Hölder Continuity for Local Minimizers of a Nonconvex Variational Problem

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1 Jounal of Convex Analysis Volume ), No., Hölde Continuity fo Local Minimizes of a Nonconvex Vaiational Poblem Giovanni Cupini Dipatimento di Matematica U. Dini, Univesità di Fienze, Viale Mogagni 67/A, Fienze, Italy cupini@math.unifi.it Anna Paola Miglioini Dipatimento di Matematica U. Dini, Univesità di Fienze, Viale Mogagni 67/A, Fienze, Italy amiglio@math.unifi.it Received Febuay, 00 Revised manuscipt eceived Febuay 05, 003 We conside integal functionals of the Calculus of Vaiations whee the enegy density is a continuous function with p-gowth, p > 1, unifomly convex at infinity with espect to the gadient vaiable. We pove that local minimizes ae α-hölde continuous fo all α < 1. Keywods: Local minimize, egulaity, nonconvex functional 1991 Mathematics Subject Classification: 49N60, 35J0 1. Intoduction Conside the integal functional of the Calculus of Vaiations Fu; Ω) := Ω fx, ux), Dux)) dx, 1) whee Ω is an open bounded subset of R N and f = fx, u, ξ) : Ω R R N [0, + ) is a Caathéodoy function, i.e. f is measuable in x and continuous in u, ξ), satisfying the p-gowth condition p > 1) ξ p fx, u, ξ) L1 + ξ p ). A function u W 1,p loc Ω) is a local minimize of F in Ω if F u; spt v u)) F v; spt v u)), fo evey v W 1,p loc Ω) such that spt v u) Ω. Well known esults due to Giaquinta and Giusti [13, 15] ensue that local minimizes of F ae locally α-hölde continuous fo some α < 1. Accoding to Meyes example in [19], when f is not continuous in Ω R R N, the α-hölde continuity fo all α < 1 cannot be achieved, even if f is twice diffeentiable and unifomly convex with espect ISSN / $.50 c Heldemann Velag

2 390 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... to ξ. Howeve, if f is of class C in ξ, unifomly elliptic in this vaiable and fo evey x, y Ω, u, v R and ξ R N fx, u, ξ) fy, v, ξ) ω x y + u v )1 + ξ p ), ) whee ωt) is a modulus of continuity that gows as t δ, δ > 0, then Du is Hölde continuous see Giaquinta-Giusti [14], Giaquinta-Modica [16], Manfedi [18]). Recent esults show that failing the diffeentiability of the integand, the Lipschitz continuity of local minimizes, o at least the Hölde continuity fo evey exponent, still holds unde suitable convexity assumptions. Fonseca and Fusco in [9] study the egulaity of local minimizes of functionals with a nondiffeentiable integand f = fξ) independent of x, u). They pove that if f has p-gowth and satisfies a unifom convexity condition, i.e. thee exists a constant ν > 0 such that fo all ξ, η in R N ) ξ + η f 1 fξ) + 1 fη) ν1 + ξ + η ) p ξ η, 3) then local minimizes ae locally Lipschitz continuous. Notice that when f is of class C, then 3) is equivalent to the usual unifom ellipticity condition D fξ)λ, λ ν ξ ) p λ, fo some ν 0 > 0. The geneal case f = fx, u, ξ) has been studied by Cupini, Fusco and Petti in []. They pove that if ) and 3) hold, then local minimizes ae locally α-hölde continuous fo all α < 1. We emak that even in the simple case f = fx, ξ), N = p =, the Lipschitz continuity cannot be achieved see Example 3. in []). Related esults can be found in [4, 7, 11]. In a ecent pape Fonseca, Fusco and Macellini [10] establish existence and egulaity of minimizes of enegy integals with f = fx, ξ) subject to Diichlet bounday conditions, whee the unifom convexity popety of f is satisfied at infinity, i.e. 3) holds fo all x Ω and evey ξ, η endpoints of a segment contained in the complement of a ball B R 0) in R N. In paticula, they pove the Lipschitz continuity of local minimizes when f is a continuous function with p-gowth, unifomly convex at infinity, such that fo ξ > R the vecto field x D ξ fx, ξ) is weakly diffeentiable with D ξx fx, ξ) L1 + ξ ). We explicitly notice that in the paticula case f = fξ) the diffeentiability assumption can be emoved. In this case the p-gowth and the unifom convexity at infinity ae sufficient conditions to the Lipschitz continuity of local minimizes see Theoem. below). In this pape we conside the geneal case of nonconvex integands f = fx, u, ξ) with p-gowth, both unifomly convex and continuous at infinity and we pove that local minimizes of F ae α-hölde continuous fo any exponent α < 1. Moe pecisely, we pove the following Theoem 1.1. Let f : Ω R R N [0, + ) be a Caathéodoy function satisfying A1) thee exist p > 1 and L > 0 such that fo a.e. x Ω, fo evey u R and ξ R N 0 fx, u, ξ) L1 + ξ p ) ;

