Construction of W 2,n (Ω) function with gradient violating Lusin (N) condition

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1 Mathematsche Nachrchte, 13 Jauary 2015 Costructo of W 2, Ω fucto wth gradet volatg Lus N codto Tomáš Roskovec 1, 1 Departmet of Mathematcal Aalyss, Charles Uversty, Sokolovská 83, Prague 8, Czech Republc Receved XXXX, revsed XXXX, accepted XXXX Publshed ole XXXX Key words Sobolev space, Lus codto, Peao curve Subject classfcato 46E35 We costruct f C 1 [ 1, 1], such that Df W 1, 1, 1, R maps [ 1, 1] {0} 1 oto [ 1, 1]. Ths shows that a mappg whch maps a set of zero measure oto the set of postve measure ca be a gradet mappg. Copyrght le wll be provded by the publsher 1 Itroducto Let Ω R be a ope set. We say that the mappg f : Ω R satsfes Lus N codto f for every A Ω, A = 0, t holds fa = 0. We ca see that ths codto s eeded for ay atural physcal model such as the deformato of a sold body space. Otherwse we ca make ew materal from othg, so the mappg s really uatural for physcal applcatos. Aother mportat applcato s the coecto betwee the area formula ad the Lus codto. If the fucto s a Sobolev fucto ad satsfes the Lus codto, the area formula holds, for more see [8]. Our work s focused o the so called Cesar example orgally wrtte [2] for = 2, later remded ad mproved [6]. That s a mappg the Sobolev space, whch maps the le segmet oto a doma wth postve -dmeso measure, so the N codto s volated. It s well kow, that for p < ad a doma Ω R wth postve measure we ca costruct such a example W 1,p Ω, R ad for p > the Lus N codto holds W 1,p Ω, R, see [3] or the orgal artcle [7]. The lmtg case W 1, s the most mportat case ad we study the volato of the codto ths case. Ths Cesar example s a specal case of the Peao curve, the mappg such that the mage of some le segmet has o-empty teror. For other results cocerg the study of the N codto for spaces close to W 1, see [5] ad [4]. I ths artcle we mprove the prevous results. We show a example of f W 2, such that Df ca be used Cesar costructo. So the restrcto to gradet mappgs does ot guaratee the Lus N codto. For the coveece of the reader we clude all detals of the costructo ad we use fgures to llustrate the dea of the costructo. Theorem 1.1 There exsts f C 1 [ 1, 1], such that Df W 1, 1, 1, R ad Df[ 1, 1] {0} 1 [ 1, 1]. Remark 1.2 We ca mprove the Theorem 1.1 ad costruct f such that D 2 f L log p L for p < 1. We expla how to modfy our costructo ths way the ed of the proof. Ths problem was orgally motvated by recet research of Mee. Our result ca be traslated to the theory of mafolds see [9] ad [10] for detals as follows. Remark 1.3 If m s a postve teger m 2, the there exsts a C 1 submafold M of dmeso m of R m+1 such that The author was supported by the ERC CZ grat LL1203 of the Czech Mstry of Educato. Correspodg author E-mal: tomas.roskovec@mff.cu.cz Copyrght le wll be provded by the publsher

