y X F n (y), To see this, let y Y and apply property (ii) to find a sequence {y n } X such that y n y and lim sup F n (y n ) F (y).

Transcription

1 Modica Mortola Fuctioal 2 Γ-Covergece Let X, d) be a metric space ad cosider a sequece {F } of fuctioals F : X [, ]. We say that {F } Γ-coverges to a fuctioal F : X [, ] if the followig properties hold: i) Limif Iequality) For every x X ad every sequece {x } X such that x x, F x) lim if F x ). ) + ii) Limsup Iequality) For every x X there exists {x } X such that x x ad lim sup F x ) F x). 2) emark A importat property of Γ-covergece is that if each F is bouded from below ad admits a miimizer x, that is, F x ) mi y X F y), if x x for some x X, ad if {F } Γ-coverges to F, the x is a miimizer of F ad mi F y) F x) lim y X F x ) + lim mi F y). + y X To see this, let y Y ad apply property ii) to fid a sequece {y } X such that y y ad lim sup F y ) F y). Usig the fact that F x ) mi y X F y) ad property i) for the sequece {x }, we have that F x) lim if F x ) lim sup F x ) lim sup F y ) F y), + which shows that x is a miimizer of F. Takig y x, gives that there exists lim + F x ) F x). 3 Compactess For > 0 cosider the fuctioal F : W,2 ; d) [0, ]

2 defied by F u) : ) W u) + u 2 dx, where the double well potetial W : d [0, ) satisfies the followig hypotheses: H ) W is cotiuous, W z) 0 if ad oly if z {α, β} for some α, β d with α β. H 2 ) There exist L > 0 ad > 0 such that for all z d with z. W z) L z. Theorem 2 Let N be a ope bouded set with Lipschitz boudary. Assume that the double-well potetial W satisfies coditios H ) ad H 2 ). Let 0 + ad let {u } W,2 ; d) be such that M : sup F u ) <. 3) The there exist a subsequece {u k } of {u } ad u BV ; {α, β}) such that u k u for i L ; d). Proof. We begi by showig that {u } is bouded i L ; d) ad equiitegrable. By 3) ad H 2 ), for every t, dx W u x)) dx M, { u t} u L { u t} which implies that {u } is equi-itegrable. Moreover, sice has fiite measure, L u dx L u dx + L u dx L { u } { u } { u <} W u x)) dx + L M + L. I view of the Vitali s covergece theorem, to obtai strog covergece of a subsequece, it suffi ces to prove covergece i measure or poitwise L N a.e. i. We divide the proof i two steps. Step : Assume first that d. Defie W z) : mi {W z), }, z. Sice 0 W W, for every N, we have F u ) W u x)) u x) ) dx 2 Φ u ) x) ) dx, 2

3 where The by 3), Moreover, sice W, Φ t) : 2 t 0 W s) ds, t. 4) sup Φ u ) ) dx M. 5) Φ u x)) u x) for L N a.e. x ad for all N. Sice {u } is bouded i L ), it follows that the sequece {Φ u } is bouded i L ). By the ellich Kodrachov theorem, there exist a subsequece {u k } ad a fuctio w BV ) such that w k : Φ u k w i L loc ). By takig a further subsequece, if ecessary, without loss of geerality, we may assume that w k x) w x) ad that W u k x)) 0 for L N a.e. x. Sice the fuctio W t) > 0 for all t α, β, it follows from 4) that the fuctio Φ is strictly icreasig ad cotiuous. Thus, its iverse Φ is cotiuous ad u k x) Φ w k x)) Φ w x)) : u x) for L N a.e. x. It follows by H ) ad the fact that W u k x)) 0 for L N a.e. x, that u x) {α, β} for L N a.e. x. I tur, w x) {Φ α), Φ β)} for L N a.e. x, ad sice w BV ), we may write for a set E of fiite perimeter. Hece, w Φ α) χ E + Φ β) χ E ) 6) u αχ E + β χ E ) 7) belogs to BV ). Step 2: Assume that d 2 ad that α β. For every t 0 defie V t) : mi z t W z). The V is upper semicotiuous, V t) > 0 for t α, β, V α ) V β ) 0, ad V t) Lt for t. For every u W,2 ; d) defie G u) : ) V u ) + u 2 dx F u). The by 3), sup G u ) sup F u ; ) <, 3

