K-UASICONVEXITY REDUCES TO UASICONVEXITY F. CAGNETTI Abstact. The elaton between quasconvexty and -quasconvexty,, s nvestgated. It s shown that evey smooth stctly -quasconvex ntegand wth p-gowth at nfnty, p > 1, s the estcton to -th ode symmetc tensos of a quasconvex functon wth the same gowth. When the smoothness condton s dopped, t s possble to pove an appoxmaton esult. As a consequence, lowe semcontnuty esults fo -th ode vaatonal poblems ae deduced as coollaes of well-nown fst ode theoems. Ths genealzes a pevous wo by Dal Maso, Fonseca, Leon and Mon, n whch the case = was teated. Keywods: quasconvexty, hghe ode vaatonal poblems. 1. Intoducton We consde hghe ode vaatonal poblems, n whch the enegy functonal has the expesson u fx, u, u,..., u)dx, 1.1) Ω whee Ω R N s open and bounded, N, ae ntege, and f s a scala functon satsfyng sutable gowth condtons. Although ou teatment can be extended to the vectoal case, to eep the fomulaton as smple as possble we wll teat the case of scala functons u : Ω R. Functonals of ths type appea n the study of elastc mateals of gade see 3), n the theoy of second ode stuctued defomatons see 1), n the Blae-Zsseman model fo mage segmentaton n compute vson see 5), n gadent theoes of phase tanstons wthn elastcty egmes see 7, 15, 0), and n the descpton of equlba of mcomagnetc mateals see 9, 6, 0, ). In ode to study lowe semcontnuty of functonals of ths type, Meyes ntoduced n 18 the noton of -quasconvexty see also 3 and 13), extendng the defnton of quasconvexty gven by Moey n 19. tmes {}}{ Let E R N... R N = R N be the set of -th ode tensos of R N that ae symmetc wth espect to all pemutatons of ndces. In patcula, E concdes wth the set of the symmetc N N matces. A functon f L 1 loc E ) s sad to be -quasconvex f fa + φ) fa) dx 0 fo evey A E and evey φ Cc ), whee = 0, 1) N s the open unt cube n R N, and Cc ) s the set of functons of class C wth compact suppot n. We ecall that a functon F L 1 loc RN ) s sad to be 1-quasconvex o smply quasconvex) f FA + ϕ) FA) dx 0 fo evey A R N and evey ϕ C 1 c; R N 1 ). In 18, the autho poved that -quasconvexty s a necessay and suffcent condton fo sequental lowe semcontnuty of 1.1) wth espect to wea convegence n the Sobolev space W,p Ω), unde appopate p-gowth and contnuty condtons on the ntegand f. Ths esult has been late extended to the case whee f s a Caathéodoy ntegand by Fusco see 13) and by Gudoz and Poggoln see 1), fo p = 1 and p > 1 espectvely. 1
F. CAGNETTI The am of ths pape s to nvestgate the elaton between -quasconvexty and quasconvexty. When = ths poblem has been studed by Dal Maso, Fonseca, Leon and Mon. In 8, they pove that evey stctly -quasconvex functon see condton a) below) of class C 1, whose gadent s locally Lpschtz contnuous, s the estcton to symmetc matces of a 1-quasconvex functon. We extend hee ths esult to the case >. Theoem 1.1. Let N,. Let f C 1 E ), and let 1 < p <, µ 0, L > 0, ν > 0. Assume that a) stct -quasconvexty) fa + φ) fa) dx ν fo evey A E and evey φ C c ); b) Lpschtz condton fo gadents) fo evey A, B E. µ + A + p φ ) φ dx fa + B) fa) L µ + A p + B ) B 1.) Then thee exsts a 1 quasconvex functon F : R N R such that fo a sutable constant c f dependng on f. FA) = fa) A E, 1.3) FA) c f 1 + A p ) A R N, 1.) Notce that the above condtons a) and b) togethe mply L ν see Poposton.8). When p, we also gve an explct expesson fo the functon F see fomula 3.9)). The poof of Theoem 1.1 see Secton 3) s obtaned by teatng 1 tmes a efned veson of 8, Theoem 1 see Lemma 3.1 fo the case 1 < p < and Lemma 3. fo the case p ). It s not clea whethe Theoem 1.1 stll holds tue by weaenng condton 1.). Howeve, f we substtute 1.) wth the mlde see Poposton.9) condton 1.5), we obtan an appoxmaton esult fo the functon f. Moe pecsely, we show that a stctly -quasconvex functon wth p-gowth at nfnty can be obtaned as pontwse lmt of a sequence of 1-quasconvex functons wth the same gowth see 8, Theoem fo the case = ). Theoem 1.. Let N,. Let 1 < p <, µ 0, ν > 0, M > 0, and let f : E R be a measuable functon such that a) stct -quasconvexty) fa + φ) fa) dx ν fo evey A E and evey φ C c ); b) p-gowth condton) fo evey A E. µ + A + p φ ) φ dx fa) M1 + A p ) 1.5) Then thee exsts an nceasng sequence {F } N of 1 quasconvex functons F : R N R, such that lm F A) = fa) A E, 1.6) + F A) M 1 + A p ) A R N, N, 1.7) whee {M } N s a sequence of postve constants dependng only on and on the constants p, µ, ν, M, but not on the specfc functon f.
K-UASICONVEXITY REDUCES TO UASICONVEXITY 3 To show ths, we use the popety that evey -quasconvex functon wth p-gowth s locally Lpschtz. We gve hee a poof of ths fact see Poposton.7), that was aleady nown n the cases = 1 see 17) and = see 1). Thans to Theoem 1., the study of lowe semcontnuty of 1.1) educes to a fst ode poblem. Thus, when f s a -quasconvex nomal ntegand see assumpton a) below) we can pove the followng esult see 8, Theoem 3 fo the case = ). Hee we use the notaton and E 1 := R N, E 1 := R E 1... E 1, SBH ) Ω) := {u W 1,1 Ω) : 1 u SBV Ω; E 1 )}. Theoem 1.3. Let N,. Let Ω R N be a bounded open set and let f : Ω E 1 E 0, + ) be a measuable functon such that: a) fx,, ) s lowe semcontnuous on E 1 E fo L N -a.e. x Ω; b) fx,v, ) s -quasconvex on E fo L N -a.e. x Ω and evey v E 1 ; c) thee exst a locally bounded functon a : Ω E 1 0, + ) and a constant p > 1 such that 0 fx,v, A) ax,v)1 + A p ) fo L N -a.e. x Ω and evey v, A) E 1 E. Then fx, u, u,..., u)dx lmnf fx, u j, u j,..., u j )dx Ω j + Ω fo evey u SBH ) Ω) and any sequence {u j } SBH ) Ω) convegng to u n W 1,1 Ω) and such that sup u j L p Ω) + θ 1 u j )dh N 1) < +, j S 1 u j) whee θ : 0, + ) 0, + ) s a concave, nondeceasng functon such that θt) lm = +, t 0 + t u s the densty of the absolutely contnuous pat of D 1 u ) wth espect to the N-dmensonal Lebesgue measue, and 1 u j denotes the jump of 1 u j on the jump set S 1 u j ). Ths extends to the -th ode settng a lowe semcontnuty popety of 1-quasconvex functons n SBV Ω; R d ) due to Amboso see ) and late genealzed by Kstensen see 16), and a lowe semcontnuty theoem fo -quasconvex ntegands n SBHΩ; R d ) poven by Dal Maso, Fonseca, Leon and Mon see 8). As a coollay, we ecove 1, Theoem 7.1. Coollay 1.. Let Ω, f, and p be as n Theoem 1.3. Then fx, u, u,..., u)dx lmnf fx, u j, u j,..., u j )dx j Ω fo evey u W,p Ω) and any sequence {u j } W,p Ω) wealy convegng to u n W,p Ω). We ema that n 1 Gudoz and Poggoln eque the functon f to be locally Lpschtz contnuous wth espect to the last vaable. As aleady mentoned, we do not need ths hypothess, snce we pove hee that ths s a dect consequence of -quasconvexty and p-gowth. Fnally, we menton that t emans stll an open poblem to pove the analogue of Theoem 1.3 fo the case p = 1, even when =, unless vey specal functons f ae consdeed see 11). Ths wll pobably eque new and ognal deas. Indeed, we thn that fo p = 1 the fundamental Kon-type nequaltes see Lemma.13 and Lemma.1) used n the poofs of Theoem 1.1 and Theoem 1. fal, although we do not have any explct counteexample. The plan of the pape s as follows. In Secton we gve the settng of the poblem. Secton 3 contans the poof of Theoem 1.1, whle Theoem 1. and Theoem 1.3 ae poved n Secton Ω