2 2 Tomáš Roskovec: W 2, Ω fucto wth gradet volatg Lus codto 1. M gves rse to a tegral varfold wth mea curvature L loc m ; fact, a curvature varfold wth secod fudametal form L loc m. 2. The cotuous Gauss map ν : M S m does ot satsfy the Lus codto wth respect to m dmesoal Hausdorff measure o both sdes. Ths shows that the assumpto [9, p. 2254, Theorem 3] caot be relaxed,.e. the codto that B C caot be replaced by B C for C = S S m {a, u: u = νa or u = νa} eve f V correspods to a C 1 submafold M. 2 Prelmares We deote by Bx, r a ope ball wth the radus r ad the ceter x. We deote by the aulus a set Ax, s, t = Bx, t \ Bx, s. We deote by, the scalar product R. We say gx = D u fx for vector u R f there exst s > 0 ad le segmet x su, x + su such that We deote x+su gyφ y x dy = x+su x su x su Dfx = D e1 fx, D e2 fx,... D e fx. gyφ y x dy for ay φ C c s, s. We use some basc propertes of Sobolev fuctos, especally we use the equalty betwee the weak dervatve ad the classcal dervatve f the classcal dervatve exsts ad we use the Sobolev embeddg theorem. For more detals see [1]. We use followg otato for Sobolev spaces. For a doma Ω R, f : Ω R measurable we defe W 2,p Ω = {f : Ω R; f p <, Df p <, D 2 f p < }, f 2,p = f p + Ω Ω =1 Ω D f p + Ω,j=1 Ω Ω D,j f p 1 p. By C we deote the geerc postve costat whose exact value may chage at each occurrece. We wrte for example Ca, b, c f C may deped o parameters a, b ad c. 3 Proof of Theorem 1.1 We splt the proof of Theorem 1.1 to fve steps. I the frst step, we prepare some dese subset of [ 1, 1], the secod step we prepare the sequece of oe dmesoal fuctos f. I the thrd step we costruct the sequece of the fuctos f + based o f ad we use them to costruct the fucto f : [ 1, 1] R. I the fourth step we verfy the cotuty of Df ad the last step we show that f W 2, 1, Frst step: The tree of cetres of dyadc sub-squares By V we deote the set of the 2 vertces of the cube [ 1, 1] ad by V k we deote the product of k copes of the set V. We use the otato w = u, v for case w V k, u V k 1 ad v V f w = u for = 1, 2,... k 1 ad w k = v. 2.1 Copyrght le wll be provded by the publsher

3 m header wll be provded by the publsher 3 z w,v1,v 1 z w,v1,v 2 z w,v2,v 1 z w,v2,v 2 z w,v1 z w,v2 z w,v1,v 4 z w,v1,v 3 z w,v2,v 3 z w,v2,v 4 z w z w,v3,v 1 z w,v4,v 1 z w,v4,v 2 z w,v3,v 2 z w,v3 z w,v4 z w,v3,v 3 z w,v3,v 4 z w,v4,v 3 z w,v4,v 4 Fg. 1 Part of structure z w, w W for = 2. We decompose [ 1, 1] to the dyadc sub-squares ad by Z we deote the set of the cetres of these squares. We beg by settg z 0 = 0 ad we fx v V. By z v we deote z v = z v. Pot z v s clearly the cetre of oe dyadc sub-square of volume 1. For ay k N we ca fx u V k ad w V k+1, w = u, w k+1. By ducto we defe z w = z u + 2 k 1 w k ad f z u Z, the clearly z w Z. Moreover, by the same argumet we ca see that ay elemet of Z ca be detfed wth some fte sequece w V m for some o egatve teger m, see Frure 1. For smplcty we deote W = V k, w l = k for w V k ad Z = {z w, w W}. k=0 3.2 Secod step: Oe dmeso C 1 bump f I ths paper we wll work wth the fucto t log ε t 1, ε < 1. We set a = e 2 We ca see that lm 2 1 ε 1 ε ad b = e 2 1 ε 1 ε 1 ε 1 ε +ε 2 1 ε 1 ε +ε = 1,, for > we choose some N, depedg oly o such that 1 ε ε 1 ε +ε > 2 1 for all ε 2 Copyrght le wll be provded by the publsher

4 4 Tomáš Roskovec: W 2, Ω fucto wth gradet volatg Lus codto We wat to prove { log + 3 } a > b > b 2 a 1 > 2 +2 a +1 for N, 0 = max,. 3.4 log2 The frst ad the secod equalty are clear by 3.2. The equalty b 2 estmate a 1 > 2 +2 a +1 follows from the logb 2 a 1 a 1 +1 > log2 + 2 for 0, whch ca be show for ad log+3 log2 logb 2 a 1 a 1 by +1 = ε ε > > ε +1 1 ε 1 ε 1 ε +ε 2 1+ > 2 1 > 4 > log We shft the dex of 3.2 order to satsfy 3.4 for all dexes 1 ε 1 ε +ε = a + 0 ad b = b We defe fucto qt ad we calculate ts dervatves for t 0, a 1 qt = logt 1 ε q t = εt 1 logt 1 ε 1 q t = εt 2 logt 1 ε 1 + ε 1εt 2 logt 1 ε We also defe the sequece d = By covexty of qt we have 2 0 qb q q b 1 b + q b b b. 3.7 qb q q b qb q q b b. So d sequece ca be estmated as d 2 0 qb q q b logb ε loga 1 ε εa 1 loga 1 ε 1 b. x y Now we ca use estmate ε max{x 1 ε,y 1 ε } xε y ε based o mea value theorem for x ε. We get 2 0 d ε loga 1 logb 1 εa 1 logb 1 1 ε loga 1 ε 1 b ε loga 1 + log b 1 1 ε log b logb 1 loga ε Copyrght le wll be provded by the publsher

