FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS: RELAXATION AND Γ-CONVERGENCE

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1 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS: RELAXATION AND Γ-CONVERGENCE ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI Abstract. In ths paper we consder postvely 1-homogeneous supremal functonals of the type F (u) := sup f(x, u(x)). We prove that the relaxaton F s a dfference quotent, that s F (u) = R d F u(x) u(y) (u) := sup for every u W 1, (), x,y, x y d F (x, y) where d F s a geodesc dstance assocated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functonals wth respect to Γ-convergence s gven exactly by the class of dfference quotents assocated to geodesc dstances. Ths class strctly contans supremal functonals, as the class of geodesc dstances strctly contans ntrnsc dstances. Keywords : Varatonal methods, Supremal functonals, Fnsler metrc, Relaxaton, Γ-convergence Mathematcs Subject Classfcaton: 47J20, 58B20, 49J45. Contents Introducton 1 1. Prelmnares on geodesc and ntrnsc dstances Geodesc dstances Intrnsc dstances 5 2. Dfference quotent functonals 7 3. Relaxaton of supremal functonals A counter example to F (u) = sup f (x, u) The relaxaton formula The dstance d F s geodesc Γ-convergence of supremal functonals 19 References 20 Introducton In ths paper we are nterested n mnmzaton problems for a class of functonals recently called supremal functonals, namely functonals F : W 1, () R of the form F (u) := sup f(x, u(x), u(x)), where sup means the essental sup on. The model case where f(x, u, u) u s related to the classcal problem of fndng the best Lpschtz constant of a functon wth prescrbed boundary data, frst consdered by McShane n [18]. 1

2 2 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI In order to apply the drect method of the calculus of varatons the man ssue s the lower semcontnuty of F. Semcontnuty propertes for supremal functonals have been studed by many authors n the last years; we refer for nstance to Barron-Jensen [3], Barron-Lu [5], Barron-Jensen-Wang [4], and to the recent papers by Prnar [19] and Gor-Magg [17]. In [4] the authors proved a lower semcontnuty result for F under the assumpton (called level convexty) that the sub levels of f(x, u, ) are convex. As necessary condtons are concerned the only results have been obtaned n the one dmensonal case n [1] and n [4] under contnuty assumptons of f wth respect to x and u. In both cases t s stated that f the supremal functonal F s lower semcontnuous, then f s level convex n the gradent varable. In the case of lack of semcontnuty an mportant step, for the characterzaton of the mnmzng sequences, s to consder the lower semcontnuous envelope of F,.e., the bggest lower semcontnuous functonal smaller than F, the so called relaxaton of F. In [5] and n [13] t s proved that f f s contnuous wth respect to x and u, then the relaxaton of F s a supremal functonal represented by the level convex envelope of f. In the case of contnuty of f, even though the structure of the functonal F s supremal, t s stll possble to adapt the technques that are usually used n the case of ntegral functonals as blow-up arguments as well as zg-zag approxmatons. Wthout the contnuty the functonal F can be affected by the values of u on arbtrarly small sets and those technques fal. Our man contrbuton to the subject s to approach the problem of relaxaton and Γ-convergence n any dmenson and wthout any contnuty assumpton, provdng some new geometrcal constructons ntrnscally related to the supremal nature of the functonal. We restrct our analyss to the case of postvely 1-homogeneous supremal functonals of the form (0.1) F (u) := sup f(x, u(x)) and we assume that f : R n R s a Carathéodory functon satsfyng (0.2) α ξ f(x, ξ) β ξ for a.e x, for every ξ R n, for some fxed postve constants α, β > 0, and (0.3) f(x, tη) = t f(x, η) for a.e. x, for every t R and for every η R n. A natural queston s whether n general the relaxaton of F s obtaned through the convex envelope (whch n the case of 1-homogeneous supremal functonals concdes wth the level-convex envelope) of the functon f whch represents F. In Example 3.2 we construct a functonal F (u) := sup f(x, u(x)) such that ts relaxaton s strctly greater than the functonal represented by the convex envelope of f, showng that n general the formula F (u) = sup f (x, u(x)) s false. A second unexpected result concerns the study of the asymptotc behavor under Γ- convergence of the class of 1-homogeneous supremal functonals. By means of an example (see Remark 4.5) we show that ths class s not closed under Γ-convergence and we prove that ts closure s gven by the class of dfference quotents assocated to geodesc dstances,.e., functonals of the form R d (u) := u(x) u(y) sup x,y, x y d(x, y) for every u W 1, (),

