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Universiä des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichung 6.1 Mahemaik Preprin Nr. 223 Differeniabiliy and higher inegrabiliy resuls for local minimizers of spliing-ype variaional

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Universiä des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichung 6.1 Mahemaik Preprin Nr. 223 Differeniabiliy and higher inegrabiliy resuls for local minimizers of spliing-ype variaional inegrals in 2D wih applicaions o nonlinear Hencky-maerials Michael ildhauer and Marin Fuchs Saarbrücken 28 Fachrichung 6.1 Mahemaik Preprin No. 223 Universiä des Saarlandes submied: Differeniabiliy and higher inegrabiliy resuls for local minimizers of spliing-ype variaional inegrals in 2D wih applicaions o nonlinear Hencky-maerials Michael ildhauer Saarland Universiy Deparmen of Mahemaics P.O. ox D-6641 Saarbrücken Germany Marin Fuchs Saarland Universiy Deparmen of Mahemaics P.O. ox D-6641 Saarbrücken Germany Edied by FR 6.1 Mahemaik Universiä des Saarlandes Posfach Saarbrücken Germany Fax: WWW: hp://www.mah.uni-sb.de/ Absrac We prove higher inegrabiliy and differeniabiliy resuls for local minimizers u: R 2 R M, M 1, of he spliing-ype energy [h 1( 1 u ) + h 2 ( 2 u ) dx. Here h 1, h 2 are raher general N-funcions and no relaion beween h 1 and h 2 is required. The mehods also apply o local minimizers u: R 2 R 2 of he funcional [h 1( div u ) + h 2 ( ε D (u) ) dx so ha we can include some varians of so-called nonlinear Hencky-maerials. Furher exensions concern non-auonomous problems. 1 Inroducion Over he las wo decades increasing aenion has been paid o he quesion of inerior regulariy (i.e. higher inegrabiliy of he gradien or even coninuiy of he firs weak derivaives) of local minimizers u: R n R M of variaional inegrals I[u, = H( u) dx (1.1) wih anisoropic energy densiy H: R nm [, ). Here, roughly speaking, H is called an anisoropic inegrand if we have λ( Z ) Y 2 D 2 H(Z)(Y, Y ) Λ( Z ) Y 2 (1.2) wih funcions λ, Λ: [, ) [, ), which do no saisfy an esimae of he form c 1 Λ λ c 2 (1.3) wih posiive consans c 1, c 2. Much of he lieraure is devoed o he invesigaion of he scalar case (i.e. M = 1) and o he closely relaed siuaion ha M 1 ogeher wih he requiremen ha H depends on he modulus of he gradien. We refer o he papers of Choe [Ch, of Fusco/Sbordone [FuS, of Marcellini [Ma1, [Ma2, of Marcellini/Papi [MP, of Mingione/Siepe [MS as well as o he work [AF and he references quoed herein, where he ineresed reader will find inerior regulariy heorems for a variey of anisoropic energies. If n 3 ogeher wih M 2, hen mainly he parial regulariy of local minimizers is discussed as done for example by Acerbi/Fusco [AF, Cupini/Guidorzi/Mascolo [CGM, Esposio/Leonei/Mingione [ELM1, Passarelli Di Napoli/Siepe [PS and he auhors [F1, [F6. In hese papers so-called anisoropic (p, q)-growh is considered, which means ha λ( Z ) Z p 2, Λ( Z ) Z q 2 (1.4) holds for exponens 1 p q , and almos everywhere regulariy follows if in addiion o (1.4) we have an esimae of he form q c(n)p, (1.5) AMS Subjec Classificaion: 49N6, 742, 74G4, 74G65 Keywords: local minimizers, vecor case, nonsandard growh, wo-dimensional problems, regulariy, nonlinear Hencky-maerials 1 where c(n) is large for low dimensions n, bu c(n) 1 as n. Le us noe ha