F. CAGNETTI. Fnally, some auxlay esults that ae extensvely used n the pape can be found n the Appendx.. Settng Thoughout the pape N and ae fxed ntege numbes, wth N,. Fo ths eason, we wll often omt to ndcate the explct dependence on N and. Also, Ω R N s an open bounded set, and = 0, 1) N denotes the open unt cube of R N. Snce N and ae fxed, Defnton.1. Let A R N. We say that A s a -th ode tenso n R N. The components of a tenso A R N wll be denoted wth the symbols A 1... 1,..., = 1,...,N. Moeove, the scala poduct of two tensos A, B R N s gven by A B := N 1,..., =1 A 1... B 1.... Accodngly, the nom of a -th ode tenso A R N s A := N A 1... 1,..., =1 Let now s {1,..., 1} be fxed. Fo any ζ C s ; R N s ), we can egad the s-th ode gadent s ζ of ζ as a -th ode tenso n R N, by settng s s ζ 1... ζ) 1... := s x s+1... x 1. 1,..., = 1...,N. Notce also that s ζ s symmetc wth espect to evey pemutaton of the last s ndces. To tae account of ths popety, we ntoduce some addtonal notaton. Defnton.. Let A R N be a -th ode tenso n R N, and let j, {1,..., }. The j, )-tanspose of A s the element A T j of R N such that assumng, fo nstance, j ) We then set A T j )1... = A 1... j 1 j+1... 1 j +1... 1,..., = 1,...,N. E N s := {A R N : A = A T j fo evey, j = s + 1,...,}. In patcula, we wll mae the dentfcaton E N 1 = R N. In ths way, fo evey ζ C s ; R N s ) we have s ζ E N s. To nclude the case s =, we defne E 1 := {A RN : A = A T j fo evey, j = 1,...,}. Vey often we wll smply wte E nstead of E 1. Hence, we have that φ E fo evey φ C ), usng the notaton φ φ) 1... := x 1... x 1,..., = 1...,N. We ae now gong to defne the symmetc pat of an element of E N s.
K-UASICONVEXITY REDUCES TO UASICONVEXITY 5 Defnton.3. The symmetzaton opeato S s+1 : E N s S s+1 A := 1 s + 1 = s A T s = s A + AT s+1 + + A T s s + 1 We wll say that S s+1 A s the symmetc pat of A. E N s 1 s defned by fo evey A E N s. The subscpt s+1 denotes the fact that the tenso S s+1 A s symmetc n the last s+1 entes. Defnton.. Accodngly, we defne the antsymmetc pat of a tenso A E N s tenso A s+1 A E N s gven by We wll use the notaton A s+1 A := A S s+1 A = s sa AT s+1 + + A T s s + 1 ). as the A s+1 E N s := {A s+1 A : A E N s } E N s. Next poposton genealzes the well-nown fact that symmetc and antsymmetc matces defne othogonal spaces. Fo the convenence of the eade, the poof s n the Appendx. Poposton.5. Thee holds A B = 0 fo evey A E N s 1 We gve now the defnton of hghe ode) quasconvexty. and fo evey B A s+1 E N s. Defnton.6. Let j {1,...,}. A functon f L 1 loc EN j ) s sad to be j-quasconvex f fa + j φ) fa) dx 0 fo evey A E N j and fo evey φ C j c ; RN j ). It s vey well-nown that evey convex functon s locally Lpschtz. Ths popety stll holds tue fo j-quasconvex functons wth p-gowth. We gve hee a poof of ths fact, that s n geneal explctly stated only fo the case j = see 1). Poposton.7. Let j {,..., }, and let f L 1 loc EN j ) be j-quasconvex. Assume, n addton, that fa) M1 + A p ) fo evey A E N j,.1) fo some M > 0 and 1 < p <. Then, thee exsts a constant L = LN, M,, j, p) > 0 such that fa + B) fa) L 1 + A p 1 + B p 1) B Poof. Let us set fo evey A, B E N j. j tmes {}}{ X := {b w... w : b R N j, w S N 1 } E N j, m = mn,, j) := dme N j. Hee, fo evey b R N j and w S N 1 the symbol b w... w denotes the element of R N such that b w... w) 1..., = b 1... j w j+1... w, 1,..., = 1...,N. It can be poven that the othogonal complement of X n E N j s zeo, so that span X = E N j.
6 F. CAGNETTI Let now {ω 1,..., ω m } X be a not necessaly othonomal) bass fo E N j, wth ω = 1 fo = 1,...,m, and let c 1,..., c m R be such that B = m =1 c ω. We have m ) m ) m 1 ) fa + B) fa) = f A + c ω fa) f A + c ω f A + c ω m 1 + f A + =1 c ω ) f =1 =1 m ) A + c ω + + fa + c1 ω 1 ) fa). =1 It wll be enough to pove that thee exsts C = CN, M,, j, p) such that fo evey l = 1,..., m f c l ω l + A + l 1 =0 ) c ω f =1 l 1 ) A + c ω C 1 + A p 1 + B p 1) B,.) =0 whee we set c 0 := 0 and ω 0 := 0. Then, the concluson wll follow by defnng L := mc. Thans to 1, Poposton 3. and Example 3.10 d), fo evey R E N j and evey ω X the functon t ft ω + R) s convex n R. Hence, defnng Gt) := f t c l ω l +A+ ) l 1 =0 c ω and usng.1), fo evey t 1 we have l 1 ) l 1 ) Gt) G0) f c l ω l + A + c ω f A + c ω = G1) G0) =0 =0 f t c l ω l + A + ) l 1 =0 c ω f A + ) l 1 =0 c ω = t M l 1 p A l 1 p) + t c l ω l + A + c ω + + c ω t M t M t =0 =0 + p 1 t p c l p + p 1 l 1 p) + 1) A + c ω + p 1 t p B p + p 1 p 1 + 1) A p + p 1 p 1 + 1)m p B p ), =0 t whee we set Let us now choose Notcng that t = m B := t p 1 B p = A p 1 + B p 1 ) B, =0 c ) 1. A p 1 + B p 1) 1 p 1 B A p t 1. A p 1 B, and usng the fact that and ae equvalent noms, we obtan.). B p t B p, Next poposton shows that condtons a) and b) of Theoem 1.1 necessaly mply L ν. Poposton.8. Let f C 1 E ) satsfy condtons a) and b) of Theoem 1.1 fo some constants µ 0, L, ν > 0 and 1 < p <. Then L ν. Poof. Let A E, φ Cc ), and let x. By the Mean Value Theoem, fa + φx)) fa) = fa + t φx)) fa) φx) + fa) φx),
K-UASICONVEXITY REDUCES TO UASICONVEXITY 7 fo some t 0, 1. Integatng last equalty, snce φ Cc ), we get fa + φx)) fa) dx = fa + t φx)) fa) φx)dx. Hence, usng popety b) fa + φx)) fa) dx L L µ + A + t p φx) ) t φx) dx µ + A + p φx) ) φx) dx, snce the functon t µ + A + t φ ) p t φ s nceasng. Compang last elaton and condton a) we conclude that L ν. We pove now that condton 1.) s stonge than 1.5). Poposton.9. Let j {,...,}, let L > 0, µ 0, 1 < p <, and let f C 1 E N j ) be such that fa + B) fa) L µ + A p + B ) B.3) fo evey A, B E N j. Then, thee exsts a postve constant c f, dependng on f, such that fa) c f 1 + A p ) A E N j. Poof. Let C E N j \ {0} be fxed. Then, by the Mean Value Theoem fo evey A E N j have fa) = fc) + fc + ta C)) fc) A C) + fc) A C), fo some t 0, 1. Thans to.3) fa) fc) + L µ + C + t p A C ) t A C + fc) A C fc) + L µ + C + A C ) p A C + fc) A C,.) snce the functon t µ + C + t A C ) p t A C s nceasng. Concenng the last tem, usng Young s nequalty we have fc) A C fc) p p + A C p p fc) p p + p 1 p we A p + C p ),.5) whee p = p p 1. Snce the functon µ + C + ) p s nceasng n R, usng nequalty we have whee Thus, snce A C A + C, µ + C p + A C ) A C µ + 3 C p + A ) A + C ) max{1, C } µ + 3 C p + A ) 1 + A ) K p max{1, C } 1 + A ) p, K = { mn{µ + 3 C, } f 1 < p <, max{µ + 3 C, } f p. 1 + A ) p C p 1 + A p ), fo some postve constant C p dependng only on p, we have L µ + C + A C ) p A C L C p K p max{1, C } 1 + A p )..6) Combnng.),.5) and.6) the concluson follows.