3 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a A) f is unifomly convex at infinity with espect to ξ, i.e. thee exist R > 0 and ν > 0 such that if the segment [ξ, η] is contained in the complement of the ball B R 0), then fo a.e. x Ω and fo evey u R f x, u, ξ + η ) 1 fx, u, ξ) + 1 fx, u, η) ν1 + ξ + η ) p ξ η ; A3) f is continuous at infinity with espect to the pai x, u), i.e. fo evey x, y Ω, fo evey u, v R and ξ R N \ B R 0) fx, u, ξ) fy, v, ξ) ω x y + u v )1 + ξ p ), whee ω : [0, + ) [0, + ) is a continuous, inceasing, bounded function such that ω0) = 0. If u W 1,p loc Ω) is a local minimize of the functional F defined in 1), then u C0,α fo all α < 1. loc Ω) To bette undestand the type of functionals we deal with, notice that A1) and A) ae equivalent to assume that fx, u, ξ) = c1 + ξ ) p + gx, u, ξ), with c > 0, g bounded fom below, with p-gowth fom above and convex at infinity, i.e. fo a.e. x Ω, fo evey u R and fo evey ξ 1, ξ R N endpoints of a segment contained in the complement of B R 0) 1 [gx, u, ξ 1) + gx, u, ξ )] g x, u, ξ ) 1 + ξ. Fo moe details, we efe to [10, Section.1], [3, Section ] and to Section below. The plan of the pape is the following. In Section we collect peliminay esults concening functions satisfying A1) and A), highe integability esults fo minimizes of functionals with p-gowth and an iteation lemma. In Section 3 we study the egulaity of a local minimize u of Iu; Ω) =: Ω fx, Dux)) dx, when f = fx, ξ) is a convex function with espect to ξ and satisfies A1) A3). We appoximate Iu; Ω) with a sequence of functionals I h ) with integands f h = f h x, ξ) of class C 1 with espect to ξ, satisfying the same assumptions as f. Fixed x 0 ) Ω, let u h be the minimize of I h w; x 0 )) in the Diichlet class u + W 1,p 0 x 0 )), whee u is the local minimize of the functional I. Fo each u h we establish an integal estimate with constants independent of h. Moeove, we pove that u h conveges in the weak topology of W 1,p to a function u, which tuns out to be a minimize of I in u + W 1,p 0 x 0 )). Passing to the limit in the integal estimate poved fo each u h, we infe that the estimate

4 39 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... is satisfied by u and, fom the unifom convexity at infinity, by u too. Fom this estimate the Hölde continuity of u fo any exponent follows by a classical agument. In Section 4 we pove Theoem 1.1. Fistly we conside the supplementay assumption of convexity of f with espect to ξ. We compae the local minimize u of F with the minimize of a functional I of the type studied in Section 3. An essential tool fo this compaison is the Ekeland vaiational pinciple. The convexity assumption on f is emoved via a elaxation agument.. Peliminay esults Let us conside the integal functional Fu; Ω) := Ω fx, ux), Dux)) dx, 4) whee Ω is a bounded open subset of R N and u : Ω R is a scala function in W 1,p loc Ω), with p > 1. We denote by x 0 ) the open ball with cente x 0 and adius ; we omit x 0 if no confusion may aise. In the sequel, c is a positive constant which may take diffeent values fom line to line. We ecalled the definition of local minimize of F in the Intoduction. Moe geneally, u is a Q-minimize of F if thee exists Q 1 such that F u; spt v u)) QF v; spt v u)), fo all v W 1,p loc Ω) such that spt v u) Ω. We state some consequences of assumptions A1) and A), poved in [10] in the case f = fξ). The genealization to ou case is staightfowad. Theoem.1. Let f = fx, u, ξ) : Ω R R N [0, + ), be a Caathéodoy function. Then A1) and A) imply: i) thee exist c 1 ν) and c p, L, R, ν) such that fo a.e. x Ω and fo evey u, ξ) R R N fx, u, ξ) c 1 ξ p c ; ii) thee exist R 0 > R and ν 0 > 0 depending only on p, L, R and ν, such that fo a.e. x Ω and fo evey u R and ξ R N \ B R0 0) thee exists q ξ x, u) R N such that and fo evey η R N q ξ x, u) cp, L)1 + ξ ), fx, u, η) fx, u, ξ) + q ξ x, u), η ξ + ν ξ + η ) p ξ η. Moeove, if ξ fx, u, ξ) is C 1 R N \ B R0 0)) then q ξ x, u)=d ξ fx, u, ξ); iii) if ξ f x, u, ξ) is the convex envelope of ξ fx, u, ξ) then f = f in Ω R R N \ B R0 0)). We undeline that Theoem.7 of [10] entails the following egulaity esult.