5 m header wll be provded by the publsher 5 f t b 2 a 1 b t Fg. 2 The graph of f. The frst part of deomator s the sequece log b ad t goes to fty by 3.5 ad 3.2, the secod part s the sequece logb 1 loga ε ad t goes to oe by 3.5, 3.2 ad 3.3, therefore 1 log b logb 1 loga ε log b. By ths ad the prevous estmate we have d ε loga 1 1 ε log b We defe p : b, R as ε 1 ε 1 +1 ε ε +ε+0 = C2 +0ε. 3.8 p t = qt q t. 3.9 Now we defe f : R + R as 2 +0 for t [0, b2 C f = 1 + d p b t b p a b t for t [ b2, b C 2 + d p t for t [b, 0 for t [,, 3.10 where C 1, C 2 are costats chose such that f C 1 0, 1. Precsely C 1 = 2 +0 d b q b b qb, C 2 = d q q Copyrght le wll be provded by the publsher

6 6 Tomáš Roskovec: W 2, Ω fucto wth gradet volatg Lus codto We check the cotuty ad the smoothess of f. The cotuty at pots b 2 a 1 ad s clear thaks to the formula descrbg C 1 ad C 2 above. The cotuty at pot b ca be verfed by 3.11, 3.9 ad 3.7 f b = C 2 + d p b = d qb q + b q = d b qb q + b q b + q = C 1 + d p b b b p = lm f t. t b The cotuty of the frst dervatve s clear at pots sde the tervals. At the edpots of the tervals we have by 3.10 ad 3.9 f b 2 a 1 =d p b p b = 0, f b = lm t b f t = d p b = d p b p f =d p = d q q = 0. b b = lm f t, t b + Let us ote that f + f b = f b f b2 a 1. Moreover, we ca see that f are C o each terval 0, b 2 a 1, b 2 a 1, b, b,,, 1. Remark 3.1 Although the defto of f s rather complcated, the smoothess ad the fgure wll be eough to follow the dea. The precse formuls eeded the proof of the cotuty of Df ad the proof of the fteess of the orm. 3.3 Thrd step: Sequece of fuctos f, summato of f ad the mage of Df Let us fx v V, see subsecto 3.1. We defe f +,v x = v x, v f x = x, v f x The support of f +,v s clearly the ball of radus ad the fucto s lear sde the smaller ball of radus b 2 a 1. Precsely, t holds Df +,v x = 2 0 v for x < b 2 a Lemma 3.2 Let {A w } w W be a set of le segmets satsfyg 1. A w [ 1, 1] {0} 1 for w W, 2. A w A v = for w, v W, w v, w l = v l, 3. A w,u A w for w W, u V. Let f W 2, 1, 1 C 1 [ 1, 1] be such that for all w W there exsts x A w wth Dfx = z w. The Df[ 1, 1] {0} 1 [ 1, 1]. P r o o f. The mage Df[ 1, 1] {0} 1 cotas Z = {z w, w W}, but Df s also cotuous. The cotuous mage of a compact set must be a compact set ad the smallest compact set cotag Z s [ 1, 1]. We have to defe the system of subsets A w ad f from the statemet of Lemma 3.2. We use the ducto by the sze of dex = w l, we start wth = 1 ad = V 0. We defe A = [ 1, 1] {0} 1. Copyrght le wll be provded by the publsher