8 F. CAGNETTI We now state some mpotant esults concenng peodc functons. Defnton.10. A functon w : R N R N s s sad to be -peodc f wx + e ) = wx) fo a.e. x R N and evey = 1,...,N, whee {e 1,..., e N } s the canoncal bass of R N. Let d, N. We wll denote wth C per N ; R d ) the space of -peodc functons of C R N ; R d ). Moeove, we wll use the notaton C c ; Rd ) fo the space of functons of class C fom to R d wth compact suppot n. Next lemma wll be extensvely used n the pape. Lemma.11 Helmholtz Decomposton). Fo evey ϕ C pe; R N s ) thee exst two functons φ C pe; R N s 1 ) and ψ C pe; R N s ) such that wth ϕ 1... s = φ) 1... s + ψ 1... s, N b =1 fo 1,..., s = 1,...,N, ψ 1... b 1 b b+1... s x b = 0 fo evey b {1,..., s}..7) Poof. By applyng the usual Helmholtz Decomposton Lemma see 8, Lemma 1) to each component ϕ 1... s of the functon ϕ, the lemma follows. Befoe statng next lemma, we need the followng defnton. Defnton.1. The s-dvegence s the opeato s-dv : C s ; R N ) C; R N s ) defned by s-dvξ) 1... s := N s+1,..., =1 s ξ 1... x s+1... x 1,..., s = 1,...,N, fo evey ξ C s ; R N ). The defnton s analogous when ξ s a Sobolev functon. We ae now eady to state a fundamental Kon-type estmate. Lemma.13. Fo evey p > 1 thee exsts a constant γ = γn, p, s) 1 such that s ψ p dx γ A s+1 s ψ p dx fo evey -peodc functon ψ : R N R N s of class C satsfyng condton.7). Poof. Notce that, fo evey = s + 1,...,, we have N N s ψ) T s s ψ 1... = s 1 x 1... x =1 x s+1... x 1 x s x +1... x =1 s N ψ 1... = s 1 = 0. x s+1... x 1 x s x +1... x x Thus, s + 1) s-dva s+1 s ψ) 1... s N s ) = s s ψ s ψ) T s s+1 + + s ψ) T s x s+1... x 1... = s = s s+1,..., =1 N s+1,..., =1 N s+1,..., =1 s x s+1... x s ψ) 1... =1 s s ψ 1... s = s s ψ 1... x s+1... x x s+1... x s,
K-UASICONVEXITY REDUCES TO UASICONVEXITY 9 whee wth s we denoted the s-th powe of the Laplace opeato. Hence, s ψ 1... s = s + 1 s-dv A s+1 s ψ) s 1... s 1,..., s = 1,...,N. The concluson follows applyng 1, Theoem 10.5 and followng ema. We wll also need the followng genealzaton of Lemma.13. Lemma.1. Fo evey p > 1 thee exsts a constant τ = τn, p, s) 1 such that µ + s ψ ) p s ψ dx τ µ + A s+1 s ψ ) p A s+1 s ψ dx fo evey constant µ 0 and evey -peodc functon ψ : R N R N s of class C satsfyng condton.7). Poof. The poof smply follows by adaptng the poof of 8, Lemma 11 and usng Lemma.13. We conclude ths secton gvng some defntons of hghe ode BV spaces. We set BH ) Ω) : = {u W 1,1 Ω) : D u s a fnte Radon measue } = {u W 1,1 Ω) : 1 u BV Ω; E 1 )}, whee D u stands fo the -th ode dstbutonal gadent of u, and SBH ) Ω) : = {u BH ) Ω) : 1 u SBV Ω; E 1 )} = {u W 1,1 Ω) : 1 u SBV Ω; E 1 )} BH ) Ω). 3. Poof of Theoem 1.1 To pove Theoem 1.1 we wll fst show that, fo evey j =,...,, evey stctly j-quasconvex functon of class C 1 can be extended to a stctly j 1)-quasconvex functon, povded we eque the gadent to be Lpschtz contnuous. In the case 1 < p <, that we pesent below, we actually have to consde a petubed stct j-quasconvexty. Lemma 3.1. Let j {,...,}, 1 < p <, µ 0, and let M j), ν j), and ε be postve constants. Let f j) C 1 E N j ) satsfy the followng condtons: a) stct j-quasconvexty up to a petubaton) f j) A + j φ) f j) A) dx εh j) A) + ν j) µ + A + j p φ ) j φ dx fo evey A E N j and evey φ Cc j; ), whee h j) : E RN j N j 0, + ); b) Lpschtz condton fo gadents) f j) A + B) f j) A) M j) µ + A p + B ) B fo evey A, B E N j. Then thee exsts a functon F j) C 1 E N j+1 ), and a postve constant L j) = L j) p, µ, M j), ν j), j), such that a ) stct j 1)-quasconvexty up to a petubaton) F j) A + j 1 ϕ) F j) A) dx νj) ε µ + A + j 1 ϕ ) p j 1 ϕ dx µ + A j A ) p A j A ε h j) S j A) fo evey A E N j+1 and evey ϕ C j 1 c ; R N j+1 );
10 F. CAGNETTI b ) Lpschtz condton fo gadents) F j) A + B) F j) A) L j) µ + A p + B ) B fo evey A, B E N j+1 ; c) F j) extends f j) ) F j) A) = f j) A) A E N j. Poof. Let β > 0 be a constant to be chosen at the end of the poof and defne F j) : E N j+1 R as F j) A) := f j) S j A) + β µ + A j A ) p µ p = f j) S j A) + β ga j A) µ p, whee g s gven by elaton 5.) wth X = E N j+1. Relaton c) s clealy satsfed. Let us show that condton a ) holds tue fo a good choce of β. Let ϕ C pe ; RN j+1 ). By Lemma.11 we can wte ϕ = φ + ψ, whee ψ C pe; R N j+1 ) satsfes condton.7) wth s = j 1, and φ C pe; R N j ). By dffeentatng j 1 tmes the pevous elaton we get j 1 ϕ = j φ + j 1 ψ, wth j 1 ϕ, j 1 ψ Cpe ; EN j+1 ), and j φ Cpe ; EN j ). We have F j) A + j 1 ϕ) F j) A) dx = f j) S j A + j φ + S j j 1 ψ) f j) S j A + j φ) dx + f j) S j A + j φ) f j) S j A) dx + β ga j A + A j j 1 ψ) ga j A)dx =: I 1 + I + I 3. Notce that f j) S j A) E N j. Then, thans to Poposton.5 and usng the fact that ψ s -peodc f j) S j A) S j j 1 ψ dx = f j) S j A) j 1 ψ dx = 0. Hence I 1 = f j) S j A + j φ + S j j 1 ψ) f j) S j A + j φ) f j) S j A) S j j 1 ψ dx. Applyng Lemma 5.6 wth ε = ν j) / thee exsts a postve constant c 1 = c 1 ν j), p, M j) ) > 0 such that I 1 µ νj) + S j A + j φ ) p j φ dx c 1 µ + S j j 1 ψ ) p S j j 1 ψ dx νj) τ c 1 µ + S j A + j φ ) p j φ dx µ + A j j 1 ψ ) p A j j 1 ψ dx,