4 ad so by the compactess i the scalar case d, there exist a subsequece {u k } ad w BV ) such that where ad Hece, w k : Φ 2 u k w i L loc ), Φ 2 t) : 2 t 0 V s) ds, t V z) : mi {V z), }, z. u k v : Φ 2 w i L ). By takig a further subsequece, if ecessary, without loss of geerality, we may assume that u k x) v x) ad that W u k x)) 0 for L N a.e. x. This implies that v BV ; {α, β}). Defie { α if v x) α, u x) : β if v x) β. We claim that u k u i L ; d). To see this, fix x such that u k x) v x) ad that W u k x)) 0. The, by H ), ecessarily, u k x) u x). Step 3: If d 2 ad α β, let e i be a vector of the caoical basis of d such that α e i β e i. The α + e i β + e i. It suffi ces to apply the previous step with W replaced by ad u by u + e i. Ŵ z) : W z e i ), z d, 4 Gamma Covergece of the Modica Mortola Fuctioal I view of the previous theorem, the metric covergece i the defiitio of Γ- covergece should be L ; d). Thus, we exted F to L ; d) by settig ) W u) + u 2 dx if u W F u) :,2 ; d), if u L ; d) \ W,2 ; d). Let 0 +. Uder appropriate hypotheses o W ad, we will show that the sequece of fuctioals {F } Γ-coverges to the fuctioal { cw Du ) if u BV ; {α, β}), F u) : if u L ; d) \ BV ; {α, β}). 4

5 We begi with the limif iequality. Cosider a sequece {u } L ; d) such that u u i L ; d) for some u L ; d). If lim if F u ), + the there is othig to prove, thus we assume that Let {k } be a subsequece of { } such that lim if F u ) <. 8) + lim if F u ) lim F k u k ) <. + k + The F k u k ) < for all k suffi cietly large. Hece, u k W,2 ; d) for all k suffi cietly large. Moreover, if hypotheses H ) ad H 2 ), the by Theorem 2, u BV ; {α, β}). Fially, by extractig a further subsequece, ot relabelled, we ca assume that u x) u x) for L N a.e. x. Hece, i what follow, without loss of geerality, we will assume that 8) holds, that {u } W,2 ; d), that u BV ; {α, β}), that lim if + F u ) is actually a limit, ad that {u } coverges to u i L ; d) ad poitwise L N a.e. i. 5 Limif Iequality, N, d We begi with the case N d ad assume that H ) ad H 2 ) hold with α < β ad that a, b). Cosider u BV ; {α, β}). Without loss of geerality we may assume that there exists a partitio a t 0 < t < < t m b such that u x) α i t 2i 2, t 2i ) ad u x) β i t 2i, t 2i ). Let 0 + ad let {u } W,2 ) be such that u u i L ) ad poitwise L a.e. i. Fix δ > 0 small. The W u ) + u 2 Cosider oe term ti+δ t i δ ) dx i ti+δ t i δ ) W u ) + u 2 dx. ) W u ) + u 2 dx. For simplicity we ca assume that t i 0 ad, by takig δ smaller, that u δ) α ad u δ) β. The δ ) δ/ W u x)) + u x) 2 dx W u y)) + 2 u y) 2) dy δ/ δ δ/ δ/ W v y)) + v y) 2) dy, 9) 5