7 m header wll be provded by the publsher 7 A w,v1 A w,v2 A w,v2 1 A w,v2 A w C w Fg. 3 Oe step costrcto A w, case = 2. C w,v1 C w,v2 C w,v2 1 C w,v2 L w,v1 L w,v2 L w,v2 1 L w,v2 We deote C w = mddle of le segmet A w We fx some w V 1. We take the le segmet of legth 2 +2 wth the mddle C w. We splt ths le segmet to 2 sub-segmets of legth 4 deoted L w,v after 2 vectors v V. We ca see that C w s ot cotaed sde ay of these tervals, t les exactly betwee two mddle oes L w,v. Let us defe ad by prevous C w,v = mddle of le segmet L w,v of legth 4a w l +1, 3.15 A w,v = [ 1, 1] {0} 1 BC w,v, b 2 a We check that the legth of A w s bgger tha the legth 2 +2 of all L w,v together. But the legth of A w s 2b 2 1 a 1 1 by 3.16 ad we get the estmate b2 1 a 1 1 > 2+2 by 3.4. We also get A w L w by 3.4, 2b 2 a 1 < 4. See Fgure 3. We ca see that C w s the mddle of both L w ad A w, we also see that legth of A w s 2a w l, so by 3.4 property 3 from Lemma 3.2 holds. Now we costruct f such that propertes from Lemma 3.2 wll be satsfed. We recall three deftos 3.12, 3.14 ad 3.16 ad defe fx = 2 0 =1 w W, w l = f +,w x C w Ths fucto s clearly cotuous ad the fourth step we wll show that f C 1 [ 1, 1]. The property that there exsts x A w such that Dfx = z w ca be prove by takg x = C w. We remd that for w W, v V, = w l the support of ay fucto s spt f + +1,v x C w,v = BC w,v, +1 ad C w does ot le sde ay terval L w,v of legth 4+1. So the dstace betwee the C w,v =mddle of L w,v ad C w satsfes C w C w,v 2+1. We prove that C w does ot belog to ths support by the tragle equalty C w x C w C w,v C w,v x > 0, for x spt f + +1,v x C w,v. Copyrght le wll be provded by the publsher

8 8 Tomáš Roskovec: W 2, Ω fucto wth gradet volatg Lus codto Because the supports of the fuctos are cotaed the supports of the fuctos prevous step, C w does ot belog eve to ay support of fucto f + u l +1,v x C u,v, u l > w l, u W, v V. Ideed, by 3.13 ad 3.1 w l w l DfC w = 2 0 Df +,w C w C w1,w 2,...,w = 2 0 w 2 0 = z w =1 3.4 Fourth step: Verfcato of f C 1 [ 1, 1] Frst, we observe that the supports of the gradets are dsjot some way ad therefore we ca swtch the summato ad the dervatve. The we check that the gradet of f s bouded aywhere o [ 1, 1] ad t s cotuous. Our clam s, that for ay x [ 1, 1] {0} 1 at most oe fucto f +,w x C w, w W, = w l from defto of fucto f 3.17 s o lear some small eghbourhood of x. Frst of all, we check that ay two fuctos =1 f +,w x C w, f +,w x C w, w, u W, = w l = w l have dsjot supports. Ths fact s obvous, because the supports of both fuctos are the balls wth radus ad C w C w 4 from The, we cosder u, w W, v V, w = u, v, = u l. We ca see that the support of f + w l,v x C w belogs to the part of the support of f + u l,u x C u, where t holds Df + u l,u x C u = 2 0 u 2 u l. Ideed, by 3.4 x C u C u C w + C w x < b 2 a 1 for ay x BC w, +1, so x BC u, b 2 a 1 ad hece 3.13 gves Df +,u x C u = 2 0 u 2. Specally, for x [ 1, 1] we have Dfx = Df +,w x C w =1 w V = Df +,w xx C w x, where w x V such that x C w <. = For every x [ 1, 1], N at most oe w x W, w x l = ca satsfy the codto x C w x < ad f there s o such w x we set f +,w x 0. We wat to estmate Df +,w x C w for fxed f +,w x C w ad x BC w,. We chage the coordates x 1, x 2... x to a base cotag the ut vector y 1 = w w to w w deoted y 2... y. By 3.12 we estmate the gradet of f +,w x C w ad 1 parwse orthogoal ut vectors orthogoal { Df + x C w, w f x C w },w x C w max j {1,2...} y j We use the formula for the dervatve of the product ad we get x C w, w f x C w x C w, w f x C w y j y j + x C w, w f x C w. y j We ca see that x Cw,w y j = 0 for all j {2, 3... } because y j, w = 0, case of y 1 = w w we have x C w, w w = w, y 1 w = 1. Copyrght le wll be provded by the publsher