3 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 3 where d s a geodesc dstance equvalent to the Eucldean dstance. Ths class of lower semcontnuous functonals contans any 1-homogeneous supremal functonal F (u) = sup f(x, u(x)) wth f convex; more precsely F = R d F, where d F s defned by } (0.4) d F (x, y) := sup u(x) u(y), u W 1, () : F (u) 1. Actually a dfference quotent R d s a supremal functonal represented by a convex functon f and only f the dstance d s geodesc and satsfes the addtonal property of beng ntrnsc (the noton of ntrnsc dstance was ntroduced by De Cecco-Palmer [15]; see Defnton 1.4). On the other hand there are geodesc (non ntrnsc) dstances such that the correspondng dfference quotent functonal can not be wrtten n a supremal form (see Example 2.6). In vew of these results the class of dfference quotents seams to be the natural class n whch to look for the relaxaton of F. Our man result n ths drecton s the followng representaton formula for the relaxaton F of F F (u) = R d F (u) for every u W 1, (), where d F s gven by (0.4). Ths relaxaton formula represents the man tool used n the characterzaton of the closure under Γ-convergence of 1-homogeneous supremal functonals: the problem reduces to fnd the closure of the class of dstances d F assocated to supremal functonals. In ths respect, the man pont s that the dstance d F s always geodesc (see Theorem 3.9). In order to show that the supremal functonals are not closed under Γ-convergence, n Remark 4.5 we exhbt a sequence of ntrnsc geodesc dstances that unformly converges to a geodesc dstance whose dfference quotent s not supremal. The same sequence s also a counterexample to Theorem 3.1, ) ff v), of [7] as t wll be explaned n [8]. Another consequence of our relaxaton formula s that whenever the dstance d F s also ntrnsc, then the relaxaton F s a supremal functonal represented by a convex functon (see Corollary 3.6). At the present the queston whether d F s n general ntrnsc s stll open. Fnally let us observe that relatons between metrc propertes and problems n L were recently used by many authors, n the case of varatonal problems (see e.g. Buttazzo-De Pascale-Fragalà [7]) and n the study of the vscosty solutons of the so called -Laplacan (see e.g. Aronsson-Crandall-Juutnen [2], Champon-De Pascale [10] and Crandall-Evans-Garepy [11]). More n general the dea that the metrc approach permts to consder stuatons wth lack of regularty s nowadays classcal and has been used n many dfferent contexts as, for nstance, to treat Hamlton-Jacob equatons wth dscontnuous Hamltonan (see Sconolf [20] and Camll-Sconolf [9]). The paper s organzed as follows. In Secton 1 we recall the man metrc notons used n the paper. In Secton 2 we ntroduce the class of dfference quotent functonals, we prove ts man propertes and show the relaton wth the supremal functonals. In Secton 3 we gve a relaxaton formula for F, and n Secton 4 we characterze the closure under Γ-convergence of 1-homogeneous supremal functonals. 1. Prelmnares on geodesc and ntrnsc dstances In ths secton we wll recall the man metrc notons that wll be used n the sequel. We frst ntroduce the noton of geodesc dstances. In partcular for our settng we wll need the defnton of ntrnsc dstance ntroduced by De Cecco and Palmer and ts man propertes (for detals see [14]-[16]).

4 4 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI 1.1. Geodesc dstances. From now on wll be a connected open bounded subset of R n wth Lpschtz contnuous boundary. We say that a dstance d : [0, + ) s geodesc f d(x, y) = nfl d (γ) : γ Γ x,y ()} for every x, y. Here Γ x,y () denotes the set of Lpschtz curves n wth end-ponts x and y, and L d (γ) denotes the length of the curve γ wth respect to the dstance d,.e., Let us denote k 1 L d (γ) := sup d(γ(t ), γ(t +1 )) : k N, 0 = t 1 <... < t <... < t k = 1}. =1 x y = nfl(γ) : γ Γ x,y ()}, where L(γ) denotes the Eucldean length of γ. Note that by the fact that s Lpschtz we deduce that there exsts a constant C > 0 such that x y x y C x y. Gven two postve constants 0 < α < β we set (1.1) D(α, β ) := d geodesc dstance: α x y d(x, y) β x y for all y, x }. Remark 1.1 (Extenson of geodesc dstances). Every dstance d defned n, satsfyng α x y d(x, y) β x y, can be unquely extended by contnuty to. Moreover f the dstance d s geodesc, then ts extenson (stll denoted by d) satsfes d(x, y) = mnl d (γ) : γ Γ x,y ()} for every x, y, where Γ x,y () denotes now the set of Lpschtz curves n wth end-ponts x and y. The followng proposton states that the class D(α, β ) s compact wth respect to unform convergence. Proposton 1.2. Let d n be a sequence of dstances n D(α, β ). Then, up to a subsequence d n unformly converge to some dstance d n D(α, β ). Defnton 1.3. A (convex) Fnsler metrc on s a functon ϕ : R N [0, + ), Borel measurable wth respect to the frst varable and contnuous wth respect to the second varable, such that the followng propertes hold: ϕ(x, tη) = t ϕ(x, η) for every x, t R, η R n ; α η ϕ(x, η) β η for every x, η ; ϕ(x, ) s convex for a.e. x ; where α and β are two fxed postve constants. To any geodesc dstance n D(α, β ), we assocate a Fnsler metrc ϕ d. Namely for every x and for every drecton η we can defne the functon ϕ d (x, η) as follows (1.2) ϕ d (x, η) := lm sup t 0 + d(x, x + tη) t.