6 where we have made the chage of variables x y ad v y) : u y). It follows that δ ) δ/ W u x)) + u x) 2 dx W v y)) + v y)) 2) dy δ δ/ δ/ δ/ 2 W v )v y) dy uδ) u δ) 2 W s) ds β α 2 W s) ds as, where we have made the chage of variables s w y). I tur, ) β lim if W u ) + u 2 dx 2 W s) ds umber of jumps of u). α 6 Limsup Iequality, N, d Assume i additio to H ) ad H 2 ) that W is of class C. solutio of the Cauchy problem Let g be the g W g), g 0) α + β. 0) 2 The g is globally defied, strictly icreasig, α < g t) < β for all t, ad lim g t) α t lim g t) β. t Defie c W : Note that by 0) we have c W lim 2 lim g) W g t)) + g t) 2) dt. ) W g t)) + g t) 2) dt 2 lim g ) Hece, if ad β W s) ds 2 W s) ds. u y) α { α if y < 0, β if y 0, the the right sequece would be x u x) : g ), W g t))g t) dt 6

7 sice ) W u x)) + u x) 2 dx x W g ) ) + 2 W g y)) + g y) 2) dy c W. ) x 2) g u x) dx I the case of a geeral u BV ; {α, β}), we eed to glue g to α ad β. Let u be as i the previous sectio. Fix ρ > 0 ad let b ρ be such that g b ρ ) β ρ ad let a ρ be such that g a ρ ) α + ρ. Defie ad g ρ y) : ) x t g 2i ρ u,ρ x) : g t2i+ x ρ u x) β if y b ρ +, ρ y b ρ ) + β ρ if b ρ < y < b ρ +, g y) if a ρ y b ρ, ρ y a ρ ) + α + ρ if a ρ < y < a ρ, α if y a ρ, ) if t 2i r < x < t 2i + r, if t 2i+ r < x < t 2i+ + r, otherwise, where 2r < mi {t i+ t i : i 0,..., m }. The usig the chage of variables x t2i y or t2i+ x y we get ) ti+r ) W u,ρ ) + u,ρ 2 dx W u,ρ ) + u,ρ 2 dx i t i r r/ r/a W gρ y)) + g ρ y) 2) dx i bρ+ i a ρ for all suffi cietly large. Sice W β) 0, by the mea value theorem W gρ y)) + g ρ y) 2) dx W t) W β) + W θ) t β) 0 + W θ) t β) for some θ betwee t ad β. Hece, i [b ρ, b ρ + ], W g ρ y)) β g ρ y)) max [α,β] W ρ y b ρ ) ρ)) max [α,β] W 2ρ max [α,β] W, g ρ y) ρ. It follows that bρ+ W gρ y)) + g ρ y) 2) dx Cρ, a ρ 7

8 where we have made the chage of variables s w y). O the other had, bρ W gρ y)) + g ρ y) 2) dx a ρ Hece, we get I tur, lim sup ρ 0 + bρ a ρ W g y)) + g y) 2) dx W g y)) + g y) 2) dy c W. ) W u,ρ ) + u,ρ 2 dx c W m + Cρm. ) lim sup W u,ρ ) + u,ρ 2 dx c W umber of jumps of u). Moreover, usig the chage of variables x t2i ti+r u,ρ u dx as, where i t i r u,ρ x) u x) dx y or t2i+ x y, r/ i bρ+ g ρ y) v y) dy C ρ 0 i a ρ v y) : { β if y > 0, α if y 0. r/a g ρ y) v y) dy 2) We ow diagoalize the sequece {u,ρ } to obtaiig a sequece {u,ρ } covergig to u i L ) ad such that ) lim W u,ρ ) + u,ρ 2 dx c W umber of jumps of u). 7 Limif Iequality, N, d I this case the costat c W should be replaced by { c W : if W g t)) + g t) 2) dt : 3) g Hloc ; d ) } such that lim g t) α lim g t) β. t t We proceed as i the case d up to 9). Fix ρ > 0 small. Sice u δ) α ad u δ) β, we have that v δ/ ) α u δ) α < ρ, v δ/ ) β u δ) β < ρ 8