5 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a Theoem.. Let f = fξ) : R N [0, + ) be a continuous function with p-gowth, satisfying the unifom convexity 3) at infinity. If u W 1,p loc Ω) is a local minimize of the functional fdux)) dx, Ω then u is locally Lipschitz continuous in Ω. Moeove, thee exists c, depending on N, p, L, R and ν, such that fo evey x 0 ) Ω [ sup Du p c / x 0 ) x 0 ) ] 1 + Du p ) dx. Now we state a simple algebaic esult that will be useful in the sequel see [17], Lemma 8.5 o [], Lemma.). Lemma.3. If p > 1, thee exists c > 0 such that fo evey ξ, η R N 1 + ξ ) p c 1 + η ) p + c 1 + ξ + η ) p ξ η. We will use some egulaity esults of Q-minimizes of integal functionals. The following esult is contained in Theoem 3.1 of [15]. Lemma.4. Let x 0 ) Ω and φ L p x 0 )). If u W 1,p loc x 0 )) is a Q- minimize of the functional w ) 1 + Dwx) p + φx) p+1 dx, 5) then thee exists τ > 1 such that u W 1,pτ loc ). Moeove, thee exists c = cn, p, Q) such that fo evey x 1 ) x 0 ) / x 1 ) Du pτ dx ) 1 τ c 1 + Du p + φ p ) dx. x 1 ) We need also an up-to-the-bounday highe integability esult see e.g. Theoem 6.8 of [17] and Lemma.7 in []). Lemma.5. Let h : x 0 ) R N R be a Caathéodoy function such that ξ p hx, ξ) L1 + ξ p ). If u 0 W 1,q x 0 )), fo a cetain q > p, and v is a minimize of the functional w hx, Dwx)) dx x 0 ) in the Diichlet class u 0 + W 1,p x 0 )), then thee exist s p, q) and c > 0, depending on N, p and L, but neithe on u 0 no, such that v W 1,s x 0 )) and [ x 0 ) ] 1 [ Dv s s dx c x 0 ) ] Du 0 q q ) dx.

6 394 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... Finally, we state an iteation lemma see the poof of Poposition 3.7 in [8]). Lemma.6. Let Z, ψ, ω : [0, T ] [0, + ) be bounded and inceasing functions, ω continuous and ω0) = 0. Suppose that thee exist α, β, γ, c 1 > 0 such that fo evey ɛ > 0 thee exist c ɛ) > 0 such that [ α ] t Zt) c 1 + ωs) + ɛ Zs) + c ɛ)s s) γ ψs) fo evey 0 < t < s T. Then fo evey 0 < δ < α, thee exists T 0 < T such that Zt) c 3 t s) α δ Zs) + c 4 t γ ψs), wheneve 0 < t < s T 0. Hee T 0 and c 3 ae positive constants depending only on α, β, δ and c 1, while c 4 depends also on γ. 3. Regulaity in the case f = fx, Du) In this section we conside the functional Iu; Ω) := Ω fx, Dux)) dx, 6) whee f = fx, ξ) : Ω R N [0, + ) is a Caathéodoy function, satisfying the assumptions A1) A3), that in this case can be stated as follows: A1) the p-gowth condition, i.e. fo a.e. x Ω and fo evey ξ R N 0 fx, ξ) L1 + ξ p ); A) the unifom convexity at infinity, i.e. thee exist R and ν > 0 such that fo a.e. x Ω if [ξ, η] R N \ B R 0), then f x, ξ + η ) 1 fx, ξ) + 1 fx, η) ν1 + ξ + η ) p ξ η ; A3) the unifom continuity in x at infinity, i.e. fo evey x, y Ω and fo evey ξ R N \ B R 0) fx, ξ) fy, ξ) ω x y )1 + ξ p ), whee ω : [0, + ) [0, + ) is a continuous, inceasing and bounded function such that ω0) = 0. Without loss of geneality, we can assume ω to be concave. Recall that by Theoem.1 ii) thee exists R 0 > R and ν 0 > 0 such that fx, η) fx, ξ) + q ξ x), η ξ + ν ξ + η ) p ξ η, 7) fo a.e. x Ω and fo evey ξ R N \ B R0 0) and η R N, whee q ξ x) R N satisfies q ξ x) cp, L)1 + ξ ).

7 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a Moeove, by Theoem.1 i) thee exist c 1, c > 0 such that fo a.e. x Ω and fo evey ξ R N fx, ξ) c 1 ξ p c. 8) With a view to the poof of Theoem 1.1, fixed x 0 ) Ω we conside the functional G ϑ0,u 0 w; ) := Iw; ) + ϑ 0 Dw Du 0 p+1 dx, 9) whee ϑ 0 0 and u 0 W 1,p ). This functional will be useful to pove the egulaity of local minimizes in the geneal case f = fx, u, ξ). We begin by poving an integal estimate fo local minimizes of G ϑ0,u 0 unde the supplementay assumption that f is of class C 1 with espect to ξ and convex in this vaiable. This assumption will be emoved in Poposition 3.. Poposition 3.1. Conside x 0 ) Ω, ϑ 0 0, u 0 W 1,p x 0 )). Let G ϑ0,u 0 be as in 9), with f of class C 1 with espect to ξ and convex in this vaiable, satisfying A1) A 3). Let u W 1,p ) be a minimize of G ϑ0,u 0 in its Diichlet class u+w 1,p 0 ). Then thee exists c = cn, p, L, ν, R) such that fo evey ρ < [ ρ ) ] N 1 + Du p ) dx c + ω) 1 + Du p ) dx p + ϑ 10) 0 Du Du [ω)] 1 0 p dx + c N. Poof. Let v be the minimize in u + W 1,p 0 x 0 )) of the fozen functional w fx 0, Dwx)) dx. The function ξ fx 0, ξ) satisfies the assumptions of Theoem., hence see [9]) v is locally Lipschitz continuous in and thee exists a constant c depending on N, p, L, R and ν such that fo all ρ < ρ ) N 1 + Dv p ) dx c 1 + Dv p ) dx. 8), the minimality of v and A1) imply 1 + Dv p ) dx c 1 + Du p ) dx, 11) so that fo any ρ < Define 1 + Dv p ) dx c ρ ) N ADu, ρ, λ) := {x : Dux) > λ}, BDu, ρ, λ) := {x : Dux) λ}. 1 + Du p ) dx. 1) 13)