5 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 5 It turns out that ϕ d s convex for a.e. x, t s postvely 1-homogeneous wth respect to η, and Borel measurable wth respect to x (thus ϕ d s a convex Fnsler metrc). Moreover t can be proved that for every γ we have (see Theorem 2.5 n [16]) Intrnsc dstances. L d (γ) = 1 0 ϕ d (γ, γ ) dt Defnton 1.4. We say that a dstance d n D(α, β ) s ntrnsc f d(x, y) = sup N nf γ Γ N x,y () 1 0 ϕ d (γ, γ ) dt, where the supremum s taken over all subsets N of such that N = 0 and Γ N x,y() denotes the set of all Lpschtz curves n wth end-ponts x and y transversal to N,.e., such that H 1 (N γ) = 0, where H 1 denotes the one dmensonal Hausdorff measure. Note that the sup over neglgble sets s actually a maxmum. We set (1.3) D(α, β ) = d D(α, β ) : d s ntrnsc}. To any Fnsler metrc ϕ we assocate an ntrnsc dstance δ ϕ through the so called support functon ϕ 0 of ϕ, defned by dualty as follows ξ η } (1.4) ϕ 0 (x, ξ) := sup. η 0 ϕ(x, η) Clearly, for every x, t satsfes the followng propertes: ϕ 0 (x, ) s convex; ϕ 0 (x, t ξ) = t ϕ 0 (x, ξ) for every t R, ξ R n ; 1 β ξ ϕ 0 (x, ξ) 1 α ξ for every ξ R N. Moreover f ϕ s convex, then ϕ 00 = ϕ. Now, f ϕ s a Fnsler metrc, then t s possble to defne a dstance δ ϕ (x, y) n the followng way (see [15], [16]): (1.5) δ ϕ (x, y) := sup u(x) u(y), u W 1, () : sup } ϕ 0 (x, u(x)) 1, for every x, y. By Theorem 3.7 n [16] δ ϕ (x, y) s a geodesc dstance and satsfes (1.6) δ ϕ (x, y) = sup N nf γ Γ N x,y () 1 0 ϕ(γ, γ ) dt. The followng example shows that n general, f ψ s a Fnsler metrc, then the dervatve ϕ δψ of δ ψ can be dfferent from ψ. Example 1.5 (Example 5.1 n [16]). Let (a h ) h be dense n R and let A R 2 be the open set defned by A := x = (x 1, x 2 ) R 2 : mn nf x 1 a h 2 h, nf x 2 a h 2 h} } < 1. h h

6 6 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI Roughly speakng the set A s gven by the unon of horzontal and vertcal thn strps. Let ψ be the Fnsler metrc on R 2 defned by v f x A, ψ(x, v) := 2 v otherwse, and let d be the assocated dstance (so that d = δ ψ ). Then the dervatve ϕ d of d s gven by v f x A, ϕ d (x, v) := v 1 + v 2 otherwse. In partcular, f x / A then ϕ d (x, v) ψ(x, v). Next proposton states that ϕ δψ s always lower then ψ. Proposton 1.6. Let ψ be a convex Fnsler metrc. Then ϕ δψ (x, ξ) ψ(x, ξ) for a.e. x, for every ξ R n. Proof. It s enough to prove the nequalty for every Lebesgue pont x for the functon z ψ(z, ξ) for every ξ R n (such ponts n fact have full measure n ). Let us fx a drecton ξ R n, and for every t > 0 let us denote by γ t the straght curve jonng x wth x + tξ. Moreover let N be the neglgble set maxmzng the rght hand sde of (1.6). Let us fx ε > 0. By Fubn s Theorem, and usng that x s a Lebesgue pont, we can easly translate the curve γ t by a vector r t such that r t < t 2, obtanng a curve γ t satsfyng the followng condtons: ) γ t s transversal to N; ) H 1 ( γ t z : ψ(z, ξ) ψ(x, ξ) > ε ξ })/t 0 as t 0. Usng ) and ) we obtan (1.7) δ ψ (x, x + tξ) δ ψ (x, x + r t ) ψ( γ t, γ t) ds + δ ψ (x + r t + tξ, x + tξ) 2β r t + tψ(x, ξ) + t o(ε), where o(ε) 0 as ε 0. Dvdng both sdes of (1.7) by t, and passng to the lmsup as t 0, n vew of the arbtrarness of ε we obtan the thess. Now we are n poston to characterze all ntrnsc dstances. Proposton 1.7. Let d D(α, β ). Then the followng are equvalent 1) d D(α, β ); 2) δ ϕd = d. Proof. 1) = 2) It follows by Theorems 4.5 and 3.7 n [16]. 2) = 1) Set ψ = ϕ d. Snce δ ψ s a geodesc dstance we have that for every x, y δ ψ (x, y) = nf γ Γ x,y() 1 and thus, by Proposton 1.6 and by (1.6), we obtan δ ψ (x, y) sup N sup N nf 0 1 γ Γ N x,y() 0 1 nf γ Γ N x,y() 0 ϕ δψ (γ, γ ) dt ϕ δψ (γ, γ ) dt ψ(γ, γ ) dt = δ ψ (x, y),

