Topology and Sobolev Spaces - Semantic Scholar
Journal of Functional Analysis 183, 321369 (2001) doi: 10.1006jfan.2000.3736, available online at http:www.idealibrary.com on
Topology and Sobolev Spaces Haim Brezis Analyse Numerique, Universite P. et M. Curie, B.C. 187, 4 pl. Jussieu, 75252 Paris Cedex 05; and Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854 E-mail: brezisccr.jussieu.fr, brezismath.rutgers.edu
and Yanyan Li Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854 E-mail: yylimath.rutgers.edu Received April 28, 2000; accepted December 19, 2000
Consider the Sobolev class W 1, p(M, N) where M and N are compact manifolds. We present some sufficient conditions which guarantee that W 1, p(M, N) is pathconnected. We also discuss cases where W 1, p(M, N) admits more than one component. There are still a number of open problems, especially concerning the values of p where a change in homotopy classes occurs. 2001 Academic Press
0. INTRODUCTION Let M and N be compact 1 connected oriented smooth Riemannian manifolds with or without boundary. Throughout the paper we assume that dim M 2 but dim N could possibly be one, for example N=S 1 is of interest. Our functional framework is the Sobolev space W 1, p(M, N) which is defined by considering N as smoothly embedded in some Euclidean space R K and then W 1, p(M, N)=[u # W 1, p(M, R K ); u(x) # N a.e.], with 1 pk+3, let [e 1 , ..., e i ] be all (k+3)-cells of the triangulation and we know from Step 1 that u 1 =Y 0 near e 1 _ } } } _ e i . Applying Proposition 3.1 (with l=k+2 and n=dim M) successively to e 1 , ..., e i , we connect u 1 to some u 2 which equals Y 0 near e 1 _ } } } _ e i . Continuing in this way (by induction), we connect u 2 to some u dim M&k&2 which equals Y 0 near T 1 _ } } } _ T l . Finally, by the technique of ``filling'' a hole, we connect u dim M&k&2 to Y 0 . This completes Step 2. We have verified that Theorem 0.3$ holds for k+1 as well. The proof of Theorem 0.3$ is complete.
6. EVIDENCE IN SUPPORT OF CONJECTURE 1: PROOF OF THEOREM 0.4. Recall the statement of Conjecture 1. Conjecture 1. Given u # W 1, p(M, N) (any 1 p3, then u # C (M, N) by the Sobolev embedding theorem and we can actually take v to be C everywhere. If p=3, then W 1, p(M, N)/VMO and we can also take v to be C everywhere (see the Appendix). On the other hand, if p2, W /C ), we may assume, after making a homotopy, that V # C (S 2, N) and v~(x, _)=V(_),
x # B 2=2 .
(6.6)
Indeed this can be achieved by the same argument as the one following formula (5.8). Step 1 is complete. Step 2. Connect u 1 to some u 2 which is W 1, p(M, N) & Lip except possibly at finite points. This step can be easily deduced by applying the following lemma successively on T 1 , ..., T l . Let B 1 denote the unit ball of R 3 centered at the origin and let 1 p0, .(Q, s) :=exp Q (s&(Q)) is a diffeomorphism from M_[0, 3=] to a neighborhood of M, where exp Q(s&(Q)) is the exponential map. By Proposition 6.1 we can connect u to some u 1 which is C except possibly at one point. Since M is connected, we easily connect u 1 to some u 2 # C (M "[P ], N) with dist(P, M)
F t(x$, x")= f 0(x$)= f 1(x$),
\0 t 1,
x$ # D$,
F 1(x$, x")= f 1(x$),
\ |x"|
0, let A = =[a # A; dist(a, A)>=]. Proposition 7.1. Let A be a smooth compact Riemannian manifold with boundary, N be a smooth Riemannian manifold with or without boundary, and let u # W 1, p(B 4 _A, N) where p 1 and B 4 is the ball in R n of radius 4 and centered at the origin. Then for all =>0, there exists a continuous path u t # C([0, 1], W 1, p(B 4 _A, N)) such that u 0 =u, u t(x, a)=u(x, a),
(x, a) # (B 4 _A)"(B 23 _A = ),
0 t 1,
(7.1)
and for some Y # W 1, p(A, N), u 1(x, a)=Y(a)
x # B 13 ,
a # A 2= .