8 396 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... Let R 0 and ν 0, depending only on p, L, R and ν, be as in Theoem.1 ii). Using Lemma.3 and 1) Du p dx = Du p dx + Du p dx BDu,ρ,R 0 ) ADu,ρ,R 0 ) c 1 + Dv p ) dx + c 1 + Du + Dv ) p Du Dv dx 14) ADu,ρ,R 0 ) ρ ) N c 1 + Du p ) dx + c 1 + Du + Dv ) p Du Dv dx. ADu,ρ,R 0 ) Let us estimate the last integal using the Eule equation fo G ϑ0,u 0. Fom Theoem.1 ii), the convexity of f and the minimality of u we have 1 + Du + Dv ) p Du Dv dx 1 ν 0 ADu,ρ,R 0 ) ADu,ρ,R 0 ) [fx, Dv) fx, Du) D ξ fx, Du), Dv Du ] dx 1 ν 0 [fx, Dv) fx, Du) D ξ fx, Du), Dv Du ] dx = 1 ν 0 = 1 ν 0 ) ] [fx, Dv) fx, Du) + ϑ 0 D ξ Du Du 0 p+1, Dv Du dx 1 ν 0 D ξ fx, Du) + ϑ 0 D ξ ) Du Du 0 p+1, Dv Du dx ) ] [fx, Dv) fx, Du) + ϑ 0 D ξ Du Du 0 p+1, Dv Du dx, whee ξ is a dummy vaiable fo the gadient. Thus, adding and subtacting fx 0,Dv)dx and fx 0, Du)dx, and using the minimality of v it follows 1 + Du + Dv ) p Du Dv dx 1 ν 0 ADu,ρ,R 0 ) Fom A1) and A3) [fx, Dv) fx 0, Dv)] dx + 1 [fx 0, Du) fx, Du)] dx ν 0 + ϑ ) 0 D ξ Du Du 0 p+1, Dv Du dx. ν 0 [fx, Dv) fx 0, Dv)] dx = [fx, Dv) fx 0, Dv)] dx + ADv,,R) ADv,,R) ω x x 0 ) 1 + Dv p ) dx + L BDv,,R) ω) 1 + Dv p ) dx + cn, p, L, R) N. BDv,,R) [fx, Dv) fx 0, Dv)] dx 1 + Dv p ) dx

9 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a Analogously, [fx 0, Du) fx, Du)] dx ω) 1 + Du p ) dx + cn, p, L, R) N. Theefoe fom 11) and the minimality of v thee exists c, depending only on N, p, L, R and ν, such that ADu,ρ,R 0 ) 1 + Du + Dv ) p Du Dv dx c ω) 1 + Du p )dx + c ϑ 0 Du Du 0 Du Dv dx + c N. This inequality, togethe with 14) and Young inequality, implies that fo evey ρ < thee exists c, depending on N, p, L, R and ν, such that [ ρ ) ] N Du p dx c + ω) 1 + Du p ) dx p + c ω) Du Dv p ϑ0 dx + c Du Du [ω)] 1 0 p dx + c N and fom 11) the thesis follows. We emak that the feezing techinque emploied in the poof of the above poposition has been fist used in [14, 18] and applied in the setting of non standad gowth conditions, up to a cetain extent, in [1]. An appoximation agument allows us to emove the diffeentiability assumption on f. Poposition 3.. Conside x 0 ) Ω, ϑ 0 0, u 0 W 1,p x 0 )). Let G ϑ0,u 0 be as in 9), with f convex with espect to ξ, satisfying A1) A 3). Let u W 1,p ) be a minimize of G ϑ0,u 0 in its Diichlet class u + W 1,p 0 ). Then fo evey ρ < [ ρ ) ] N Du p dx c + ω) 1 + Du p + Du 0 p ) dx with c depending on N, p, L, R and ν. p ϑ0 + c N + c N, [ω)] p+1 Poof. Let σ Cc B 1 0), [0, + )) be a adially symmetic mollifie such that σz) dz = 1, and define B 1 0) f h x, ξ) := B 1 0) f h ) h N satisfies the following assumptions: σz)f x, ξ + 1h ) z dz.