7 .e., FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 7 d(x, y) = δ ψ (x, y) = sup N nf γ Γ N x,y () 1 0 ϕ d (γ, γ ) dt. We conclude wth an example gven n [6] whch shows that not all geodesc dstances are ntrnsc. Example 1.8. Let = ( 1, 1) 2 and consder the segment S = ( 1, 1) 0}. Consder the Fnsler metrc ϕ defned by β ξ f x \ S ϕ(x, ξ) = α ξ f x S, wth 0 < α < β. Consder now the dstance assocated to ths metrc,.e., d(x, y) = 1 nf γ Γ x,y() 0 ϕ(γ, γ ) dt. Ths dstance s clearly dfferent from β tmes the Eucldean dstance, n partcular for many pars of ponts (x, y) near S we have d(x, y) < β x y. On the other hand the dervatve ϕ d of d concdes wth β ξ n \ S and hence the dstance δ ϕd (x, y) concdes wth β x y. In concluson the class D(α, β ) s strctly contaned n D(α, β ). In the sequel we wll use the dstance defned n (1.5) also n the case where the functonal sup ϕ 0 (x, u) s replaced by some postvely 1-homogeneous supremal functonal F (u) = sup f(x, u(x)), wth f possbly non convex, satsfyng (0.2). In ths case we wll denote t by d F,.e., (1.8) d F (x, y) := sup u(x) u(y), u W 1, () : F (u) 1 }. By the growth condton on f we clearly have 1 (1.9) β x y d F (x, y) 1 α x y. 2. Dfference quotent functonals In ths secton we ntroduce the class of dfference quotent functonals, whch s the natural settng n the study of relaxaton and Γ-convergence of supremal functonals. For every dstance d equvalent to the Eucldean dstance let R d : W 1, () R be defned by (2.1) R d (u) := sup x,y, x y u(x) u(y). d(x, y) The functonal R d s referred to as the dfference quotent assocated to d. Remark 2.1. Let d 1, d 2 be two dstances equvalent to the Eucldean dstance. It s easy to see that (2.2) R d 1 = R d 2 f and only f d 1 = d 2. In fact for every fxed z the functons u( ) := d 1 (, z) and v( ) := d 2 (, z) belong to W 1, (). To show (2.2) t s suffcent to test the functonals R d 1 and R d 2 on these functons.

8 8 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI Proposton 2.2. The dfference quotent R d s lower semcontnuous wth respect to the strong convergence n L. Proof. Let u W 1, () and let u n } W 1, () be a sequence convergng to u n L (). We have that for every x, y, x y, u(x) u(y) d(x, y) = lm n u n (x) u n (y) d(x, y) Takng the supremum as x, y we get the thess. lm nf n From now on we wll consder supremal functonals of the form (2.3) F (u) := sup f(x, u(x)), where f satsfes the followng growth condton R d (u n ). (2.4) α ξ f(x, ξ) β ξ for a.e x, for every ξ R n for some fxed postve constants 0 < α < β, and t s postvely 1-homogeneous,.e., (2.5) f(x, tη) = t f(x, η) for a.e. x, for every t R and for every η R n. Remark 2.3. We wll see n Remark 3.1 that a supremal functonal does not admt a unque representatve. Thus we may not expect that f the functonal F (u) := sup f(x, u(x)) s postvely 1-homogeneous (.e., F (λu) = λ F (u)), then f s postvely 1-homogeneous. Nevertheless t s easy to see that the functon f(x, ξ) := ξ f (x, ξ/ ξ ) ξ R n, for a.e. x s postvely 1-homogeneous and always represents F. In the sequel, any postvely 1-homogeneous functonal wll be understood to be represented by a postvely 1-homogeneous functon. The followng proposton gves a frst mportant relaton between supremal functonals and dfference quotent functonals. Proposton 2.4. Let F be a 1-homogeneous supremal functonal assocated to a Carathéodory functon f satsfyng (2.4), (2.5), and convex wth respect to ξ. Then R d F = F, where d F s the dstance defned by (1.8). Proof. Usng that both functonals are postvely 1-homogeneous, we have to prove that u(x) u(y) sup f(x, u(x)) 1 f and only f sup 1. x,y, x y d F (x, y) By the defnton of d F we have sup f(x, u(x)) 1 mples that u(x) u(y) d F (x, y) for all x, y,.e., R d F (u) 1. Conversely, let ϕ := f 0. Snce f s convex, then ϕ 0 = f and thus d F concdes wth the dstance δ ϕ defned by (1.5). If R d F (u) = R δϕ (u) 1, then by usng Proposton 1.6 we obtan that for a.e. x and any z R n, u(x + tz) u(x) δ ϕ (x + tz, x) u(x) z = lm lm sup = ϕ δϕ (x, z) ϕ(x, z). t 0 t t 0 t Fnally, by the defnton of ϕ 0 we deduce f(x, u(x)) = ϕ 0 (x, u(x)) 1 for a.e. x.

