ON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS

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1 ON THE WEA LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS FENGBO HANG Absrac. We denfy all he weak sequenal lms of smooh maps n W (M N). In parcular, hs mples a necessary su cen opologcal condon for smooh maps o be weakly sequenally dense n W (M N).. Inroducon Assume M N are smooh compac Remannan manfolds whou boundary hey are embedded no R l R l respecvely. The followng spaces are of neres n he calculus of varaons: W (M N) = n u W M R l : u (x) N a.e. x M W (M N) = u W (M N) : here exss a sequence u C (M N) such ha u * u n W (M N) : For a bref hsory dealed references on he sudy of analycal opologcal ssues relaed o hese spaces, one may refer o [, 3, 7]. In parcular, follows from heorem 7. of [3] ha a necessary condon for W (M N) = W (M N) s ha M sas es he -exenson propery wh respec o N (see secon. of [3] for a de non). I was conjecured n secon 7 of [3] ha he -exenson propery s also su cen for W (M N) = W (M N). In [, 7], was shown ha W (M N) = W (M N) when (M) = 0 or (N) = 0. Noe ha f (M) = 0 or (N) = 0, hen M sas es he -exenson propery wh respec o N. In secon of [4], was proved ha he above conjecure s rue under he addonal assumpon ha N sas es he -vanshng condon. The man am of he presen arcle s o con rm he conjecure n s full generaly. More precsely, we have Theorem.. Le M n N be smooh compac Remannan manfolds whou boundary (n 3). Take a Lpschz rangulaon h : M, hen W (M N) = u W (M N) : u # (h) has a connuous exenson o M w.r.. N = u W (M N) : u may be conneced o some smooh maps : In addon, f [M N] sas es hj j j = u # (h), hen we may nd a sequence of smooh maps u C (M N) such ha u * u n W (M N), [u ] = du du a.e.. o

2 FENGBO HANG Here u # (h) s he -homoopy class de ned by Whe [] (see also secon 4 of [3]) [M N] means all homoopy classes of maps from M o N. I follows from Theorem. ha Corollary.. Le M n N be smooh compac Remannan manfolds whou boundary n 3. Then smooh maps are weakly sequenally dense n W (M N) f only f M sas es he -exenson propery wh respec o N. For p [3 n ] beng an naural number, remans a challengng open problem o nd ou wheher he weak sequenal densy of smooh maps n W p (M N) s equvalen o he condon ha M sas es he p exenson propery wh respec o N. Ths was ver ed o be rue under furher opologcal assumpons on N (see secon of [4]). However, even for W 3 S 4 S, s sll no known wheher smooh maps are weakly sequenally dense. Some very neresng recen work on hs space can be found n [5]. The paper s wren as follows. In Secon, we wll presen some echncal lemmas. In Secon 3, we wll prove he above heorem corollary. Acknowledgmens. The research of he auhor s suppored by Naonal Scence Foundaon Gran DMS Some preparaons The followng local resul, whch was proved by Pakzad Rvere n [7], plays an mporan role n our dscusson. Theorem. ([7]). Le N be a smooh compac Remannan manfold. Assume n 3, B = B n, f W (@B N) \ C (@B N), f cons, u W (B N), = f, hen here exss a sequence u W (B N) \ C B N such ha u = f, u * u n W (B N) du du a.e.. In addon, f v W (B nb N) \ C B nb N sas es = f cons, hen we may esmae jdu j dh n c (n N) jduj dh n + jdvj dh n : B B B nb For convenence, we wll use hose noaons conceps n secon, 3 4 of [3]. The followng lemma s a rough verson of Luckhaus s lemma [6]. For reader s convenence, we skech a proof of hs smpler verson usng resuls from secon 3 of [3]. Lemma.. Assume M n N are smooh compac Remannan manfolds whou boundary. Le e > 0, 0 < <, A > 0, hen here exss an " = " (e A M N) > 0 such ha for any u v W (M N) wh jduj L (M) jdvj L (M) A ju vj L (M) ", we may nd a w W (M (0 ) N) such ha, n he race sense w (x 0) = u (x), w (x ) = v (x) a.e. x M jdwj L (M(0)) c (M) p jduj + jdvj L(M) L (M) + e : Proof. Le " M > 0 be a small posve number such ha V "M (M) = x R l : d (x M) < " M s a ubular neghborhood of M. Le M : V "M (M) M be he neares pon projecon. Smlarly we have " N, V "N (N) N for N. Choose a Lpschz