10 398 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... H1) thee exists L, depending on L and p, but not on h, such that fo evey h, fo a.e. x Ω and fo evey ξ R N 0 f h x, ξ) L1 + ξ p ) ; H) thee exists ν 1 > 0, depending on p and ν, but not on h, such that if [ξ, η] R N \ B R+1 0), then fo a.e. x Ω f h x, ξ + η ) 1 f hx, ξ) + 1 f hx, η) ν ξ + η ) p ξ η ; H3) thee exists c = cp) such that fo evey x, y Ω and fo evey ξ with ξ R + 1 f h x, ξ) f h y, ξ) c ω x y )1 + ξ p ). Moeove, fom 8) thee exist c 1, c > 0, independent of h, such that f h x, ξ) c 1 ξ p c, x, ξ) Ω R N. 15) Let u h be a minimize in u + W 1,p 0 ) of G h w; ) := f h x, Dw) dx + ϑ 0 Dw Du 0 p+1 dx. Using 15), the minimality of u h, H1) and Young inequality, it is not difficult to pove that thee exists c = cn, p, L, R, ν, ϑ 0 ), such that Du h p dx c 1 + Dw p + Du 0 p+1 ) dx, fo evey w u + W 1,p 0 ). In paticula Du h p dx c 1 + Du p + Du 0 p ) dx 16) and u h is a Q-minimize of w ) 1 + Dw p + Du 0 p+1 dx. Theefoe, fom Lemma.4 thee exist τ > 1 and c > 0 such that u h W 1,pτ loc ) fo evey h and fo evey x 1 ) / x 1 ) Du h pτ dx ) 1 τ c 1 + Du h p + Du 0 p ) dx. x 1 ) This inequality, togethe with 16), implies that fo evey ρ < thee exists c = cρ, ) such that ) 1 τ Du h pτ dx c 1 + Du p + Du 0 p ) dx. 17)

11 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a Moeove, 16) implies that the sequence u h ) is bounded in W 1,p ). Up to a subsequence, we may assume that thee exists u u + W 1,p 0 ) such that u h u in the weak topology of W 1,p ). Let us pove that u is a minimize of G ϑ0,u 0 defined in 9). We shall use the notations in 13). Since G ϑ0,u 0 is lowe semicontinuous with espect to the weak topology of W 1,p, then fo evey ρ < and k N G ϑ0,u 0 u ; ) lim inf lim sup + lim sup ADu h,ρ,k) { fx, Du h ) dx ] [fx, Du h ) + ϑ 0 Du h Du 0 p+1 dx } fx, Du h )dx + ϑ 0 Du h Du 0 p+1 dx. BDu h,ρ,k) Since f is convex with espect to ξ and A1) holds, then fo a.e. x Ω and fo any ξ and η in R N, fx, ξ) fx, η) cp, L)1 + ξ + η ) ξ η ; 18) theefoe lim 1 c lim h BDu h,ρ,k) BDu h,ρ,k) fx, Du h ) f h x, Du h ) dx 1 + Du h ) dx 1 c lim h 1 + k) BDu h, ρ, k) = 0. 19) Thus, fom 18) and 19) Gu ; ) lim sup + lim sup ADu h,ρ,k) fx, Du h ) dx [f h x, Du h ) + ϑ 0 Du h Du 0 p+1 Fom A1), Hölde inequality, 17) and the minimality of u h we get that G ϑ0,u 0 u ; ) L lim sup + lim sup c k p1 τ) lim sup ] dx. ) 1 τ 1 + Du h p τ )dx ADu h, ρ, k) τ 1 τ ] [f h x, Du) + ϑ 0 Du Du 0 p+1 dx 1 + Du h pτ ) dx + [fx, Du) + ϑ 0 Du Du 0 p+1 ] dx.

12 400 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... The last inequality and 17) imply G ϑ0,u 0 u ; ) c k p1 τ) + G ϑ0,u 0 u; ), whee c = cn, p, L, u, u 0, ϑ 0, ρ, ). Letting k go to infinity and then ρ go to we infe G ϑ0,u 0 u ; ) = G ϑ0,u 0 u; ), so that u is a minimize of G ϑ0,u 0 w; ). Now we compae u with u. Fo evey x we apply Theoem.1 ii) with ξ = 1 Dux) + Du x)) and η = Dux). Thus, fom 7) we have ADu+Du,,R 0 ) ADu+Du,,R 0 ) + ν ADu+Du,,R 0 ) [ fx, Du) f q ξ x), Du Du x, Du + Du dx 1 + Du + Du )] dx )p + Du Du Du dx. 0) Analogously, Theoem.1 ii) with η = Du x) yields [ fx, Du ) f x, Du + Du )] dx ADu+Du,,R 0 ) ADu+Du,,R 0 ) + ν adding 0) and 1) ADu+Du,,R 0 ) ADu+Du,,R 0 ) ADu+Du,,R 0 ) + ν + ν By convexity 1 ADu+Du,,R 0 ) ADu+Du,,R 0 ) BDu+Du,,R 0 ) BDu+Du,,R 0 ) + 1 q ξ x), Du Du dx )p 1 + Du + Du + Du Du Du [fx, Du) + fx, Du )] dx f x, Du + Du ) dx 1 + Du + Du BDu+Du,,R 0 ) ) p + Du Du Du )p 1 + Du + Du + Du Du Du f x, Du + Du ) dx +ϑ 0 fx, Du) dx + ϑ 0 fx, Du ) dx + ϑ 0 Du + Du Du Du 0 p+1 dx dx; dx dx. Du 0 p+1 dx Du Du 0 p+1 dx, 1) )