9 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 9 The followng result shows that the class of supremal functonals represented by a convex functon actually concdes wth the class of dfference quotents assocated to an ntrnsc dstance. Proposton 2.5. Let d be a geodesc dstance n D(α, β ) and ϕ d be ts dervatve accordng to (1.2). The followng facts are equvalent 1) d D(α, β ); 2) R d (u) = sup ϕ 0 d (x, u(x)); 3) R d (u) = sup f(x, u(x)), where f s a Carathéodory functon, convex wth respect to ξ, satsfyng (2.5) and (2.4) wth α = 1 β and β = 1 α. Proof. 1) = 2) We have to prove that sup ϕ 0 u(x) u(y) d (x, u(x)) 1 f and only f sup 1. x,y, x y d(x, y) Snce d D(α, β ), by Proposton 1.7 we have that δ ϕd = d. Thus by the defnton of δ ϕd sup ϕ 0 d (x, u(x)) 1 mples that u(x) u(y) δ ϕ d (x, y) = d(x, y) for all x, y,.e., R d (u) 1. Conversely, f R d (u) 1 then for a.e. x and any z R n we have u(x + tz) u(x) d(x + tz, x) u(x) z = lm lm sup = ϕ d (x, z), t 0 t t 0 t and hence, by the defnton of ϕ 0 d, we get ϕ0 d (x, u(x)) 1 for a.e. x. 2) = 3) The proof of ths mplcaton trvally follows by the propertes of ϕ 0 d. 3) = 1) It s easy to check that d(x, y) = supu(x) u(y), u W 1, () : R d (u) 1} = supu(x) u(y), u W 1, () : g(x, u(x)) 1 a.e. n }. Snce g s convex, d = δ ϕ wth ϕ = g 0 and hence by Proposton 1.7 d s ntrnsc. Ths concludes the proof. A natural queston s whether a dfference quotent R d assocated to a dstance d can be expressed as a supremal functonal of the type (2.3), wth f possbly non convex n ξ. In the next example we wll show that the fact that d s geodesc s not enough to ensure the supremalty of R d. Example 2.6. Consder the dstance d D(α, β ) gven n Example 1.8. It s easy to check that α sup u R d (u) β sup u for every u W 1, (), wth α = 1 β and β = 1 α. We now prove that R d can not be wrtten as a supremal functonal. Assume by contradcton that R d (u) = sup g(x, u(x)), for some Carathéodory functon g.

10 10 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI Clam: There exsts a set N, wth N = 0, such that g(x, ξ) α ξ for every x \ N and for every ξ R N. From ths clam the concluson follows mmedately takng the functon u(x) = x 1. In fact we have R d (u) = β, whch s n contradcton wth g(x, u(x)) α for a.e. x. It remans to prove the clam. By the defnton of d t s easy to see that for every x \S there exsts a radus r(x) > 0 such that d(x, y) = β x y n B r(x) (x) B r(x) (x) (where B r (x) denote the ball of radus r and center x) and hence (2.6) R d (u) = α sup u u W 1, (), wth supp u B r(x) (x). In order to prove the clam t s enough to prove that for any ξ S 1 there exsts a set N ξ, wth N ξ = 0, such that (2.7) g(x, ξ) α n \ N ξ. Let then assume by contradcton that there exsts a vector ξ S 1 and a set M ξ, wth M ξ > 0, such that (2.8) g(x, ξ) > (α + ε) n M ξ, for some ε > 0. Now fx a pont x 0 n M ξ such that for every postve r the set M ξ B r (x 0 ) has postve measure (ths s always possble because a.e. x M ξ s of densty one for M ξ ). We defne the functon ξ(x x 0 ) r f x B r (x 0 ) u(x) = mnnf y Br(x0 )(ξ(y x 0 ) r + ξ x y ), 0} otherwse. Clearly f r s small enough, then supp u B r(x0 )(x 0 ) and thus by (2.6) we have R d (u) = α sup u = α, whle by (2.8) sup g(x, u(x)) α + ε, whch gves a contradcton and concludes the proof. 3. Relaxaton of supremal functonals In ths secton we gve a relaxaton formula for postvely 1-homogeneous supremal functonals wth respect to the strong convergence n L (), n terms of dfference quotent functonals. Gven a postvely 1-homogeneous supremal functonal of the type (2.3), we denote by F ts relaxaton,.e., ts lower semcontnuous envelope wth respect to the L -convergence gven by (3.1) F (u) := nf lm nf n F (u n ) : u n u n L () }. We start observng that the convexty of f s not a necessary condton for the lower semcontnuty of F. Remark 3.1. A postvely 1-homogeneous supremal functonal has n general not a unque representaton. In fact, under the notaton of Example 1.5, usng Proposton 2.4 and Proposton 1.7 we deduce that G(u) := sup ψ 0 (x, u) = sup ϕ 0 d (x, u) for every u W 1, (),