3 WEA LIMITS OF SMOOTH MAPS 3 cubeulaon h : M. We may assume each cell n s a cube of un sze. For B" l M, x jj, le h (x) = M (h (x) + ). Assume " M s small enough such ha all h s are b-lpschz maps. Se m = +, usng [0 ] = [ m = m m, we may dvde each k-cube n no m k small cubes. In parcular, we ge a subdvson of, called m. I follows from secon 3 of [3] ha for a.e. B" l M, u h v h W ( m N). Applyng he esmaes n secon 3 of [3] o each un sze k-cube n m k, we ge dh l () d u h j j k B" l j k M m j m j dh k c (M) k n jduj L (M) dh l () d v h j j k B" l m j dh k c (M) k n jdvj L (M) M j k m j ju h v h j L (j B" l m j) dhl () M c ( M) jd (u v)j 3 4 L (M) ju vj 4 L (M) + ju vj L (M) c ( A M) " 4 : By he mean value nequaly, we may nd a B l " M such ha u h v h W ( m N), ju h v h j L (j m j) c ( A M) " 4 < "N when " s small enough, d d u h j j k jm k j m j + v h j j k m j dh k c (M) k n jduj L (M) + jdvj L (M) for k n. Fx a C (R R) such ha 0, j ( 3) = j ( ) = 0. Leng f = u h, g = v h, we wll de ne : jj [0 ] N 3 nducvely. Frs se (x 0) = f (x) (x ) = g (x) for x jj. For mn m, 0 on [0 ], we le (x ) = N f (x) + g (x) x 0 : For mn m, le y be he cener of, de ne on [0 ] as he homogeneous degree zero exenson of wh respec o y. Nex we hle each 3-cube, 4-cube,, n-cube n a smlar way. Calculaons show ha jdj dh n+ jj[0] c (n) nx k= n+ k j k m j L c (M) jduj (M) + jdvjl (M) + e d d u h j j k m j + v h j j k m j dh k + c ( A M) "

4 4 FENGBO HANG when " s small enough. Fnally w : M [0 ] N, de ned by w (x ) = (x), s he needed map. h Lemma.. Assume N s a smooh compac Remannan manfold, n, B = B n, u v W (B N) such ha = De ne w : B (0 ) N by u (x) x B nb w (x ) = u x x B nb >: v x x B hen w W (B (0 ) N) jdwj L (B (0)) c (n) jduj L (B ) + jdvj L (B ) : Proof. Noe ha jdu (x)j < jdw (x )j c (n) du x < < >: c (n) dv x < : Hence jdw (x )j dh n+ (x ) The lemma follows. 0<< << c (n) = c (n) 0 0 d d dr c (n) jduj L (B ) r 4 du r x r 4 dhn (x) (n ) jdu (n ) (y)j dh n (y) jdw (x )j dh n+ (x ) 0<< < x c (n) d dv 4 dhn (x) 0 B c (n) jdvj L (B ) : 3. Idenfyng weak lms of smooh maps In hs secon, we shall prove Theorem. Corollary.. Proof of Theorem.. Le h : M be a Lpschz cubeulaon. We may assume each cell n s a cube of un sze. Le " M > 0 be a small number such ha V "M (M) = x R l : d (x N) < " M s a ubular neghborhood of M. Denoe M : V "M (M) M as he neares pon projecon. For B" l M, we le h (x) = M (h (x) + ) for x jj, he polyope of. We may assume " M s small enough such ha all h are b-lpschz maps. Replacng h by h when necessary, we may assume f = u h W ( N).