Moreover, if for some $>0, u is Lip in B 4 _(A"A 2$ ), then u t can be taken to satisfy in addition u 1 # Lip(B 4 _(A"A $ ), N). The proof of Proposition 7.1 is a variant of the proof of Proposition 3.2. We point out one modification, since the others are more obvious. What we will need is a variant of Lemma 3.2. Let \ # C (A), 0 \ 1, \(a)=1 for a # A 2= , \(a)=0 for a # A"A = . Under the hypotheses of Lemma 3.1, set, for 0
Topology and Sobolev Spaces Haim Brezis Analyse Numerique, Universite P. et M. Curie, B.C. 187, 4 pl. Jussieu, 75252 Paris Cedex 05; and Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854 E-mail: brezisccr.jussieu.fr, brezismath.rutgers.edu
and Yanyan Li Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854 E-mail: yylimath.rutgers.edu Received April 28, 2000; accepted December 19, 2000
Consider the Sobolev class W 1, p(M, N) where M and N are compact manifolds. We present some sufficient conditions which guarantee that W 1, p(M, N) is pathconnected. We also discuss cases where W 1, p(M, N) admits more than one component. There are still a number of open problems, especially concerning the values of p where a change in homotopy classes occurs. 2001 Academic Press
0. INTRODUCTION Let M and N be compact 1 connected oriented smooth Riemannian manifolds with or without boundary. Throughout the paper we assume that dim M 2 but dim N could possibly be one, for example N=S 1 is of interest. Our functional framework is the Sobolev space W 1, p(M, N) which is defined by considering N as smoothly embedded in some Euclidean space R K and then W 1, p(M, N)=[u # W 1, p(M, R K ); u(x) # N a.e.], with 1 pk+3, let [e 1 , ..., e i ] be all (k+3)-cells of the triangulation and we know from Step 1 that u 1 =Y 0 near e 1 _ } } } _ e i . Applying Proposition 3.1 (with l=k+2 and n=dim M) successively to e 1 , ..., e i , we connect u 1 to some u 2 which equals Y 0 near e 1 _ } } } _ e i . Continuing in this way (by induction), we connect u 2 to some u dim M&k&2 which equals Y 0 near T 1 _ } } } _ T l . Finally, by the technique of ``filling'' a hole, we connect u dim M&k&2 to Y 0 . This completes Step 2. We have verified that Theorem 0.3$ holds for k+1 as well. The proof of Theorem 0.3$ is complete.
6. EVIDENCE IN SUPPORT OF CONJECTURE 1: PROOF OF THEOREM 0.4. Recall the statement of Conjecture 1. Conjecture 1. Given u # W 1, p(M, N) (any 1 p3, then u # C (M, N) by the Sobolev embedding theorem and we can actually take v to be C everywhere. If p=3, then W 1, p(M, N)/VMO and we can also take v to be C everywhere (see the Appendix). On the other hand, if p2, W /C ), we may assume, after making a homotopy, that V # C (S 2, N) and v~(x, _)=V(_),
x # B 2=2 .
(6.6)
Indeed this can be achieved by the same argument as the one following formula (5.8). Step 1 is complete. Step 2. Connect u 1 to some u 2 which is W 1, p(M, N) & Lip except possibly at finite points. This step can be easily deduced by applying the following lemma successively on T 1 , ..., T l . Let B 1 denote the unit ball of R 3 centered at the origin and let 1 p0, .(Q, s) :=exp Q (s&(Q)) is a diffeomorphism from M_[0, 3=] to a neighborhood of M, where exp Q(s&(Q)) is the exponential map. By Proposition 6.1 we can connect u to some u 1 which is C except possibly at one point. Since M is connected, we easily connect u 1 to some u 2 # C (M "[P ], N) with dist(P, M)
F t(x$, x")= f 0(x$)= f 1(x$),
\0 t 1,
x$ # D$,
F 1(x$, x")= f 1(x$),
\ |x"|
0, let A = =[a # A; dist(a, A)>=]. Proposition 7.1. Let A be a smooth compact Riemannian manifold with boundary, N be a smooth Riemannian manifold with or without boundary, and let u # W 1, p(B 4 _A, N) where p 1 and B 4 is the ball in R n of radius 4 and centered at the origin. Then for all =>0, there exists a continuous path u t # C([0, 1], W 1, p(B 4 _A, N)) such that u 0 =u, u t(x, a)=u(x, a),
(x, a) # (B 4 _A)"(B 23 _A = ),
0 t 1,
(7.1)
and for some Y # W 1, p(A, N), u 1(x, a)=Y(a)
x # B 13 ,
a # A 2= .
Moreover, if for some $>0, u is Lip in B 4 _(A"A 2$ ), then u t can be taken to satisfy in addition u 1 # Lip(B 4 _(A"A $ ), N). The proof of Proposition 7.1 is a variant of the proof of Proposition 3.2. We point out one modification, since the others are more obvious. What we will need is a variant of Lemma 3.2. Let \ # C (A), 0 \ 1, \(a)=1 for a # A 2= , \(a)=0 for a # A"A = . Under the hypotheses of Lemma 3.1, set, for 0