11 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 11 and t s easy to check that ψ 0 v f x A, (x, v) := 1 2 v otherwse, whle ϕ 0 d (x, v) := v f x A, v 2 f v 1 < v 2 and x A, v 1 otherwse. In partcular the functonal G s lower semcontnuous and can be represented by any functon g, possbly non convex, such that ψ 0 g ϕ 0 d A counter example to F (u) = sup f (x, u). Now we gve an example showng that n general the relaxaton of F s not obtaned through the convexfcaton f of f wth respect to ξ. Example 3.2. Let us call G the set of all contnuous functons g : R n R, postvely 1-homogeneous and satsfyng α ξ g(ξ) β ξ for all ξ R n, and let (3.2) C := C R n : C = ξ R n : g(ξ) 1} for some g G}. Note that the sets n C are closed, star-shaped (wth respect to the orgn), and that by defnton to every C C s assocated a functon g G, whch we denote by g C. Moreover C s closed for ntersecton and unon. Let now B be the unt ball n R n centered at 0. Then B C wth g B (ξ) = ξ. Let H C be satsfyng the followng propertes: 1) H s not convex; 2) H \ B and B \ H ; 3) B s contaned n the convex hull of H. Fnally let us construct an open and dense set A wth 0 < A < as follows. Let v } N be a dense subset of B and let p j } j N be dense n. For a gven postve constant δ > 0 we defne A := x : dst(x, p + sv j, s R}) < δ } 2 j.,j N Clearly, f δ s small enough, we have that 0 < A <. Roughly speakng the set A s gven by a countable unon of thn strps along a dense set of drectons. We consder the functons f, f + : R n R defned by f(x, ξ) := g B (ξ) f x A; g H (ξ) f x \ A. The assocated supremal functonals are F (u) := sup f + (x, ξ) := g B (ξ) f x A; g H B (ξ) f x \ A. f(x, u(x)) and F + (u) := sup f + (x, u(x)). Clam: F = F +. Once the clam s proved we can conclude the argument as follows. By the fact that H B s closed, star shaped and strctly contaned n B, t s possble to prove that there exsts a vector ξ B such that ξ s not n the convex hull of H B. By property 3) above and by the defnton of f + and the choce of ξ we have (3.3) f (x, ξ) 1 a.e. on whle f + (x, ξ) > 1 a.e. on \ A.

12 12 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI Snce by the clam sup f+ (x, ( )) F ( ), and sup f+ (x, ( )) s lower semcontnuous, we have sup f+ (x, ( )) F ( ). On the other hand, by (3.3) we have sup f+ (x, (ξ x)) > sup f (x, (ξ x)), and therefore F s not represented by f. It remans the proof of the clam. By constructon we have that F + F, and so let us assume by contradcton that for some u W 1, () we have (3.4) F + (u) > 1 whle F (u) < 1. Ths wll mply that u H \ B on a set of postve measure. Therefore there exsts a pont x of dfferentablty for u wth u(x) > 1. To smplfy the notaton we can assume x = 0 and u(0) = 0. Let ρ n } be a sequence convergng to zero, and for every n let us consder the functon u n : B R defned by u n (x) := 1 ρ n u(ρ n x) for every x B. By the defnton of A, for every n and for every ε > 0 we can fnd an open strp L ε n n B such that ρ n L ε n A and such that L ε n contans two ponts a ε n and b ε n wth (3.5) aε n u(0) ( u(0) + b ε n u(0) ) ε. u(0) By Proposton 2.5, we have that sup u n (x) = L ε n u n (x) u n (y) sup x,y L ε x y n u n(a ε n) u n (b ε n) a ε n b ε. n Usng that, by the dfferentablty of u at 0, u n } converges to u(0) x unformly, by (3.5) we deduce that, for n bg enough, sup u n (x) u(0) + o(ε), L ε n where o(ε) 0 as ε 0. Therefore, recallng that u(0) > 1, we can fnd ε and n such that sup L ε n u n (x) > 1. We conclude that F (u) sup u(x) sup A ρ nl ε n whch s n contradcton wth (3.4). u(x) = sup u n (x) > 1, L ε n Remark 3.3. The man dea of Example 3.2 s that the dense set A does not allow to perform a zg-zag approxmaton on \ A of an affne functon ξ x, whch uses two gradents ξ 1, ξ 2 H \ B, of whch ξ B s a convex combnaton. Note that n Example 3.2 we don t even know whether the relaxaton of F s a supremal functonal.