under some exra assumpions on he srucure of H (1.5) can be replaced by weaker resricions a leas if he case of locally bounded local minimizers is considered. For an overview on he hisory as well as for a collecion of recen conribuions mainly concerning anisoropic (p, q)-growh we refer o [i. A very naural class of anisoropic problems arises if we consider inegrands H( u) which spli ino a sum of sricly convex funcions, each of hem depending on differen parial derivaives, for example H( u) = H 1 ( u) + H 2 ( n u), u := ( 1 u,..., n 1 u), (1.6) where H 1 and H 2 migh be of power growh wih differen growh raes p and q in he sense ha D 2 H 1 ( ξ) ξ p 2, D 2 H 2 (ξ n ) ξ n q 2, ξ = ( ξ, ξ n ) R nm. (1.7) Le 2 p q. Then from (1.6) and (1.7) we deduce he validiy of (1.2) and (1.4) wih p := 2 and q := q, and (1.5) reads as q 2c(n), which means ha we canno benefi in any way from he value of p if we reduce he seing described above hrough (1.6) and (1.7) o he unsrucured requiremen (1.2) ogeher wih (1.4) and (1.5). In he papers [F5, [F6, [F7 and [FZ2 and [FZ3 we showed how o ge much beer resuls by working wih echniques based on he spliing srucure of he inegrand, for example in he scalar case and under he naural hypohesis ha he local minimizer is locally bounded we could show inerior C 1,α -regulariy for local minimizers of he energy wih densiy H( u) = n i=1 (1 + iu 2 ) p i/2 independen of he choices of p i 1. In he presen paper we now concenrae on spliing inegrals (1.6) in wo dimensions including he vecorial siuaion (i.e. M 1) and working wih he following hypoheses: le for Z R 2M H(Z) = h 1 ( Z 1 ) + h 2 ( Z 2 ) (1.8) wih funcions h 1, h 2 : [, ) [, ) of class C 2 s.. for h = h 1 and h = h 2 i holds h is sricly increasing and convex ogeher wih h h() () and lim = ; here is a consan k s.. h(2) kh() for all ; (A1) (A2) for an exponen ω and a consan a i holds h () h () a(1 + 2 ) ω h () 2 for all. Le us draw some conclusions from (A1) (A3): (A3) i) (A1) implies ha h() = = h () and h () for . From (A3) i follows ha h ()/ is increasing, moreover we ge h() h () 2 /2. In paricular h is a N-funcion (see [Ad) of a leas quadraic growh. 2 ii) The ( 2)-propery saed in (A2) implies h() c( m + 1) for some exponen m 2, hence by he convexiy of h h () c( m 1 + 1), where here and in he following c denoes a consan whose value may vary from line o line. iii) Combining (A2) wih he convexiy of h we see ha kh () h() h (),. (1.9) iv) For Y = (Y 1, Y 2 ), Z = (Z 1, Z 2 ) R 2M we have [ 2 min h i ( Z i ), h i( Z i ) Y i 2 D 2 H(Z)(Y, Y ) Z i=1 i [ 2 max h i ( Z i ), h i( Z i ) Y i 2, Z i so ha by (A3) 2 i=1 h i( Z i ) Y i 2 D 2 H(Z)(Y, Y ) Z i and for a suiable exponen q 2 i follows i=1 2 a(1 + Z i 2 ) ω 2 i=1 h i( Z i ) Y i 2, (1.1) Z i c Y 2 D 2 H(Z)(Y, Y ) C(1 + Z 2 ) q 2 2 Y 2, (1.11) he firs inequaliy being a consequence of i). Definiion 1.1 Le R 2 and le H from (1.8) saisfy (A1) (A3). Then a funcion u W 1 1,loc (; RM ) is called a local minimizer of he funcional I from (1.1) iff I[u, and I[u, I[v, for all v W 1 1,loc (; RM ) such ha sp(u v), where is any subdomain of wih compac closure in. For a definiion of he Sobolev classes W k p,loc (; RM ) and relaed spaces we refer he reader o [Ad. Our firs resul is he following Theorem 1.1 Le n = 2 and le H saisfy (1.8) ogeher wih (A1) (A3). Suppose furher ha u W 1 1,loc (; RM ) locally minimizes he funcional I from (1.1). Then we have: i) u belongs o L loc (; R2M ) for any finie. ii) If (A3) holds wih ω 2, hen u C 1,α (; R M ) for any α 1. Remark 1.1 We emphasize ha in i) no resricion on he value of ω is required. 