13 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a theefoe, fom ) we get u + u ) G ϑ0,u 0 ; 1 G ϑ 0,u 0 u; ) + 1 G ϑ 0,u 0 u ; ) ν )p 1 + Du + Du + Du Du Du ν ADu+Du,,R 0 ) ADu+Du,,R 0 ) )p 1 + Du + Du + Du Du Du The minimality of u, togethe with the pevious inequality, yields u + u ) G ϑ0,u 0 ; G ϑ0,u 0 u; ) ν )p 1 + Du + Du + Du Du Du ν ADu+Du,,R 0 ) ADu+Du,,R 0 ) which implies that the Lebesgue measue of the set )p 1 + Du + Du + Du Du Du {x : Dux) + Du x) > R 0 } {x : Dux) Du x) > 0} is zeo. Theefoe, fo evey ρ < Du p dx c Du p dx + c Du + Du p dx c Du p dx + c p R p 0 ρ N + c Du + Du p dx ADu+Du,ρ,R 0 ) = c Du p dx + c ρ N + c Du p dx, ADu+Du,ρ,R 0 ) { Du Du =0} thus thee exists c = cn, p, L, R, ν) such that Du p dx c 1 + Du p ) dx. 3) Let us estimate the ight-hand side. Since f h is of class C 1 with espect to ξ and satisfies H1) H3), fom Poposition 3.1 estimate 10) holds with u eplaced by u h. Hence, fo all ρ <, Du p dx lim inf B ρ [ ρ lim sup { c + lim sup 1 + Du h p ) dx ) N ] } + ω) 1 + Du h p ) dx + c N p c ϑ0 Du [ω)] 1 h Du 0 p dx, dx dx dx. dx,

14 40 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... whee c is independent of h, ρ and. Young inequality implies p cϑ0 Du [ω)] 1 h Du 0 p dx p = cϑ [ ] p [ ] p 0 ϑ0 ) ω) ) Duh Du [ω)] 1 0 p dx ω) ϑ 0 p ϑ c ω) c ω) Du h Du 0 p dx + c Du h p + Du 0 p ) dx + c 0 [ω)] p+1 N p ϑ0 [ω)] p+1 Theefoe, fom 16), thee exists c, depending only on N, p, L, R and ν, such that fo evey ρ < [ ρ ) ] N Du p dx c + ω) 1 + Du p + Du 0 p ) dx Finally, by 3) the thesis follows. + c p ϑ0 [ω)] p+1 N + c N. If ϑ 0 = 0 we get a egulaity esult fo local minimizes of the functional I in 6) when f is convex in ξ. Theoem 3.3. Let u be a local minimize of the functional I, whose integand fx, ξ) is a Caathéodoy function, convex with espect to the last vaiable and satisfies A1) A3). Then u is locally in C 0,α Ω) fo all α < 1. Moeove, fo all α < 1 thee exists c > 0, depending on N, p, L, R, ν and α, such that fo evey Ω and ρ < ρ ) N p+pα Du p dx c 1 + Du p ) dx. 4) Poof. When ϑ 0 = 0, Poposition 3. implies that fo evey ρ < [ ρ ) ] N Du p dx c + ω) 1 + Du p ) dx + c N. A standad iteation agument see [1], p.170) leads to estimate 4). The Hölde continuity of u follows fom a chaacteization of Campanato spaces see [17], p.57). 4. Poof of Theoem 1.1 In this section we pove Theoem 1.1. Fist we shall suppose that f is convex with espect to ξ. This supplementay assumption will be emoved obseving that a local minimize of F is a local minimize fo the elaxed functional, too. N.

15 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a The Ekeland vaiational pinciple enables us to compae the local minimize of F with a minimize of a functional whose integand does not explicitly depend on u. Theoem 4.1 Ekeland vaiational pinciple [5]). Let V, d) be a complete metic space and let H : V, + ] be a lowe semicontinuous functional with finite infimum. If u V is such that Hu) inf V H + ɛ fo some ɛ > 0, then thee exists v 0 V such that i) du, v 0 ) 1, ii) Hv 0 ) Hu), iii) v 0 minimizes in V the functional Ew) := Hw) + ɛ dv 0, w). Poof of Theoem 1.1. Without loss of geneality we assume ω in A3) to be a concave function. Fom Theoem.1 i) a local minimize of F is a Q-minimize of 5) with φ = 0, theefoe fom Lemma.4 thee exists q > p such that Du L q loc Ω). Since we aim to pove a local esult, we can assume that u W 1,q Ω) and that fo any ball x 1 ) Ω [ / x 1 ) Du q dx ] 1 q [ c x 1 ) ] Du p p ) dx. 5) Moeove see [15]) we can assume u C 0,γ Ω) fo some 0 < γ < 1. We denote by [u] γ the Hölde constant of u in Ω. Let us fix x 0 ) such that B 4 x 0 ) Ω. Step 1. Assume that f is convex in ξ. Conside the space V = u + W0 ) and let v be the minimize in V of the functional Hw; ) := hx, Dwx)) dx, 6) whee 1, p+1 hx, ξ) := fx, ux), ξ). 7) By Theoem.1 i), the minimality of v and A1) Dv p dx c 1 + Du p ) dx. 8) Using notations 13), the minimality of u and A3) we have Hu; ) = Fu; ) Fv; ) = Hv; ) + [fx, v, Dv) fx, u, Dv)] dx + ADv,,R) Hv; ) + L inf V Hw; ) + BDv,,R) [fx, v, Dv) fx, u, Dv)] dx BDv,,R) 1 + Dv p ) dx + ADv,,R) ω v u )1 + Dv p ) dx + c N. ω v u )1 + Dv p )dx 9)