13 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS The relaxaton formula. In ths paragraph we characterze the relaxaton of postvely 1-homogeneous supremal functonals n terms of assocated dfference quotent functonals R d F, where d F s the dstance assocated to F as n (1.8). The key step s the the followng approxmaton lemma. Lemma 3.4. Let F be a postvely 1-homogeneous supremal functonal on W 1, () represented by a Carathéodory functon f : R n R satsfyng (2.4). Let v W 1, () be such that R d F (v) < 1. Then there exsts a sequence v n } W 1, () convergng to v n L () wth F (v n ) 1. Navely the dea for the constructon of the sequence v n would be to consder a lattce of ponts on, then n a neghborhood of each pont p of the lattce we would lke the functon v n to look lke a small cone gven by v(p) + d F (x, p). Unfortunately, the dfference quotent of the above functon s less then 1 but t s not clear whch s the value of F on t. So, startng from ths dea, the rght constructon wll requre a fner argument. Proof. Let us fx a postve radus r > 0. By the fact that R d F (v) < 1, for every x, y wth x y = r (3.6) v(y) v(x) < d F (x, y) γ, for a postve constant γ dependng on r. Let us fx 0 < ε < γ 3. For every x and for every y B r (x) (where B r (x) denotes the ball of radus r centered at x), by the defnton of d F there exsts a functon wr x,y W 1, () such that 1) F (wr x,y ) 1; 2) wr x,y (y) wr x,y (x) + d F (x, y) ε; 3) wr x,y (x) = v(x); the thrd property beng possble thanks to the translaton nvarance of the frst two. By propertes 2), 3) and by (3.6), for every y B r (x) (3.7) wr x,y (y) v(x) + d F (x, y) ε > v(y) + γ ε. Note that by property (2.4) we have that sup wr x,y < 1/α, and hence there exsts δ > 0 (dependng only on ε) such that (3.8) wr x,y (z) > v(z) + γ 2ε > v(z) + ε for every z B r (x) : z y δ. Moreover, snce wr x,y (x) = v(x), there exsts 0 < r < r (dependng only on ε) such that (3.9) wr x,y (z) < v(z) + ε for every z B r (x). For every x, let us fx a fnte set of ponts y 1,..., y N } on B r (x) such that N B r (x) B δ (y ), and let us set the functon w x r : B r (x) R defned by (3.10) w x r (z) := max w x,y r (z) for every z B r (x). By constructon and by (3.8) and (3.9), we have =1

14 14 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI 1) sup Br(x) f(z, w x r ) 1; 2) w x r (z) > v(z) + ε for every z B r (x) ; 3) w x r (z) < v(z) + ε for every z B r (x). Now let Z r be a fnte set of ponts of such that z Z r B r (z), and consder the functon w r : R defned by (3.11) w r (x) := mn z Z r B r(x) wz r(x). By propertes 2) and 3) above t follows that w r s contnuous. Moreover, for almost every x n, w r (x) concdes wth w z r(x) for some z Z r and ths mples that w r W 1, () and F (w r ) 1. Now let us prove that w r v L () 0. To ths am, let us fx x, and let z B r (x) be such that w r (x) = w z r(x). Recallng that by constructon w z r(z) = v(z), and usng (1.9) we conclude w r (x) v(x) w z r(x) w z r(z) + w z r(z) v(x) = w z r(x) w z r(z) + v(z) v(x) 2d F (x, z) 2C α r. Therefore, for every r n } 0, the sequence v n := w rn does the job. We are now n a poston to gve the representaton formula for the relaxaton. Theorem 3.5. Let F be a postvely 1-homogeneous supremal functonal represented by a functon f satsfyng property (2.4). Moreover let d F be the dstance assocated to F as n (1.8) and let R d F be the correspondng dfference quotent (see (2.1), wth d replaced by d F ). Then the relaxaton F of F wth respect to the strong convergence n L () s gven by F (u) = R d F (u) u W 1, (). Proof. By Proposton 2.2, the functonal R d F s lower semcontnuous, and hence by the defnton of the relaxaton we get R d F F. In order to prove the nverse nequalty, let v W 1, () such that R d F (v) < 1. By Lemma 3.4 there exsts a sequence v n } convergng to v n L () wth F (v n ) 1. In partcular, F (v) = nf lm nf F (u n ) lm nf F (v n ) 1. u n v n n By the postvely 1-homogenety of F and R d F, and by the arbtrarness of the functon v satsfyng R d F (v) < 1, the proof s concluded. The representaton formula for the relaxaton gven by Theorem 3.5 s n accordance wth the stablty, n terms of Γ-convergence, of the class of dfference quotents assocated to geodesc dstances, consdered n the next secton. On the other hand a natural queston s whether the relaxaton of F can be represented as a supremal functonal. In vew of Proposton 2.5 ths s assured by the fact that d F s an ntrnsc dstance and ths s precsely stated n the followng Corollary. We suspect that the dstance d F assocated to any supremal functonal F s always ntrnsc but at the moment ths queston s stll open.