K-UASICONVEXITY REDUCES TO UASICONVEXITY 11 whee τ = τn, p, j 1) s gven by Lemma.1. The petubed stct j-quasconvexty of f j) gves I ν µ j) + S j A + j φ ) p j φ dx εh j) S j A), so that I 1 + I νj) τ c 1 µ + S j A + j φ ) p j φ dx εh j) S j A) µ + A j j 1 ψ ) p A j j 1 ψ dx. We apply now Lemma 5.3 to the fst ntegal of the last expesson wth µ = µ + S j A, x = j φ, and y = j 1 ψ. Recallng that j φ + j 1 ψ = j 1 ϕ we get I 1 + I νj) νj) τ c 1 Usng the fact that 1 < p < and Lemma.1 νj) µ + S j A + j 1 ϕ ) p j 1 ϕ dx εh j) S j A) µ + S j A + j 1 ψ ) p j 1 ψ dx µ + A j j 1 ψ ) p A j j 1 ψ dx. 3.1) µ + S j A + j 1 ψ ) p j 1 ψ dx νj) τ νj) Hence, collectng 3.1) and 3.) I 1 + I νj) νj) µ + j 1 ψ ) p j 1 ψ dx µ + A j j 1 ψ ) p A j j 1 ψ dx. 3.) µ + S j A + j 1 ϕ ) p j 1 ϕ dx εh j) S j A) ) τ c 1 + νj) µ + A j j 1 ψ ) p A j j 1 ψ dx µ + A + j 1 ϕ ) p j 1 ϕ dx εh j) S j A) ) τ c 1 + νj) µ + A j j 1 ψ ) p A j j 1 ψ dx, whee we used once agan the fact that 1 < p <. Snce ga j A) A j E N j+1 -peodc, ga j A) A j j 1 ψ dx = ga j A) j 1 ψ dx = 0, so that I 3 = β gaj A + A j j 1 ψ) ga j A) ga j A) A j j 1 ψ dx. and ψ s
1 F. CAGNETTI Let 0 < δ < 1 to be chosen at the end of the poof. Thans to Lemma 5.1 I 3 βθ p µ + A j A + A j j 1 ψ ) p A j j 1 ψ dx βθ p δ p βθ p δ µ + A j j 1 ψ ) p A j j 1 ψ dx µ + A j A ) p A j A, whee n the second nequalty we used Lemma 5.3 wth X = E N j+1, µ = µ, x = A j A and y = A j j 1 ψ. Choosng β = β j) > 0 and δ = δ j) 0, 1) such that we obtan β j) θ p δ j) ) p τ I 1 + I + I 3 νj) c 1 + νj) ), β j) θ p δ j) ε, µ + A + j 1 ϕ ) p j 1 ϕ dx εh j) S j A) ε µ + A j A ) p A j A, so that a ) holds. To chec condton b ), we obseve that the functon g satsfes the hypotheses of Lemma 5.5. Then, fo evey A, B E N j+1 ga + B) ga) C p µ + A + B ) p B, whee C p s a postve constant dependng only on p. Usng last elaton, b), and the fact that β j) depends on ν j), τ and c 1, we conclude that b ) holds fo some postve constant L j) = L j) p, µ, M j), ν j), j). We pass now to the case p. Lemma 3.. Let j {,...,}, p, µ 0, M j) > 0, ν j) > 0, and let θ p and Θ p be gven by Lemma 5.1. Let f j) C 1 E N j ) satsfy the followng condtons: a) stct j-quasconvexty) f j) A + j φ) f j) A) dx ν j) µ + A + j p φ ) j φ dx fo evey A E N j and evey φ C j c; R N j ); b) Lpschtz condton fo gadents) f j) A + B) f j) A) M j) µ + A p + B ) B fo evey A, B E N j. Then thee exsts a functon F j) C 1 E N j+1 ), and a postve constant L j) = L j) p, µ, M j), ν j), j), such that a ) stct j 1)-quasconvexty) F j) A + j 1 ϕ) F j) A) dx ν j) θ p Θ p µ + A + j 1 ϕ ) p j 1 ϕ dx fo evey A E N j+1 and evey ϕ C j 1 c ; R N j+1 );
K-UASICONVEXITY REDUCES TO UASICONVEXITY 13 b ) Lpschtz condton fo gadents) F j) A + B) F j) A) L j) µ + A p + B ) B fo evey A, B E N j+1 ; c) F j) extends f j) ) F j) A) = f j) A) A E N j. Poof. Let λ 0, ν j) /Θ p and β > 0 be two constants to be detemned at the end of the poof. We defne F j) : E N j+1 R as F j) A) := f j) S j A) λ µ + S j A ) p + λ µ + S j A + β A j A ) p Let g and g β be defned by 5.) and 5.3) espectvely, wth X = E N j Settng fo evey B E N j we have f j) λ B) := fj) B) λgb), F j) A) = f j) λ S ja) + λg β S j A, A j A).. and Y = A j E N j+1. Condton c) s clea fom the defnton of F j). In ode to chec a ), let ϕ C pe ; RN j+1 ). By epeatng the agument of the pevous poof, we can wte j 1 ϕ = j φ + j 1 ψ, wth j 1 ϕ, j 1 ψ Cpe ; EN j+1 ), and j φ Cpe ; EN j ), whee ψ Cpe ; ) RN j+1 satsfes condton.7) wth s = j 1. Hence, F j) A + j 1 ϕ) F j) A) dx = f j) λ S ja + S j j 1 ϕ) f j) λ S ja + S j j 1 ϕ S j j 1 ψ) dx + f j) λ S ja + j φ) f j) λ S ja) dx + λ gβ S j A + S j j 1 ϕ, A j A + A j j 1 ϕ) g β S j A, A j A) dx =: I 1 + I + I 3. Concenng the second ntegal, snce by peodcty gs j A) j φdx = 0, usng condton a) and Lemma 5.1 we have I = fsj A + j φ) fs j A) dx λ = fsj A + j φ) fs j A) dx λ ν j) λθ p ) gsj A + j φ) gs j A) dx gsj A + j φ) gs j A) + gs j A) j φ dx µ + S j A + j p φ ) j φ dx 0. 3.3) Let us pass to the fst ntegal. Notcng that f j) λ S ja) E N j, thans to Poposton.5 and usng the fact that ψ s -peodc, f j) λ S ja) S j j 1 ψ dx = f j) λ S ja) j 1 ψ dx = 0.