5 WEA LIMITS OF SMOOTH MAPS 5 Then we may nd a g C (jj N) \ W ( N) such ha g h = gj j j = fj j j (see he proof of heorem 5.5 heorem 6. n [4]). For each cell, le y be he cener of. For x, le be he Mnkowsk norm wh respec o y, ha s = nf > 0 : y + : Sep : For every n, we may nd a sequence C ( N)\W ( N) such ha = fj n W ( N) d d (fj ) a.e. (see lemma 4.4 n [3]). For x, le f (x) = >: (x) y + g y + ( ) : I s clear ha f * fj n W ( N), df d (fj ) a.e. on, jdf j L () jd c j L () + jd (gj )j c (f g) L () f C N. In addon, f we de ne h : [0 ] N by (x) + y + + h (x ) = + >: g y + + : + Then by Lemma., we know h W ( [0 ] N), jdh j L ([0]) jd c j L () + jd (gj )j c (f g) L () + h C [0 ] N. Sep : Assume for some k n, we have a sequence f C k N \ W k N h k C k [0 ] N such ha for each k, f * fj n W ( N), h k W ( [0 ] N), (3.) jd (f j )j L () c (f g) jdh kj L ([0]) c (f g) h k (x 0) = f (x), h k (x ) = g (x) for x k. Snce for every k+ n k, f * n W (@ N), for xed j by Lemma. we may nd a n j j such ha for each k+ n k, here exss a w j 0 j N wh w j (x 0) = f (x), w j x = j fnj (x) jdw j j L (@(0 )) c (n) jd )j L (@) + dfnj L (@) + c (f g) : j j Whou loss of generaly, we may replace f by f n h k by h kn. Fx a k+ n k. For x, le < f y + ( ) (x) = : w y + : j

6 6 FENGBO HANG Then j jk j = f fj n W ( N) as for each k+ n k. By Theorem. (3.) (use h k g for he needed v n Theorem., one may refer o lemma 9. of [4]), for every k+ n k, we may nd C ( N)\ W ( N) such ha = f j j L () <, jd j L () c (f g) jd d j + jd d j dhk+ : M Afer passng o subsequence, we may assume d d (fj ) a.e. on. Fx a k+ n k, for any x, de ne ( h g k+ (x) = k y + x y + g (y + ( )) f (x) = e hk+ (x ) = e hk+ (x ) = >: (x) y + g k+ y + ( ) y + (x) >: g k+ y < h k y + x y : g y + + ( ) + ( ehk+ (x ) 0 h k+ (x ) = e hk+ (x ) : Smple calculaons show ha for any k+ n k, f * fj n W ( N), df d (fj ) a.e. on, h k+ W ( [0 ] N), jdf j L () c (f g) jdh k+j L ([0]) c (f g) h k+ (x 0) = f (x), h k+ (x ) = g (x) for x k+. Hence we nsh when we reach f C (jj N) \ W ( N) h n C (jj [0 ] N). Le v = f h. Then s clear ha v C (M N) \ W (M N), [v ] =, jv uj L (M) 0, jdv j L (M) c (u g) dv du a.e. on M. Hence, we may nd u C (M N) such ha ju uj L (M) 0, jdu j L (M) c (u g), [u ] = du du a.e. on M. In parcular, hs shows W (M N) u W (M N) : u # (h) has a connuous exenson o M w.r.. N : The oher drecon of ncluson was proved n secon 7 of [3]. To see W (M N) = u W (M N) : u may be conneced o some smooh maps we only need o use he above proved equaly proposon 5. of [3], whch shows u W (M N) : u # (h) has a connuous exenson o M w.r.. N = u W (M N) : u may be conneced o some smooh maps :

7 WEA LIMITS OF SMOOTH MAPS 7 We remark ha many consrucons above are movaed from secon 5 secon 6 of [4]. Proof of Corollary.. Ths follows from Theorem. corollary 5.4 of [3]. References [] P. Hajlasz. Approxmaon of Sobolev mappngs. Nonlnear Anal (994), no., [] F. B. Hang F. H. Ln. Topology of Sobolev mappngs. Mah Res Le (00), no. 3, [3] F. B. Hang F. H. Ln. Topology of Sobolev mappngs II. Aca Mah 9 (003), no., [4] F. B. Hang F. H. Ln. Topology of Sobolev mappngs III. Comm Pure Appl Mah 56 (003), no. 0, [5] R. Hard T. Rvere. Connecng opologcal Hopf sngulares. Annal Sc Norm Sup Psa, (003), no., [6] S. Luckhaus. Paral Holder connuy for mnma of ceran energes among maps no a Remannan manfold. Indana Unv Mah J 37 (9), [7] M. R. Pakzad T. Rvere. Weak densy of smooh maps for he Drchle energy beween manfolds. Geom Func Anal 3 (003), no., [] B. Whe. Homoopy classes n Sobolev spaces he exsence of energy mnmzng maps. Aca Mah 60 (9), no., 7. Deparmen of Mahemacs, Prnceon Unversy, Fne Hall, Washngon Road, Prnceon, NJ 0544,, School of Mahemacs, Insue for Advanced Sudy, Ensen Drve, Prnceon, NJ 0540 E-mal address: fhang@mah.prnceon.edu