16 404 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... Let s p, q) be the exponent of the highe integability of Dv which follows fom Lemma.5. Using the Hölde inequality, 5), the boundedness and the concavity of ω, we estimate the last integal in 9) as follows: ω v u )1 + Dv p ) dx ) p c 1 + Dv s s ) dx ) p c 1 + Du q q ) dx ω v u ) dx ) c 1 + Du p ) dx ω 1 p s v u dx B 4 ) c 1 + Du p ) dx ω σ v u dx, B 4 ) 1 p ω s s s p v u ) dx ) 1 p s with σ = 1 p. Caccioppoli inequality fo the minimize u see e.g. [15]) gives s Du p dx c 1 + u u ) x 1,ρ p dx, x 1 ) x 1 ) which holds fo evey x 1 ) x 0 ) hee u x1,ρ stands fo using the Poincaé inequality and 8), we get that v u dx c p [ c p 1 + u u x 0, p p ρ p x 1 ) ) 1 Dv Du p p dx c p 1 + Du p ) dx ) ] 1 p dx c p + c [u] p γ pγ) 1 p c γ, whee c depends also on [u] γ ; then ) ω σ v u dx ω σ c γ ). 30) ux)dx). Thus, This estimate, togethe with 9), 30) and 8), leads to Hu; ) inf Hw; ) + c ω σ c γ ) 1 + Du p )dx + c N. 31) V B 4 Step. Let us define H) := c ω σ c γ ) 1 + Du p ) dx + c N B 4 3) and apply Theoem 4.1 with V endowed with the distance [ H) dw 1, w ) := N ] p+1 p N ) 1 p Dw 1 Dw p+1 dx, 33)

17 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a and H defined in 6). Thus, by 31), thee exists v 0 V such that B Du Dv 0 p+1 dx [ H) N ] p+1 p N, 34) Hv 0 ; ) Hu; ), 35) v 0 is a minimize of Ew; ) in V, whee [ H) Ew; ) := Hw; ) + N ] p B Dw Dv 0 p+1 dx. 36) The minimality of v 0 implies that fo evey ϕ W 1,p 0 ) [ ] H) p Hv 0 ; spt ϕ) Hv 0 + ϕ; spt ϕ) ϕ) Dv 0 spt ϕ Dv p+1 dx. Using Young inequality and noting that p p+1 ) = p, we obtain Hv 0 ; spt ϕ) Hv 0 + ϕ; spt ϕ) + c 1 Dv 0 p dx + c spt ϕ N spt ϕ Dv 0 + Dϕ p dx + c H) spt ϕ. N Fom this inequality, Theoem.1 i) and A1), it follows that Dv 0 p dx c 1 + Dv 0 + Dϕ p + H) ) spt ϕ spt ϕ N theefoe v 0 is a Q-minimize with Q depending on N, p, L, R and ν) of the functional w 1 + Dw p + H) ) dx. N dx, Thus, thee exist λ p, q) and c > 0, independent of v 0, such that Dv 0 λ dx / ) p λ c 1 + Dv 0 p ) dx + c c 1 + Du p ) dx, B 4 [ 1 + H) ] N 37) whee the last inequality follows fom 3), Theoem.1 i), 35) and A1). Notice that h defined in 7) satisfies assumptions A1) A3), with a slightly diffeent modulus of continuity. Pecisely, fo any 0 < α < 1, h satisfies A3) with ω eplaced by ω α t) := max {[ω σ ct γ )] p, ωt + [u]γ t γ ), t p1 α) p+1 }. 38)