1 F. CAGNETTI Hence, I 1 = f j) λ S ja + S j j 1 ϕ S j j 1 ψ) f j) λ S ja + S j j 1 ϕ) f j) λ S ja) S j j 1 ψ dx. As obseved n the pevous poof the functon g satsfes condton 5.6), and so by Lemma 5.5 condton b) stll holds fo the functon f λ fo a sutable constant M = Mp, M j), λ) n place of M j). Thus, applyng Lemma 5.6 wth ε = λθ p /, thee exsts a postve constant σ = σp, M j), λ) such that I 1 λθ p σ µ + S j A + S j j 1 ϕ ) p S j j 1 ϕ dx µ + S j A ) p Thans to Lemma.13 and usng 3.3) we get I 1 + I λθ p σ γn,, j 1) S j j 1 ψ dx σ S j j 1 ψ p dx. µ + S j A + S j j 1 ϕ ) p S j j 1 ϕ dx 3.) µ + S j A ) p A j j 1 ψ dx σ γn, p, j 1) A j j 1 ψ p dx. Snce ϕ s -peodc, 0 = g β S j A, A j A) j 1 ϕdx = x g β S j A, A j A) S j j 1 ϕ + y g β S j A, A j A) A j j 1 ϕ dx, so that I 3 = λ gβ S j A + S j j 1 ϕ, A j A + A j j 1 ϕ) g β S j A, A j A) x g β S j A, A j A) S j j 1 ϕ y g β S j A, A j A) A j j 1 ϕ dx. 3.5) We ae now gong to splt I 3 nto two tems. We wll use the fst tem to compensate the sum I 1 + I, and the emanng one to get the stct j 1)-quasconvexty. Relaton 5.5) of Lemma 5. gves I 3 λθ p + λθ pβ µ + S j A + S j j 1 ϕ ) p S j j 1 ϕ dx µ + S j A ) p If we choose β = β j) > 0 so lage that A j j 1 ψ dx + λθ pβ p A j j 1 ψ p dx. 3.6) λθ p β j) ) σγn,, j 1) and λθ p β j) ) p σγn, p, j 1), usng elatons 3.) and 3.6), we have I 1 + I + I 3 I 3.
K-UASICONVEXITY REDUCES TO UASICONVEXITY 15 Let us estmate the last tem. Wthout any loss of genealty we can assume β j) 1. Then, ecallng 3.5) and usng nequalty 5.) of Lemma 5. F j) A + j 1 ϕ) F j) A) dx = I 1 + I + I 3 λθ p λθ p = ν j) θ p Θ p whee we chose µ + S j A + S j j 1 ϕ + β j) ) A j A + β j) ) A j j 1 ϕ ) p µ + A + j 1 ϕ ) p j 1 ϕ dx µ + A + j 1 ϕ ) p j 1 ϕ dx, λ = νj) Θ p. S j j 1 ϕ + β j) ) A j j 1 ϕ ) dx One can show that F j) satsfes condton b ) as t was done n the poof of Lemma 3.1. We can now pass to the poof of Theoem 1.1. Poof of Theoem 1.1. Step 1. Case 1 < p <. To smplfy the notaton, fo evey B R N we set PB) := µ + B ) p B, GB) := µ + B ) p µ p. Let ε > 0 be fxed. We stat the poof by applyng Lemma 3.1 wth j =, ν ) = ν, h ) 0 and f ) A) = fa), fo evey A E. Then, we apply agan tmes Lemma 3.1 wth j = 1,,..., espectvely, wth ν j) = ν j, and f j) A) = F j+1) A), fo evey A E N j, whle the functons h j) : E N j 0, + ) wll be chosen as h 1) A) = PA A), h j) A) = PA j+1 A) + =j+ PA S 1... S j+1 A) j =,...,. In ths way afte the last step, coespondng to j =, we obtan a functon F ) : R N R gven by F ) A) := fs S 1... S ) + β ) GA A) + β ) GA S 1... S A). 3.7) Hee, fo evey j =,...,, the constant β j) s gven by the poof of Lemma 3.1 wth the coespondent ndex j. F ) has the followng popetes: a ) stct 1-quasconvexty up to a petubaton) =3 F ) A + ϕ) F ) A) dx εpa A) εh ) S A) + ν 1 µ + A + ϕ ) p ϕ dx
16 F. CAGNETTI fo evey A R N and evey ϕ C 1 c ; R N 1 ); b ) Lpschtz condton fo the gadent) F ) A + B) F ) A) L µ + A p + B ) B, fo evey A, B R N, wth L = Lp, µ, M, ν); c) F ) extends f) Now, let us defne F ) A) = fa) A E. { } FA) := nf F ) A + ϕx))dx : ϕ Cpe; R N 1 ) fo evey A R N. Popety a ) mples that fo evey A R N F ) A) εpa A) εh ) S A) FA) F ) A). 3.8) Snce fo evey A E PA A) = h ) S A) = 0, fom popety c) and elaton 3.8) equalty 1.3) follows. Let us chec 1.). Thans to Poposton.9, fom condton b ) we nfe that thee exsts a postve constant c, dependng on the functon F ) and n tun on f, such that F ) A) c 1 + A p ) A R N. Recallng the defntons of the functons P and h ), last elaton and 3.8) gve 1.). Step. Case p. Repeatng the stategy used fo the case 1 < p <, we fst apply Lemma 3. wth j =, ν ) = ν and f ) A) = fa), fo evey A E. Then, we apply agan tmes Lemma 3. wth j = 1,,..., espectvely, wth ) j+1 ν j) θp = ν, Θ p and f j) A) = F j+1) A), fo evey A E N j. Fnally, when j = we obtan a functon F ) : R N R gven by F ) A) = fs... S A) + L ) S A, A A) + whee we set L ) S S 1... S A, A S 1... S A), 3.9) =3 L ) A, B) := ν) µ + A ) p + ν) µ + A + β ) ) B ) p, =,...,, Θ p Θ p and fo evey j =,...,, the constant β j) s gven by the poof of Lemma 3. wth the coespondent ndex j. The functon F ) just defned s such that a ) stct 1-quasconvexty) F ) A + ϕ) F ) A) dx ν ) θp Θ p fo evey A R N and evey ϕ C 1 c ; R N 1 ); µ + A + ϕ ) p ϕ dx
K-UASICONVEXITY REDUCES TO UASICONVEXITY 17 b ) Lpschtz condton fo the gadent) F ) A + B) F ) A) L µ + A p + B ) B, fo evey A, B R N, wth L = Lp, µ, M, ν); c) F ) extends f) F ) A) = fa) A E. We clam that the poof s concluded by settng F := F ). Indeed, condton c) gves 1.3), whle 1.) follows by applyng Poposton.9 to F ).. Poof of Theoem 1. To pove the theoem, we fst need two pelmnay lemmas. Lemma.1. Let j {,..., }, 1 < p <, µ 0, ν j) > 0, and let {M j) } N be a sequence of postve constants. Let {f j) } N be a sequence of functons f j) : E N j R satsfyng the followng condtons: a) stct j-quasconvexty up to a petubaton) f j) A + j φ) f j) A) + ν j) dx h j) A) µ + A + j p φ ) j φ dx fo evey A E N j, fo evey φ Cc j; ), and fo evey N, whee {h j) RN j } N s a sequence of functons h j) : E N j 0, + ); b) p-gowth condton) f j) A) M j) 1 + A p ) A E N j, N. : E N j+1 Then thee exsts an nceasng sequence {F j) } N of functons F j) sequences {L j) } N and {λ j) } N of postve numbes, dependng on ν j), M j) a ) stct j 1)-quasconvexty up to a petubaton) F j) A + j 1 ϕ) F j) A) dx νj) R, and two, j, p, µ, such that µ + A + j 1 ϕ ) p j 1 ϕ dx 1 1 S ja p λ j) A j A p h j) S j A) fo evey A E N j+1, fo evey ϕ Cc j 1 ; R N j+1 ), and fo evey N; b ) p-gowth condton) c) F j) extends f j) ) F j) A) L j) 1 + A p ) A E N j+1, N; F j) A) = f j) A) A E N j, N. Poof. Fst we obseve that, thans to Poposton.7, thee exsts a postve constant L =, j, p) we do not stess hee the dependence on N and ), such that LM j) fo evey A, B E N j A E N j+1, we defne f j) A + B) f j) A) L 1 + A p 1 + B p 1) B.1). Let β > 0 be a constant to be chosen at the end of the poof. Fo evey F j) A) := f j) S j A) + β A j A p..)