3 Remark 1.2 From he proof i will become clear ha he resuls of Theorem 1.1 are also rue for local minimizers of [h 1( u ) + h 2 ( 2 u ) dx or of [h 1( 1 u ) + h 2 ( u ) dx provided (A1) (A3) hold for h 1 and h 2. Remark 1.3 Le us compare Theorem 1.1 o our previous works on spliing funcionals on plane domains: i) In [F4 we discussed he case of densiies H 1 ( 1 u) + H 2 ( 2 u) wih funcions H i : R M [, ) s.. for i = 1, 2 and Y, Z R M λ(1 + Z 2 ) p i 2 2 Y 2 D 2 H i (Z)(Y, Y ) Λ(1 + Z 2 ) p i 2 2 Y 2 for exponens 2 p 1 p 2 and proved par ii) of Theorem 1.1 under he assumpion p 2 2p 1. ii) This resul was improved in [F5, Theorem 1, c) and Remark 4, by showing ha he hypohesis p 2 2p 1 can be dropped in case ha 2 p 1 p 2 . iii) In [F7, Theorem 2.2, we considered he densiy H( u) = h 1 ( 1 u ) + h 2 ( 2 u ), where h 1, h 2 saisfy (A1) (A3) wih ω = and where h 1 () h 2 () for large values of is required. Then we obained he resul of Theorem 1.1, i). Now, in he presen seing, we impose no ordering relaion like h 1 h 2 on h 1 and h 2, moreover a leas for par i) of he heorem here is also no limiaion on he value of ω. Nex we pass o non-auonomous densiies of he form H(x, Z) = h 1 (x, Z 1 )+h 2 (x, Z 2 ), x, Z = (Z 1, Z 2 ) R 2M, wih funcions h i (x, ) saisfying (A1) (A3) uniformly in x (replacing h i by h i, ec.) and for which (α, i = 1, 2) x α h i(x, ) c h i(x, ), x, (A4) holds. Then we have Theorem 1.2 Le H(x, Z) saisfy he modified se of assumpions (1.8), (A1) (A3) and le (A4) hold. Then, if u W 1 1,loc (; RM ) locally minimizes H(x, u) dx = h 1 (x, 1 u ) dx + he saemens of Theorem 1.1 coninue o hold. h 2 (x, 2 u ) dx, Remark 1.4 A ypical example o which Theorem 1.2 applies is he energy [ [ (1 + 1 u 2 ) p(x) [(1 + 2 u 2 ) q(x) 2 1 dx wih funcions p(x), q(x) 2 having (locally) bounded gradiens. We noe ha he isoropic case u p(x) dx was discussed earlier by Coscia/Mingione [CM. 4 Remark 1.5 Since we deal wih local minimizers and discuss inerior regulariy, i is sufficien o know ha in he non-auonomous case he bounds (A1) (A4) are uniform in x for subdomains. As an applicaion of he argumens used for he proof of Theorem 1.1 we also obain regulariy resuls for a cerain class of nonlinear elasic maerials in 2D. Le n = M = 2. Then, according o [Ze, he energy funcional of a nonlinear Hencky maerial is given by [ λ E[u, := 2 (div u)2 + ϕ( ε D (u) ) dx, where λ denoes a posiive consan and where ε(u) is he symmeric par of he gradien of he deformaion u: R 2. ε D (u) := ε(u) 1 div u1 is he deviaoric par of ε(u), and 2 since he above model is used as an approximaion for plasiciy, he densiy ϕ usually is of nearly linear growh which means ϕ() = ln(1 + ) or ϕ() = (1 + 2 ) s/2 1 for some s 1 close o 1. From he work of Frehse and Seregin [FrS he inerior C 1,α - regulariy of local minimizers of he funcional E follows for he logarihmic case as well as for he power growh case wih s 2. In [F3 we gave a sligh exension up o s 4 and for any s under he addiional hypohesis ha (for some reason) we have he informaion div u L s loc (). Now we can remove hese resricions, which