9 m header wll be provded by the publsher 9 We estmate the secod term x C w, w f x C w y j x Cw f x C w, because a drectoal dervatve of radal mappg ca be estmated by the bggest drectoal dervatve. We estmate by 3.20 ad by prevous estmates appled for every j = {1, 2... } Df +,w x C w f x C w + x Cw f x C w By 3.10, 3.8, 3.9 ad 3.6 we ca estmate the value of fucto f ad ts dervatve for b < x C w < as f x C w 2 0, x C w f x C w = x C w d p x C w 2 0 x C w 2 +0ε ε x C w 1 log x C w 1 ε 1 a 1 loga 1 2 ε 0ε b b 1 logb 1 ε 1 2 ε. ε These estmates are vald eve for 0 x C w b, because by 3.10 for 0 x C w b t holds f x C w f b see Fgure 2. We apply 3.22 to 3.21 ad we get Df +,w x C w 2 ε. By ths estmate ad 3.19 we ca see that Dfx s the lmt of the Cauchy sequece of cotuous fuctos the Baach space C[ 1, 1], R. 3.5 Ffth step: Verfcato of f W 2, 1, 1 We recall Sobolev embeddg theorem, 1, 1 s Lpschtz doma ad therefore D 2 f L 1, 1 Df L 1, 1 ad f L 1, 1. It remas to prove D 2 f L 1, 1. Although f s defed as the sum of the fuctos wth the odsjot supports, the supports of the secod dervatves of these fuctos are parwse dsjot thaks to the observato about the learty of the fuctos o some part of the supports at the begg of the fourth step. So by 3.17 ad 3.13 we express 1,1 D 2 fx = 2 0 =1 w W, w l = 1,1 D 2 f +,w x p w. We ca shft ad rotate the fucto wthout chagg the sze of the tegral. So we ca detfy all copes of fucto D 2 f +,w x C w for the same sze of dex w l regardless o the oretato, we get 2 w l tegrals wth the same value. We cosder ay v V ad we express by 3.12 =1 w W, w l = 1,1 D 2 f +,w x C w = 2 1,1 D 2 x 1 v f x We cosder the fucto x 1 f x ad splt ts support to the three parts followg The frst part s the ball B0, b 2 a 1, fucto s lear o ths doma because f x = 2 0 from The secod part s the aulus A0, b 2 a 1, b ad the thrd part s the aulus A0, b,. The tegral over the frst part s clearly 0 because the secod dervatve of a lear fucto s costat 0. We look at the secod part, by tragle =1 Copyrght le wll be provded by the publsher

10 10 Tomáš Roskovec: W 2, Ω fucto wth gradet volatg Lus codto equalty ad 3.10 we estmate 2 D 2 x 1 v f x = =1 = 2 =1 =1 A0, b2,b A0 b2,b D 2 x 1 v d p b x b 2 D 2 x 1 v d p b x + A0, b2 a,b p x b A0, b2,b D 2 d x 1 v b p x. b We estmate the frst tegral, by drect calculato D 2 d v p b x 1 x { = d v p xj b max j,k {1,2,...} x, x 1x j x k } x 3 d v p b. We apply ths estmate o the frst tegral wth 3.9, 3.8, 3.5 ad 3.2 ad hece D 2 x 1 v d p b x b d p b b 2 +0ε q b q A0,b 2 a 1,b b 2 +0ε q b 2 +0ε b 2 +0ε +0. logb 1 ε We ca see that the sum of these tegrals multpled by 2 s stll fte. Now we work wth the secod tegral from 3.24 over the secod part ad we wll see that there s a close coecto wth the tegral over the thrd part. By 3.2 we have b a 1 < 1 ad by 3.9 we get D 2 x 1 d v b p x { x j d v max j,k {1,2,...} b x q x q, b x 1x j x k q x 3 x q + x q x }. b b b We use smple equalty xj x 1 ad formula 3.6 ad we estmate D 2 x 1 d v b p x { q d v max x, q x + x q x } b b b b b d b x 1 log b x 1 ε 1. We estmate the tegral by ths ad we trasfer t to a oe-dmesoal case by the chage of varables to the sphercal system of coordates. The we use the chage of varables t = b a 1 s ad we get D 2 b d v x 1 p x A 0, b2 a,b b d x log x 1 ε 1 dx A 0, b2 a,b b b b b 2 a 1 a b a b t 1 b t log b t 1 ε 1 dt b s 1 s logs 1 ε 1 b ds s 1 logs 1 ε 1 ds Copyrght le wll be provded by the publsher