18 F. CAGNETTI Condton c) s clealy satsfed. In ode to show a ), let us consde a functon ϕ C pe; R N j+1 ). We can wte j 1 ϕ = j φ + j 1 ψ, wth j 1 ϕ, j 1 ψ Cpe; E N j+1 ), and j φ Cpe; E N j ), whee ψ Cpe; R N j+1 ) satsfes condton.7) wth s = j 1. Hence, F j) = A + j 1 ϕ) F j) A) f j) dx S j A + j φ + S j j 1 ψ) f j) S j A + j φ) dx + f j) S j A + j φ) f j) S j A) dx Aj + β A + A j j 1 ψ p A j A p dx =: I 1 + I + I 3. By.1) and Young s nequalty, fo evey δ > 0 thee exsts a constant C = CM j), j, p, δ) such that I 1 L 1 + Sj A + j φ p 1 + S j j 1 ψ p 1) S j j 1 ψ dx δ δ S j A p δ j φ p dx C S j j 1 ψ p dx. Usng Lemma.13 I 1 δ δ S j A p δ Thans to Lemma 5., fo evey 0 < ε < 1 I 1 δ1 + εµ p ) δ1 + ε) S j A p C γ 8 δ ε p p j φ p dx C γ A j j 1 ψ p dx..3) A j j 1 ψ p dx µ + S j A + j φ ) p j φ dx. Then, applyng Lemma 5.3 wth µ = 0, x = A j A, and y = A j j 1 ψ, I 1 δ1 + εµ p ) δ1 + ε) S j A p C γ ε p Aj A p C γ ε p A j A + A j j 1 ψ ) p A j j 1 ψ dx 8 δ ε p p µ + S j A + j φ ) p j φ dx. Thus, thee exsts a sequence of postve numbes {λ j) } N, such that fo evey N I 1 µ νj) + S j A + j φ ) p j φ dx λ j) A j A + A j j 1 ψ ) p A j j 1 ψ dx 1 1 S ja p λ j) A j A p. Hee, fo evey fxed N, λ j) = λ j) ν j), M j), j, p, µ). By condton a) I ν j) µ + S j A + j p φ ) j φ dx h j) S j A),
K-UASICONVEXITY REDUCES TO UASICONVEXITY 19 so that I 1 + I νj) λ j) µ + S j A + j φ ) p j φ dx A j A + A j j 1 ψ ) p A j j 1 ψ dx 1 1 S ja p λ j) A j A p h j) S j A),.) We focus now on the fst tem of last expesson. Applyng Lemma 5.3 wth µ = µ + A, x = j φ, and y = j 1 ψ, and ecallng that j φ + j 1 ψ = j 1 ϕ we get ν j) µ + S j A + j φ ) p j φ dx µ νj) + A + j φ ) p j φ dx νj) νj) µ + A + j 1 ϕ ) p j 1 ϕ dx µ + A + j 1 ψ ) p j 1 ψ dx, whee n the fst lne we used the fact that 1 < p <. By Lemma.1 last nequalty becomes ν j) µ + S j A + j φ ) p j φ dx µ νj) + A + j 1 ϕ ) p j 1 ϕ dx µ νj) τ + A + A j j 1 ψ ) p A j j 1 ψ dx νj) νj) τ µ + A + j 1 ϕ ) p j 1 ϕ dx A j A + A j j 1 ψ ) p A j j 1 ψ dx,.5) agan explotng that 1 < p <. Collectng.) and.5) we have I 1 + I µ νj) + A + j 1 ϕ ) p j 1 ϕ dx λ j) + νj) τ ) A j A + A j j 1 ψ ) p A j j 1 ψ dx 1 1 S ja p λ j) A j A p h j) S j A)..6) Concenng I 3, usng the peodcty of ψ and thans to Lemma 5.1 wth µ = 0 Aj I 3 = β A + A j j 1 ψ p A j A p p A j A p A j A A j j 1 ψ dx Choosng β = β j) β θ p > 0 such that A j A + A j j 1 ψ ) p A j j 1 ψ dx..7) β j) θ p λ j) + νj) τ,
0 F. CAGNETTI fom.6) and.7) we obtan I 1 + I + I 3 νj) µ + A + j 1 ϕ ) p j 1 ϕ dx 1 1 S ja p λ j) so that a ) holds. Fom.) condton b ) follows. A j A p h j) S j A), The second lemma addesses the case p. Lemma.. Let j {,...,}, p, µ 0, ν j) > 0, and let {M j) } N be a sequence of postve constants. Let moeove θ p and Θ p be gven by Lemma 5.1, and let {f j) } N be a sequence of functons f j) : E N j R satsfyng the followng condtons: a) stct j-quasconvexty up to a petubaton) f j) A + j φ) f j) A) + ν j) dx h j) A) µ + A + j p φ ) j φ dx fo evey A E N j, fo evey φ Cc j; RN j ), and fo evey N, whee {h j) } N s a sequence of functons h j) : E N j 0, + ); b) p-gowth condton) f j) A) M j) 1 + A p ) A E N j, N. Then thee exsts an nceasng sequence {F j) } N of functons F j) : E N j+1 sequence {L j) } N of postve numbes, dependng on ν j), M j), j, p, µ, such that a ) stct j 1)-quasconvexty up to a petubaton) F j) A + j 1 ϕ) F j) A) + ν j) θ p Θ p dx h j) S j A) 1 1 S ja p µ + A + j 1 ϕ ) p j 1 ϕ dx fo evey A E N j+1, fo evey ϕ Cc j 1 ; R N j+1 ), and fo evey N; b ) p-gowth condton) c) F j) F j) extends f j) ) F j) A) L j) 1 + A p ) A E N j+1, N; F j) A) = f j) A) A E N j, N. A) := f j) S j A) α R, and a Poof. Let α 0, ν j) /Θ p and β > 0 to be detemned at the end of the poof. We defne µ + S j A ) p µ + S j A + β A j A ) p Condton c) s clealy satsfed. Let now ϕ C pe; R N j+1 ). As usual, we can wte + α j 1 ϕ = j φ + j 1 ψ,..8) wth j 1 ϕ, j 1 ψ Cpe; E N j+1 ), and j φ Cpe; E N j ), whee ψ Cpe; R N j+1 ) satsfes condton.7) wth s = j 1. Settng f j) ) α B) := f j) B) α µ + B ) p
K-UASICONVEXITY REDUCES TO UASICONVEXITY 1 fo evey B E N j, we have F j) A + j 1 ϕ) F j) A) dx = f j) ) α S j A + j φ + S j j 1 ψ) f j) ) α S j A + j φ) dx + f j) ) α S j A + j φ) f j) ) α S j A) dx + α gβ S j A + S j j 1 ϕ, A j A + A j j 1 ϕ) g β S j A, A j A) dx =: I 1 + I + I 3, wth g β defned by 5.3), wth X = E N j and Y = A j E N j+1. By epeatng the chan of nequaltes 3.3), one can show f j) ) α s j-quasconvex. In addton, applyng Lemma 5.5 and Poposton.9 to the functon B α µ + B ) p, we have that f j) j) ) α satsfes condton b), fo some postve constant M = place of M j). Thus, by applyng Poposton.7, we can stll conclude that elaton.1) holds M j) α, µ, M j) ) n ) α, fo a sutable constant L = LN, M j),, j, p, α). By epeatng the tue fo the functon f j) same agument of the pevous poof, we get that fo evey δ > 0 thee exsts a postve constant c = cm j), j, p, α, µ, δ) such that I 1 δ δ S j A p δ δ δ S j A p δ c γ A j j 1 ψ p dx. j φ p dx c γ A j j 1 ψ p dx µ + S j A + j φ ) p j φ dx Hence, we can fnd a sequence of postve numbes {λ j) } N such that fo evey N Hee λ j) 3.3) we get = λ j) ) I 1 ν j) α Θ p µ + S j A + j φ ) p j φ dx λ j) A j j 1 ψ p dx 1 1 S ja p. M j), j, p, α, µ) fo evey fxed N,. Adaptng to the pesent stuaton nequalty I = + f j) ) α S j A + j φ) f j) ) α S j A) dx h j) S j A) ) ν j) α Θ p µ + S j A + j p φ ) j φ dx.