enables us o discuss energies having raher general growh w.r.. div u and ε D (u), precisely: Theorem 1.3 Le n = M = 2, le (A1) (A3) hold for he funcions h 1, h 2, and consider a local minimizer u of he energy [ h 1 ( div u ) + h 2 ( ε D (u) ) dx. Then u is in he space L loc (; R2 2 ) for any finie exponen. If ω 2 holds in (A3), hen his can be improved o u C 1,α (; R 2 ) for any α 1. In paricular we have inerior differeniabiliy for he choices h 1 () = 2, h 2 () = (1 + 2 ) s/2 1 wih s 2. Remark 1.6 Of course a non-auonomous varian of Theorem 1.3 can be obained in he spiri of Theorem 1.2. Our paper is organized as follows: in Secion 2 we give he proof of Theorem 1.1, he necessary adjusmens concerning he non-auonomous case are presened in Secion 3. In Secion 4 we briefly skech he siuaion for funcionals relaed o he energy modeling nonlinear Hencky-maerials. A class of energies saisfying our hypoheses is shorly discussed in he appendix. 2 Proof of Theorem 1.1 Le (A1) (A3) hold and consider a local minimizer u of he funcional I from (1.1). As oulined for example in [F4 he following calculaions can be jusified by working wih a local regularizaion wih exponen q inroduced in (1.11) having a sufficien degree of regulariy, which follows from he resuls of [GM or [Ca1. 5 Le η C (). Then we have (from now on summaion w.r.. indices repeaed wice and his convenion is used boh for Greek and for Lain indices) = α [DH( u) : (η 2 α u) dx, hence an inegraion by pars yields D 2 H( u)( α u, α u)η 2 dx = α [DH( u) : ( η 2 α u) dx = DH( u) : α [ η 2 α u dx. (2.1) Here : is he scalar produc of marices and denoes he ensor produc of vecors. From he firs inequaliy in (1.11) we deduce l.h.s. of (2.1) c 2 u 2 η 2 dx. (2.2) For he r.h.s. of (2.1) we observe (w.l.o.g. η 1) DH( u) : α [ η 2 α u dx [ c h i( i u ) 2 u η η dx + h i( i u ) u 2 η 2 dx ε η 2 2 u 2 dx + c(ε) η 2( h 1( 1 u ) 2 + h 2( 2 u ) 2) dx +c [h 1( 1 u ) 2 + h 2( 2 u ) 2 + u 2 2 η dx, where ε is arbirary and where we have used Young s inequaliy several imes. If ε is small enough and if we use (2.2), he ε-erm can be absorbed in he l.h.s. of (2.1). Recalling he lower bound h i () c 2, we arrive a η 2 D 2 H( u)( α u, α u) dx c ( η η ) ( h 1( 1 u ) 2 + h 2( 2 u ) 2 + H( u) ) dx. (2.3) The r.h.s. of (2.3) is handled using ideas of [Fu: le us fix a subdomain and consider discs r (z) R (z). Le furher η 1 on r (z), sp η R (z) and l η c(r r) l, l = 1, 2. Denoing by c( ) consans depending on he (finie) energy of u over, we ge from (2.3) D 2 H( u)( α u, α u) dx r (z) (R r) [c( 2 ) + c R (z) 6 ( h 1 ( 1 u ) 2 + h 2( 2 u ) 2) dx. (2.4) For any L we have using (1.9) h 1( 1 u ) 2 dx = h 1( 1 u ) 2 dx + h 1( 1 u ) 2 dx R (z) R (z) [ 1 u L R (z) [ 1 u L h 1(L) 2 πr 2 + cl 2 h 1 ( 1 u ) 2 dx R (z) [ 1 u L πr 2 h 1(L) 2 + cl 2 h 1 ( 1 u ) 2 dx, R (z) and he same esimae is rue for h 2. Le L := 1 λ 1 R r for some λ . Recalling h i(l) 2 cl 2m 2 we deduce from (2.4) and he above inequaliies for a suiable posiive exponen β D 2 H( u)( α u, α u) dx r (z) ( c(, λ)(r r) β + cλ 2 h1 ( 1 u ) 2 + h 2 ( 2 u ) 2) dx, (2.5) R (z) and (2.5) is valid for all λ and all discs R (z). Le ρ (, R) and define r = (ρ + R)/2. Wih η C ( r (z)), η 1, η 1 on ρ (z) and η c/(r ρ) (= 2c/(R ρ)) we find wih Sobolev s inequaliy ( h1 ( 1 u ) 2 + h 2 ( 2 u ) 2) dx ρ (z) r (z) ( ηh1 ( 1 u ) ) 2 dx + r (z) [ c η h i ( i u ) dx + r(z) c(r ρ) 2 [ R (z) H( u) dx r(z) 2 [ +c h 1( 1 u ) 1 u dx + r(z) c( )(R ρ) 2 + c[ ( ηh2 ( 2 u ) ) 2 dx h 1( 1 u ) 1 u dx + r(z) h 2( 2 u ) 2 u dx r(z) 2 h 2( 2 u ) 2 u dx 2 7 In [... 