11 m header wll be provded by the publsher 11 Now we skp the summato ad we go back to the thrd tegral over aulus A0, b,. After short calculato we wll see, that the estmate goes the same way as Precsely, we estmate the secod partal dervatve by the same argumets as before ad we get D 2 d v x 1 p x { x d v max d x 1 log x 1 ε 1. x q x q, x 1 x x j x 3 q x q + x 1x x j x 2 q x } 3.27 We estmate the tegral over the last aulus D 2 d v x 1 p x d x 1 log x 1 ε 1 dx A0,b, A0,b, a t 1 t 1 logt 1 ε 1 dt b a b t 1 logt 1 ε 1 dt We get exactly the same tegral as We use 3.2 ad 3.5 to straghtforward estmate log 1 t 1 ε 1 ε for ay t b, By 3.23 ad 3.24 we have [ 1,1] D 2 fx 2 =1 A0,b a 1,b A0,b 2 a 1,b A0,b, D 2 x 1 x v d p b D 2 d x 1 b D 2 x 1 v p x. v p x b Now by estmates 3.25, 3.26, 3.28 ad 3.29 we get [ 1,1] D 2 fx 2 a 2 +0ε +0 + t 1 logt 1 ε 1 dt b a 2 ε + C 2 t 1 logt 1 ε 1 dt =1 =1 + C + C =1 1 0 a b =1 b t 1 logt 1 ε 1+1 ε 1 ε dt t 1 logt 1 1 ε dt < Remark 3.3 Let us cosder ay p, p < 1 ad ε > 0 such that p ε < 1 ε. We sketch that eve D 2 f L log p L ca be prove ths way. At frst, we redefe 3.2 as a = e 2 1 ε p 1 ε, b = a e 2 1 ε p 1 ε +ε. We follow the dea of the prevous proof ad oly the ffth step of the proof would be slghtly dfferet. We sketch the calculato oly o aulus A0, b,, where we have the estmate Copyrght le wll be provded by the publsher

12 12 Tomáš Roskovec: W 2, Ω fucto wth gradet volatg Lus codto I the last estmate 3.28 we replace the expresso D 2 by D 2 log D 2 p ad usg early the same techque we get D 2 d v x 1 p x a t 1 logt 1 ε 1+p dt. A0,b, The, sted of 3.29 we get log 1 t 1 ε p 1 ε for ay t b,. If we use ths patter also o 3.25 ad 3.26, we get the estmate aalogous to 3.30, therefore D 2 fx log D 2 fx p [ 1,1] a + C t 1 logt 1 ε 1+p+1 ε p 1 ε dt + C =1 1 0 b b t 1 logt 1 1 ε dt <. Ackowledgemets We would lke to thak Ulrch Mee, for potg our terest to ths problem. We also thak to Ja Malý, who has some useful deas about the problem. Ad we thak to the author s supervsor, Staslav Hecl, for valuable commets. Refereces [1] Adams R. A., Fourer J. J. F.: Sobolev Spaces, Pure ad Appl. Math., vol. 140 Academc Press, Oxford, [2] Cesar L.: Sulle trasformazo cotue. A. Mat. Pura Appl 21, [3] Hecl, S.; Koskela, P.: Lectures o mappgs of fte dstorto. Lecture Notes Mathematcs, 2096 Sprger, Cham, 2014 [4] Kauhae, J.; Koskela, P.; Malý J.: O fuctos wth dervatves a Loretz space. Mauscrpta Math. 100, o. 1, [5] Koskela, P.; Malý J.; Zrcher, T.: Luz s codto N ad Sobolev mappgs. Att Accad. Naz. Lce Cl. Sc. Fs. Mat. Natur. Red. Lce 9 Mat. Appl. 23 o. 4, [6] Malý, J.; Marto, O.: Lus s codto N ad mappgs of the class W 1,. J. Ree Agew. Math. 458, [7] Marcus, M., Mzel, V. J.: Trasformatos by fuctos Sobolev spaces ad lower semcotuty for parametrc varatoal problems. Bull. Amer. Math. Soc. 79, [8] Marto, O.; Zemer, W. P.: Lus s codto N ad mappgs wth oegatve Jacobas. Mchga Math. J. 39, o. 3, [9] Mee, U.: A sharp lower boud o the mea curvature tegral wth crtcal power for tegral varfolds. I abstracts from the workshop held July 2228, Orgazed by Camllo De Lells, Gerhard Huske ad Robert Jerrard, Oberwolfach Reports., Vol. 9, o [10] Mee, U.: Weakly dfferetable fuctos o varfolds. preprt, submtted Copyrght le wll be provded by the publsher