F. CAGNETTI Moeove, assumng wthout any loss of genealty that β 1, I 3 α θ p µ + A + j 1 ϕ ) p j 1 ϕ dx + αθ p µ + S j A + S j j 1 ϕ ) p S j j 1 ϕ dx + αθ pβ α θ p Let us now choose and β = β j) we obtan µ + S j A ) p > 0 such that A j j 1 ψ dx + αθ pβ p µ + A + j 1 ϕ ) p j 1 ϕ dx + αθ pβ p F j) α = α j) = νj) Θ p, α j) θ p β j) ) p λ j), A + j 1 ϕ) F j) A) dx = I 1 + I + I 3 ν j) θ p Θ p A j j 1 ψ p dx µ + A + j 1 ϕ ) p j 1 ϕ dx h j) S j A) 1 1 S ja p, so that a ) holds. Fnally, condton b ) follows by.8). We ae now eady to pove Theoem 1.. Poof of Theoem 1.. A j j 1 ψ p dx. Step 1. Case 1 < p <. We stat by applyng Lemma.1 wth j =, ν ) = ν and f ) A) = fa), h ) A) = 0 fo evey A E, N. Then, we apply agan tmes Theoem.1 wth j = 1,,..., espectvely, wth ν j) = ν j, and, fo evey A E N j and N, Accodngly, the functons h j) and, fo j =,...,, h j) A) = j + 1 + f j) wll be chosen as A) = F j+1) A). h 1) A) = 1 + 1 S A p + λ ) A A p, =j+ =j+1 S S 1... S j+1 A p + λ j+1) A j+1 A p λ ) A S 1... S j+1 A p,
K-UASICONVEXITY REDUCES TO UASICONVEXITY 3 whee the sequences {λ j) } N ae gven by Lemma.1. In ths way afte the last step, coespondng to j =, we obtan a sequence {F ) } N of functons F ) : R N R gven by F ) A) = fs... S A) + β ) A A p + =3 β ) A S 1... S A p. Hee fo =,...,, the sequence {β ) } N s that one gven n the poof of Lemma.1. The functons F ) just defned have the followng popetes: a ) stct 1-quasconvexty up to a petubaton) F ) A + ϕ) F ) A) dx h ) S A) 1 1 S A p + ν 1 µ + A + ϕ ) p h ) S A) 1 1 S A p λ ) A A p ϕ dx λ ) A A p fo evey A R N, fo evey ϕ C 1 c ; RN 1 ), and fo evey N; b ) gowth condton) F ) A) L ) 1 + A p ) A R N, N, wth L ) = L ) ν, M, p, µ) fo evey fxed N; c) F ) extends f) F ) A) = fa) A E, N. Now, fo evey A R N and N, we set { } F A) := nf F ) A + ϕx))dx : ϕ Cpe; R N 1 ). Fom popety a ) t follows that fo evey A R N and fo evey N F ) A) h ) S A) 1 1 S A p λ ) A A p F A) F ) A)..9) Notcng that fo evey A E lm + h) S A) = lm + + 1 S... S A p = 0, fom popety c) and.9) we have 1.6). Fnally, 1.7) follows fom b ) and.9). =3 Step. Case p. We fst apply Lemma. wth j =, ν ) = ν and f ) A) = fa), h ) A) = 0 fo evey A E, N. At ths pont, we apply agan tmes Lemma. wth j = 1,,..., espectvely, wth ) j+1 ν j) θp = ν, Θ p
F. CAGNETTI and, fo evey A E N j and N, f j) A) = F j+1) A), h j) A) = j + 1 S S 1... S j+1 A p. =j+1 Fnally, when j =, we obtan a sequence {F ) } N of functons F ) : R N R gven by whee we set F ) A) = fs... S A) + =3 L ) A, B) := ν) µ + A ) p + ν) Θ p Θ p L ) S S 1... S A, A S 1... S A) + L ) S A, A A),.10) µ + A + β ) The functons F ) just defned have the followng popetes: a ) stct 1-quasconvexty up to a petubaton) ) B ) p, =,...,. F ) A + ϕ) F ) A) dx h ) S A) 1 1 S A p + ν ) θp Θ p h ) S A) 1 1 S A p µ + A + ϕ ) p ϕ dx fo evey A R N, fo evey ϕ C 1 c ; R N 1 ), and fo evey N; b ) gowth condton) c) F ) extends f) F ) A) L ) 1 + A p ) A R N, N; F ) A) = fa) A E, N. Fo evey N, we defne now F as the quasconvexfcaton of the functon F ) { } F A) := nf F ) A + ϕx))dx : ϕ Cpe; R N 1 ) fo evey A R N. Fom popety a ) and by the defnton of F, we have F ) A) h ) S A) 1 1 S A p F A) F ) A), A R N..11) : Notcng that lm + h) S A) = 0 fo all A R N, fom popety c) and.11) we have 1.6). Fnally, 1.7) follows fom b ) and.11)..1. Poof of Theoem 1.3. To conclude the secton, we gve the poof of Theoem 1.3. Poof of Theoem 1.3. It s enough to adapt the poof of 8, Theoem 3 and to use Theoem 1.3.
K-UASICONVEXITY REDUCES TO UASICONVEXITY 5 5. Appendx Ths secton contans some auxlay esults used n the est of the pape. Fst, we gve the poof of Poposton.5. Poof of Poposton.5. Snce A E N s 1 and B A s+1 E N s, we can wte A = S s+1 A and B = A s+1 C, fo some C E N s. As a fst step, let us pove that fo evey, l { s+1,..., } wth l we have A T s C T s l = A C T s To fx the deas, let us assume < l. By defnton of tanspose opeatos A T s 1... C T s l = A T s C. 5.1) 1... = A 1... s 1 s+1... 1 s +1... C 1... s 1 l s+1... l 1 s l+1..., fo evey 1,,..., = 1,...,N. In the last expesson, snce A E N s 1 and, l > s, we can exchange the ndces n -th and l-th poston n the fst facto, obtanng A T s 1... C T s l 1... = A 1... s 1 s+1... 1 l +1... l 1 s l+1... C 1... s 1 l s+1... l 1 s l+1.... Summng last elaton wth espect to 1,..., and enumeatng the ndces A T s C T s l = = 1,N A T s 1... C T s l 1... 1..., 1,N 1..., A 1... s 1 s+1... 1 l +1... l 1 s l+1... C 1... s 1 l s+1... l 1 s l+1... 1,N = A 1... C 1... s 1 s+1... 1 s +1... = A 1... C T 1... = A C T s 1..., 1..., In the same way one can pove the second equalty n 5.1). Let us now pove the poposton. We have s + 1) A B = s + 1) S s+1 A A s+1 C) = A + A T ) s s+1 + + A T s s C C T s s+1 + + C T s ) = sa C + s = s+1 = s+1 A T s A T s C C T s = s+1 l= s+1 l = s+1 1,N A C T s A T s C T s l Snce the sum of the fst two tems s zeo, usng elaton 5.1) we get s + 1) A B = s 1) = s 1) = s+1 = s+1 A C T s A C T s s 1) l= s+1 l = s+1 = s+1 A C T s s A C T s = 0. In the emanng pat of the secton we state some lemmas that ae poved n 8. Lemma 5.1. Let X be a Hlbet space, and let g : X R be gven by gx) := µ + x ) p. 5.).