2 we can apply Hölder s inequaliy o ge (again using (1.9)) [... 2 h 1( 1 u ) 1 u 2 dx 1 u h r(z) 1 u 1( 1 u ) dx r(z) h + 2( 2 u ) 2 u 2 dx 2 u h r (z) 2 u 2( 2 u ) dx r (z) { h c H( u) dx 1( 1 u ) 1 u 2 dx r (z) r (z) 1 u + r (z) c( ){... }. h 2( 2 u ) 2 u 2 dx 2 u If we use he firs inequaliy in (1.1) wih he choices Z = u and Y = 1 u, 2 u, hen {... } D 2 H( u)( α u, α u) dx, and from (2.5) we finally deduce ( h1 ( 1 u ) 2 + h 2 ( 2 u ) 2) dx ρ (z) r(z) c( )(R ρ) 2 + c(, λ)(r r) β + c( )λ 2 } R (z) ( h1 ( 1 u ) 2 + h 2 ( 2 u ) 2) dx. Since R r = 1 (R ρ) and since we may assume ha β 2, he above inequaliy implies 2 afer appropriae choice of λ ( h1 ( 1 u ) 2 +h 2 ( 2 u ) 2) dx c( )(R ρ) β + 1 ( h1 ( 1 u ) 2 +h 2 ( 2 u ) 2) dx, ρ(z) 2 R (z) which means (see [Gi, Lemma 3.1, p. 161) ha h 1 ( 1 u ) 2 + h 2 ( 2 u ) 2 is in he space L 1 loc () (uniformly w.r.. he hidden approximaion). u hen (2.5) shows he same for D 2 H( u)( α u, α u) and as remarked before (2.2) his yields u W2,loc 2 (; RM ) (again uniform w.r.. he approximaion). Sobolev s heorem finally implies par i) of Theorem 1.1. For proving ii) we proceed similar o Theorem 1, c) in [F5 by reducing he siuaion o a lemma on higher inegrabiliy esablished in [FZ1. Wih η C () and P R 2M we have = α [DH( u) : (η 2 α [u P (x)) dx and from his equaion we obain η 2 Φ 2 dx = 2 ηd 2 H( u)( α u, α [u P (x) η) dx, (2.6) where we have abbreviaed Φ := D 2 H( u)( α u, α u) Noe ha by he foregoing calculaions Φ is in he space L 2 loc (). On he r.h.s. of (2.6) we apply he Cauchy-Schwarz inequaliy o he bilinear form D 2 H( u) and ge from (2.6) afer choosing η s.. η 1 on r (z ), η 1, sp η 2r (z ), η c/r for a disc 2r (z ) Φ 2 dx c Φ D 2 H( u) 1 2 u P dx. (2.7) r (z ) r 2r (z ) The second inequaliy in (1.1) shows [ D 2 H( u) 1 2 c (1 + 1 u 2 ) ω 4 =: c [ ψ1 + ψ 2, h 1( 1 u ) + (1 + 2 u 2 ) ω h 2( 2 u ) 4 1 u 2 u and if we le ψ := ( ψ ψ 2 2) 1/2, hen exacly he same argumens leading o (3) in [F5 enable us o derive from (2.7) he inequaliy [ Φ 2 dx 1 2 c[ ( ψφ) 4 3 dx 3 4. (2.8) r (z ) 2r (x ) Noe ha during he proof of (2.8) one needs he informaion ha 2 u cφ cφ ψ which follows from our assumpions concerning h 1, h 2. In order o proceed as in [F5 we have o check ha exp(β ψ 2 ) L 1 loc( ) (2.9) is rue for any β . Le us define 1 u h ψ 1 := 1 () d, ψ 2 := The firs inequaliy in (1.1) shows 2 u ψ ψ 2 2 cφ 2, h 2 () d. so ha ψ 1 and ψ 2 belong o W 1 2,loc () and herefore ψ := (ψ2 1 + ψ 2 2) 1/2 is in he same space. y Trudinger s inequaliy (see Theorem 7.15 of [GT) we find β s.. for discs ρ we have ρ exp(β ψ 2 ) dx c(ρ). (2.1) We have a.e. on [ 1 u 1 (recalling (1.9)) ψ 1 c 1 u ω h 1( 1 u ) 2 = c 1 u ω 1( 2 1 u h 1 u 1( 1 u ) ) 1 2 c 1 u ω 2 1 h 1 ( 1 u ) 1 2, whereas ψ 1 1 u 1 u /2 h 1 () d ch 1 ( 1 u ) 1 2 9 (see (A2), (1.9)), hence ψ 1 c 1 u ω 2 1 ψ 1 on [ 1 u 1. A he same ime i holds and for small δ we obain ψ 1 ch 1 ( 1 u ) 1 2 c 1 u m 2, ψ 1 cψ 1 δ 1 1 u ω 2 1+δ m 2 (2.11) on [ 1 u 1. Since we assume ω 2 in par ii) of Theorem 1.1, we can fix δ s.. we have ω 1 + δ m . Young s inequaliy applied on he r.h.s. of (2.11) hen gives for any 2 2 µ ψ 1 2 µψ1 2 + c(µ) on [ 1 u 1. (2.12) On [ 1 u 1 we jus observe ψ 2 1 c c + ψ 1 µψ c(µ), hence he inequaliy (2.12) holds on, and obviously he same argumens apply o ψ 2, ψ 2. This shows ψ 2 µψ 2 + c(µ) a.e. on (2.13) for any µ . Le us fix β . Then (by (2.13)) exp(β ψ 2 ) dx c(µ, β) exp(βµψ 2 ), ρ ρ and if we choose µ = β /β, hen he desired claim (2.9) follows from (2.1). Now we can complee he proof of Theorem 1.1, ii) as done in [F5. 