9 for all large. We ow exted v to by settig β if y δ/ +, β v δ/ )) y δ/ ) + v δ/ ) if δ/ < y < δ/ +, w y) : v y) if δ/ y δ/, v δ/ ) α) y + δ/ ) + v δ/ ) if δ/ < y < δ/, α if y δ/, Sice W β) 0 by the mea value theorem W z) W β) + W θ) z β) 0 + W θ) z β) for some θ i the segmet betwee z ad β. Hece, i [δ/, δ/ + ], W w y)) v y) β max W Cρ, Bβ,2) w y) ρ. A similar estimate ca be made i the iterval [ δ/, δ/ ]. It follows that δ/ W v y)) + v y) 2) dy W w ) + w 2) dy Cρ, δ/ I tur, lim if c W Cρ. ) W u ) + u 2 dx c W umber of jumps of u) Cρ. Lettig ρ 0 gives ) lim if W u ) + u 2 dx c W umber of jumps of u). 8 Limsup Iequality, N, d Fix η > 0 ad by 3) fid g H ) loc ; d such that lim t g t) α, lim t g t) β ad W g t)) + g t) 2) dt c W + η. 4) 9

10 Let u be as i the previous sectio. Fix ρ > 0 ad let b ρ >> be such that g b ρ ) β < ρ ad let a ρ << be such that g a ρ ) α < ρ. Defie β if y b ρ +, β g b ρ )) y b ρ ) + g b ρ ) if b ρ < y < b ρ +, g ρ y) : g y) if a ρ y b ρ, 5) g a ρ ) α) y a ρ ) + g a ρ ) if a ρ < y < a ρ, α if y a ρ, ad ) x t g 2i ρ u,ρ x) : g t2i+ x ρ u x) ) if t 2i r < x < t 2i + r, if t 2i+ r < x < t 2i+ + r, otherwise, where 2r < mi {t i+ t i : i 0,..., m }. sectios we get ) ti+r W u,ρ ) + u,ρ 2 dx i t i r r/ i bρ+ i m easoig as i the previous ) W u,ρ ) + u,ρ 2 dx r/a W gρ y)) + g ρ y) 2) dx a ρ W gρ y)) + g ρ y) 2) dx W g t)) + g t) 2) dt + Cρ c W + η) m + Cρ for all suffi cietly large. Note that we ca take η ρ. It follows that ) lim sup lim sup W u,ρ ) + u ρ 0 +,ρ 2 dx c W m ad as i 2), u,ρ u dx 0 as. Agai we ca diagoalize to get a sequece {u,ρ }. 9 Limif Iequality, N, d 0 Limif Iequality, N, d Limsup Iequality, N, d For N, give u BV ; {α, β}), we write u αχ E + βχ \E. 0

11 Assume first that E is a regular set, that is, that E is a ope set with E of class C 2, ad that E meets the boudary of trasversally, that is, H N E ) 0. Let g ad g ρ be as i 4) ad 5). I this case we take α if dist x, E) < L ρ u,ρ x) : g ρ dist x, E) / ) if dist x, E) L ρ, β if dist x, E) > L ρ, where dist is the siged distace, ad L ρ > 0 is such that g ρ L ρ ) β ad g ρ L ρ ) α. Note that the fuctio dist x, E) is of class C 2 i the set { x N : dist x, E) < r } for r small, with dist x, E). Moreover, lim { r 0 HN x N : dist x, E) r }) H N E). + Hece, by the coarea formula ) W u,ρ ) + u,ρ 2 dx W gρ dist x, E) / )) + g ρ dist x, E) / ) 2) dx distx, E) L ρ Lρ L ρ L ρ W gρ r/ )) + g ρ r/ ) 2) H N { x N : dist x, E) r }) dr L ρ W gρ r)) + g ρ r) 2) H N { x N : dist x, E) r }) dr Lρ I tur, lim sup L ρ W gρ r)) + g ρ r) 2) drh N E). ) W u,ρ ) + u,ρ 2 dx W g t)) + g t) 2) dt H N E) + CρH N E), ad so takig η ρ ad usig 4), lim sup ρ 0 + lim sup W u,ρ ) + u,ρ 2 ) dx c W H N E).