6 F. CAGNETTI Fo evey p > 1, thee exst two constants θ p > 0 and Θ p > 0 such that fo evey µ 0 the functon g defned n 5.) satsfes the followng nequaltes fo evey x, y X. θ p µ + x + y ) p y gx + y) gx) gx) y Θ p µ + x + y ) p y Lemma 5.. Let X, Y be Hlbet spaces and let p > 1, µ 0, β 0. Let g β : X Y R be gven by g β x, y) := µ + x + β y ) p. 5.3) Then g β x + ξ, y + η) g β x, y) x g β x, y) ξ y g β x, y) η 5.) θ p µ + x + ξ + β y + β η ) p ξ + β η ) fo evey x, ξ X, y, η Y, whee θ p s the fst constant n Lemma 5.1. Theefoe, f p, we have g β x + ξ, y + η) g β x, y) x g β x, y) ξ y g β x, y) η 5.5) θ p µ + x + ξ ) p ξ + θ pβ µ + x ) p η + θ pβ p η p fo evey x, ξ X, y, η Y. Lemma 5.3. Let X be a Hlbet space and let 1 < p. Then fo evey µ 0 and evey 0 < δ < 1 we have µ + x + y ) p x + y µ + x ) p x + µ + y ) p y, δ p fo evey x, y X. µ + y ) p y Lemma 5.. Let 1 < p. Then fo evey a 0, b 0, µ 0, and 0 < ε < 1. µ + x + y ) p y + δ b p 8ε p p µ + a + b ) p b + εa p + εµ p µ + x ) p Lemma 5.5. Let X be a Hlbet space, and let f C 1 X) C X \ {0}). Assume that thee exst p > 1, C > 0, and µ 0 such that fo evey x X \ {0}. Then x fx) C µ + x ) p 5.6) fx + y) fx) K p C fo evey x, y X, whee K p 1 s a constant dependng only on p. µ + x + y ) p y 5.7) Lemma 5.6. Let X be a Hlbet space and let f C 1 X). Assume that thee exst p > 1 and µ 0 such that fx + y) fx) µ + x + y ) p y fo evey x, y X. If 1 < p, then fo evey ε > 0 thee exsts a constant c 1 = c 1 ε, p) > 0, dependng only on ε and p, such that fx + y + z) fx + y) fx) z ε µ + x + y ) p y + c 1 µ + z ) p z
K-UASICONVEXITY REDUCES TO UASICONVEXITY 7 fo evey x, y, z X. If p, then fo evey ε > 0 thee exsts a constant c = c ε, p) > 0, dependng only on ε and p, such that fx + y + z) fx + y) fx) z ε µ + x + y ) p y + c µ + x ) p z + c z p fo evey x, y, z X. 6. Acnowledgments The autho gatefully acnowledges vey useful convesatons wth Iene Fonseca and Govann Leon on the subject of the pape. He also thans the Cente fo Nonlnea Analyss NSF Gants No. DMS-00533 and DMS-0635983) fo ts suppot dung the pepaaton of ths pape. Refeences 1 Agmon S., Dougls A., Nenbeg L.: Estmates nea the bounday fo solutons of ellptc patal dffeental equatons satsfyng geneal bounday condtons II. Comm. Pue Appl. Math., 17 196), 35 9. Amboso L., Fusco N., Pallaa D.: Functons of bounded vaaton and fee dscontnuty poblems. Oxfod Unvesty Pess, New Yo, 000. 3 Ball J.M., Cue J.C., Olve P.: Null Lagangans, wea contnuty, and vaatonal poblems of abtay ode. J. Funct. Anal. 1 1981), No., 135 17. Ball J.M., Kchhem B., Kstensen J.: Regulaty of quasconvex envelopes. Calc. Va. Patal Dffeental Equatons, 11/ 000), 333 359. 5 Caeo M., Leac A., Tomaell F.: Stong mnmzes of Blae & Zsseman functonal. Ann. Scuola Nom. Sup. Psa Cl. Sc. 15 1997), No. 1-, 57 85. 6 Chos R., Kohn R.V., Otto F.: Doman banchng n unaxal feomagnets: a scalng law fo the mnmum enegy. Comm. Math. Phys. 01 1999), No. 1, 61 79. 7 Cont S., Fonseca I., Leon G.: A Γ-convegence esult fo the two-gadent theoy of phase tanstons. Comm. Pue Appl. Math. 55 00), No. 7, 857 936. 8 Dal Maso G., Fonseca I., Leon G., Mon M.: Hghe-ode quasconvexty educes to quasconvexty. Ach. Raton. Mech. Anal., 171 00), 55 81. 9 DeSmone A.: Enegy mnmzes fo lage feomagnetc bodes. Ach. Raton. Mech. Anal., 15 1993), 99 13. 10 Fonseca I., Leon G.: Moden methods n the calculus of vaatons: L p spaces. Spnge, New Yo, 007. 11 Fonseca I., Leon G.: A note on Meyes theoem n w,1. Tans. Ame. Math. Soc., 35 00), 373 371. 1 Fonseca I., Mülle S.: A-quasconvexty, lowe semcontnuty, and Young measues. SIAM J. Math. Anal. 30 1999), 1355 1390. 13 Fusco N.: uasconvexty and semcontnuty fo hghe-ode multple ntegals Italan). Rceche Mat. 9 1980), No., 307 33. 1 Gudoz M., Poggoln L.: Lowe semcontnuty fo quasconvex ntegals of hghe ode. NoDEA Nonlnea Dffeental Equatons Appl. 6 1999), No., 7 6. 15 Kohn R.V., Mülle S.: Suface enegy and mcostuctue n coheent phase tanstons. Comm. Pue Appl. Math. 7 199), No., 05 35. 16 Kstensen J.: Lowe semcontnuty n spaces of wealy dffeentable functons. Math. Ann. 313 1999), No., 653 710. 17 Macelln P.: Appoxmaton of quasconvex functons and lowe semcontnuty of multple ntegals. Manuscpta Math., 51 1985), 1 8. 18 Meyes N.G.: uas-convexty and lowe sem-contnuty of multple vaatonal ntegals of any ode. Tans. Ame. Math. Soc., 119 1965), 15 19. 19 Moey, C.B.: uas-convexty and the lowe semcontnuty of multple ntegals. Pacfc J. Math., 195), 5 53. 0 Mülle S.: Vaatonal models fo mcostuctues and phase tanstons. Lectue Notes, MPI Lepzg, 1998. 1 Owen D.R., Paon R.: Second-ode stuctued defomatons. Ach. Raton. Mech. Anal., 155 000), No 3, 15 35. Rvèe T., Sefaty S.: Lmtng doman wall enegy fo a poblem elated to mcomagnetcs. Comm. Pue Appl. Math. 5 001), No. 3, 9 338. 3 Toupn, R.A.: Theoes of elastcty wth couple-stess. Ach. Raton. Mech. Anal., 17 196), 85 11.
8 F. CAGNETTI Depatment of Mathematcal Scences, Canege Mellon Unvesty, Pttsbugh, PA 1513, USA E-mal addess, Flppo Cagnett: cagnett@andew.cmu.edu