3 Proof of Theorem 1.2 Le us firs assume ha u is sufficienly regular so ha we do no have o argue wih soluions of regularized problems or wih difference quoiens. Then, wih he noaion H = H(x, P ), he counerpar of (2.1) reads as DpH(, 2 u)( α u, α u)η 2 dx [ + D p H (, u) : (η 2 α u) dx x α = D p H(, u) : α ( η 2 α u) dx, where he second erm on he l.h.s. is he new one. However, due o assumpion (A4), he behavior of his erm is of he same qualiy as of he r.h.s. and we herefore have (2.5). The nex sep in Secion 2 is o make use of Sobolev s inequaliy, which in he non-auonomous case jus gives uncriical new erms and as before we arrive a u L loc () for all . Now, following he argumens of Secion 2 leading o par ii) of Theorem 1.1, we again obain some exra erms in he non-auonomous case under consideraion. 1 u in Secion 4 of [FZ1 and Secion 6 of [F4 i is described in deail how hese exra erms can be handled leading o a generalized version of (2.8) o which Lemma 1.2 of [FZ1 sill is applicable. Thus, as skeched in Secion 2, he proof of Theorem 1.2 would be complee if our smoohness assumpion can be guaraneed. As oulined in [ELM2 he usual local regularizaion procedure canno be applied, which means ha if we fix a disk compacly conained in and consider he mollificaion (u) ε of our local minimizer, hen he convergence H(x, (u) ε ) dx H(x, u) dx as ε (3.1) may fail o hold due o he possibiliy of he occurrence of Lavreniev s phenomenon. In he auonomous case (3.1) easily follows from Jensen s inequaliy and enables us o sudy he regularized problems (as done in Secion 2) H δ ( w) dx min in (u) ε + W q 1 (; R M ), where H δ = δ(1 + 2 ) q/2 + H wih q from (1.11) and δ = δ(ε) being defined in a suiable way. In fac, (3.1) is he key ingredien for proving ha he (regular) soluions u δ of he auxiliary problems converge owards our local minimizer u on he disk so ha all uniform esimaes obained for he sequence {u δ } finally coninue o hold for u. In he non-auonomous case we now follow ideas of Marcellini [Ma2 and of Cupini, Guidorzi and Mascolo [CGM by inroducing a regularizaion from below, which means he inegrand H(x, P ) is replaced by an appropriae sequence H Θ (x, P ) of inegrands having quadraic growh w.r.. P R 2M and s.. H Θ (x, P ) H(x, P ) as Θ. We noe ha a relaed ype of approximaions also occurs in Secion 3 of [F2 bu we canno refer o his since now he seing is differen. Le us pass o he deails by recalling ha H(x, P ) = h 1 (x, P 1 ) + h 2 (x, P 2 ), where h(x, ) := h i (x, ), i = 1, 2, saisfies (A1) (A4). Since he approximaion procedure is done w.r.. he second variable we jus wrie h(). Then we define g() := h (), i.e. h() = sg(s) ds, and by (A1) and (A3) g is increasing and saisfies g() = h () . Now we fix Θ and consider η = η Θ C 1 ([, )) s.. η 1, η, η c/θ, η 1 on [, 3Θ/2 and η on [2Θ, ). Moreover, for all we le g Θ () :=

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