v ‹>] q W À ¬ œ  H  H q n XÁÀ ‹ ‹ 7 ] n H H H ¥ § à v ¥ § à v ¥ v q ¬ q TjW k Þ ó 7    Ÿ Q  H3ß  Ÿ H  q n X  7 ] ¬ n X ó ‹ ] q W 7 v H ó v ‹ ‹ ž ¥ q 7 v H ¬ ó ‹ q ‹ ¥ TjW v  ó  7  k  Ò Ÿ   H   H q n X‘Ý ] ¬ v n H X‘Ý ¬ v 7 ž ó ‹ ] q W q 7 v H ¬ ó v ‹ v q H‹ ¥ TjW v  ó  7  Ý k  Ò Ÿ   q n  ¼X Ý ½ ] ¬ n ¼X Ý ] q W ½ ž Ý ¥ q ½Fq ‹ TjW   “k °*Ën ° W t ³ ž È É8Ê Ë Ò Ÿ Ò Ÿ On some questions of topology for

Lemma E.1

for some

Let

-valued fractional Sobolev spaces

and

. Then

independent of .

We next define the norm

by the formula

Lemma E.2 Let . Then, for a) for a.e. ; b) there is a fat set (i.e., with positive measure)

, we have

such that

Remark E.1. Here, are restrictions, not traces. However, when by traces, by a standard argument. We obtain Corollary E.1

Let . Moreover, for a.e.

, tr

we may replace restrictions

. Let . Then, for a.e. , tr has a trace on which belongs to

, and so on. P ROOF OF L EMMA E.2. In order to avoid long computations, we treat only the case . The general case does not bring any additional difficulty. Let ; denote its lower (resp. upper, left, right) edge by (resp. ). By (E.1), we have for a.e. and, for in a fat set, const. . Similar statements hold for the other edges. It remains to control the cross - integrals in the Gagliardo norm, e.g. to prove const.

(here, we take the usual Gagliardo norm in

). We have

The proof of Lemma E.2 is complete.

Acknowledgement. The first author (H.B.) warmly thanks Yanyan Li for useful discussions. He is partially supported by a European Grant ERB FMRX CT980201, and is also a member of the Institut Universitaire de France. This work was initiated when the second author (P.M.) was visiting Rutgers University; he thanks the Mathematics Department for its invitation and hospitality. It was completed while both authors were visiting the Isaac Newton Institute in Cambridge, which they also wish to thank

141

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[5] Bourgain, J., Brezis, H. and Mironescu, P. (2000). On the structure of the space â C. R. Acad. Sci. Paris, S´er. I, 321, 119-124.

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H. Brezis ´ ANALYSE NUMERIQUE UNIVERSITE´ P. ET M. CURIE, B.C. 187 4, Pl. Jussieu PARIS CEDEX 05 [email protected] RUTGERS UNIVERSITY DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS 110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854 [email protected]

P. Mironescu ´ ´ DEPARTEMENT DE MATHEMATIQUES UNIVERSITE´ PARIS-SUD 91405 ORSAY 75252 [email protected]

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