18 406 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... Apply Poposition 3. to Ew; ) defined in 36), with ϑ 0 and u 0 eplaced by and v 0, espectively. Then fo evey ρ / Du p dx Dv 0 p dx + Du Dv 0 p dx / [ ρ ) N c + ωα )] 1 + Dv 0 p ) dx H) + c + c N + Du Dv [ ω α )] p+1 0 B p dx / [ ρ ) N c + ωα )] 1 + Du p ) dx + c ωσ c γ ) 1 + Du p ) dx [ ω α )] p+1 B 4 [ ] H) p N 39) + c N p+pα + c N + / Du Dv 0 p dx [ ρ ) N+ c ωα )] 1 + Du p )dx+c N p+pα + Du Dv 0 p dx. B 4 / In ode to estimate the last integal, let ϑ 0, 1) be such that ϑ + 1 ϑ) = 1, whee λ λ p+1 p is the exponent in 37). Using 5), 37), 34) and 35), we get Du Dv 0 p dx / ) ϑ p λ ) 1 ϑ) p c Du Dv 0 λ dx Du Dv 0 p+1 p+1 dx / ) ϑ [ ] 40) 1 ϑ H) c N 1 + Du p ) dx B 4 N ) ϑ c [ω σ c γ )] 1 ϑ 1 + Du p ) dx + c N1 ϑ) 1 + Du p ) dx. B 4 B 4 Finally, collecting 39), 40) and using Young inequality, we deduce that fo evey ɛ > 0 thee exists cɛ) such that [ ρ ) N Du p dx c + ωα ) + ɛ] 1 + Du p ) dx B 4 p 1 ϑ) + c[ ω α )] 1 + Du p ) dx + c N p+pα + cɛ) N, B 4 which implies [ ρ ) N Du p dx c + [ ωα )] δ + ɛ] 1 + Du p ) dx + c ɛ N p+pα, B 4 fo a cetain δ > 0 independent of ρ and. Fom this inequality the thesis follows by Lemma.6 and by a standad iteation agument see [1], p.170).

19 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a Step 3. Now we conside a geneal f non convex with espect to the last vaiable. We obseve that local minimizes of F ae also local minimizes of the elaxed functional { } F u; x 0 ))= inf lim inf fx, u h, Du h ) dx : u h u in W 1,p 0 x 0 )) h + which is equal to x 0 ) x 0 ) f x, ux), Dux)) dx, whee ξ f x, u, ξ) is the bipola of ξ fx, u, ξ), see [6, Coollay 3.8]. The esult follows immediately fom the fact that by Theoem.1 ii) thee exists R 0 such that in Ω R R N \ B R0 0)). fx, u, ξ) = f x, u, ξ) Refeences [1] E. Acebi, G. Mingione: Regulaity esults fo a class of functionals with non-standad gowth, Ach. Rat. Mech. Anal ) [] G. Cupini, N. Fusco, R. Petti: Hölde continuity of local minimizes, J. Math. Anal. Appl ) [3] G. Cupini, M. Guidozi, E. Mascolo: Regulaity of minimizes of vectoial integals with p q gowth, Nonlinea Anal., Se. A: Theoy Methods ) [4] G. Cupini, R. Petti: Hölde continuity of local minimizes of vectoial integal functionals, NoDEA ) [5] I. Ekeland: Nonconvex minimization poblems, Bull. Am. Math. Soc. 13) 1979) [6] I. Ekeland, R. Temam: Convex Analysis and Vaiational Poblems, Noth Holland, Amstedam 1976). [7] L. Esposito, F. Leonetti, G. Mingione: Regulaity esults fo minimizes of iegula integals with p, q) gowth, Foum Mathematicum 14 00) [8] V. Feone, N. Fusco: Continuity popeties of minimizes of integal functionals in a limit case, J. Math. Anal. Appl ) 7 5. [9] I. Fonseca, N. Fusco: Regulaity esults fo anisotopic image segmentation models, Ann. Sc. Nom. Sup. Pisa ) [10] I. Fonseca, N. Fusco, P. Macellini: An existence esult fo a nonconvex vaiational poblem via egulaity, ESAIM; Contol, Optim. Calc. Va. 7 00) [11] N. Fusco, G. Mingione, C. Tombetti: Regulaity of minimizes fo a class of anisotopic fee discontinuity poblems, J. Convex Analysis 8 001) [1] M. Giaquinta: Multiple Integals in the Calculus of Vaiations and Non Linea Elliptic Systems, Annals of Math. Studies 105, Pinceton Univ. Pess, Pinceton NJ 1983). [13] M. Giaquinta, E. Giusti: On the egulaity of the minima of vaiational integals, Acta Math ) [14] M. Giaquinta, E. Giusti: Diffeentiability of minima of non-diffeentiable functionals, Invent. Math )

20 408 G. Cupini, A. P. Miglioini / Hölde Continuity fo Local Minimizes of a... [15] M. Giaquinta, E. Giusti: Quasi-minima, Ann. Inst. H. Poincae, Non Lineaie ) [16] M. Giaquinta, G. Modica: Remaks on the egulaity of the minimizes of cetain degeneate functionals, Manuscipta Math ) [17] E. Giusti: Metodi Dietti nel Calcolo delle Vaiazioni, Unione Matematica Italiana, Bologna 1994). [18] J. J. Manfedi: Regulaity fo minima of functionals with p-gowth, J. Diffeential Equations ) [19] N. G. Meyes: An L p -estimate fo the gadient of solutions of second ode elliptic divegence equations, Ann. Scuola Nom. Sup. Pisa ) [0] K. Uhlenbeck: Regulaity fo a class of nonlinea elliptic systems, Acta Math )