15 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 15 Corollary 3.6. Let F be a postvely 1-homogeneous supremal functonal represented by a functon f satsfyng (2.4). Assume that the dstance d F s ntrnsc. Then the relaxaton F of F n the strong topology of L s gven by F (u) = sup ϕ 0 d F (x, u(x)) u W 1, (), where ϕ 0 d F s the support functon of the dervatve of the dstance d F assocated to F as n (1.8), accordng to (1.4) and (1.2). Moreover ϕ 0 d F satsfes (2.4). Proof. The proof s an mmedate consequence of Theorem 3.5 and Proposton The dstance d F s geodesc. In ths paragraph we prove the remarkable fact that the dstance d F s geodesc n. Ths fact wll be crucal n our man results of Secton 4, on the characterzaton of the Γ-lmt of sequences of supremal functonals. We need some prelmnary constructons. Let d be a dstance equvalent to the Eucldean dstance. We denote by d the dstance defned by (3.12) d (x, y) = nf γ Γ x,y() L d(γ) for every x, y, where L d (γ) s the length of γ wth respect to the dstance d, and Γ x,y () denotes the set of all Lpschtz curves γ : [0, 1], wth γ(0) = x and γ(1) = y. Notce that by the fact that s connected t follows that d (x, y) < + for any x, y. Moreover snce s Lpschtz, there exsts a constant C 1 such that, chosen two arbtrary ponts x, y, we can fnd a curve γ whch connect them such that the Eucldean length of γ, L(γ), satsfes L(γ) C x y. From ths we can easly deduce that d s equvalent to the Eucldean dstance and n partcular d s bounded. Proposton 3.7. Let d be a dstance equvalent to the Eucldean dstance and let d be the dstance defned by (3.12). Then d s the smallest geodesc dstance greater than or equal to d. Proof. Let us show that d s a geodesc dstance. By defnton, we have that d d and thus d (x, y) = nf L d(γ) nf L d (γ). γ Γ x,y() γ Γ x,y() On the other hand, fx ε > 0 and γ Γ x,y () such that L d ( γ) d (x, y) + ε. By the defnton of L d ( γ) there exst k N, and t 1 = 0, t k = 1, t < t +1, such that denoted by γ the restrcton of γ to the nterval [t, t +1 ], we have nf L k 1 d (γ) L d ( γ) d ( γ(t ), γ(t +1 )) + ε γ Γ x,y() =1 k 1 L d ( γ ) + ε = L d ( γ) + ε d (x, y) + 2ε. =1 Then the reverse nequalty follows by the arbtrarness of ε. It s also very easy to check that d s the smallest geodesc dstance greater than or equal to d.

16 16 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI To smplfy the notaton n what follows and accordng wth Remark 1.1 t s convenent to extend d and d from to. It s then easy to check that (3.13) d (x, y) = nf L d (γ) = mn L d (γ) for every x, y. γ Γ x,y( ) γ Γ x,y( ) The advantage of ths extenson s that the nfmum n (3.13) s always acheved. Now for every postve δ R we construct an approxmaton d δ : R of d. For every x we defne recursvely a partton of as follows: we set C δ,x 0 := x}, (3.14) C δ,x 1 := y : d(x, y) δ and, assumng to have defned C δ,x 0,..., Cδ,x 1, we set 1 (3.15) C δ,x := y \ We defne the functon j=1 C δ,x j (3.16) d δ (x, y) := δ ( 1) + d(y, C δ,x 1 ) }, : d(y, C δ,x 1 ) δ }. f y Cδ,x. Lemma 3.8. The sequence of functons d δ } unformly converges to d. Proof. We prove the lemma n two steps. Step 1. For every δ > 0 we have d δ d. Let x, y, let us fx δ, and let γ Γ x,y (). Moreover let k N be such that y C δ,x k. Let us set t := nft [0, 1] : γ(t) C δ,x } for every 1 k, and t k+1 = 1. Notce that t < t +1, for every 1 k. Therefore we have that k (3.17) d δ (x, y) d(γ(t ), γ(t +1 )) L d (γ). By the arbtrarness of γ Γ x,y and by defnton of d the step s proved. =1 Step 2. Up to a subsequence, d δ converges unformly to some d 0 wth d 0 d. Snce the boundary of s Lpschtz regular, the dstance d s bounded. By the prevous step the sequence d δ s unformly bounded; therefore by Ascol-Arzelà Theorem d δ converges unformly, up to a subsequence, to some functon d 0. It remans to show that d 0 d. Fx x, y. The concluson follows f we construct a curve γ Γ x,y ( ) wth d 0 (x, y) L d (γ). For every δ > 0, let C δ,x 1,... Cδ,x N δ be the decomposton of defned n (3.14) and (3.15) wth N δ N such that y C δ,x N δ. Let us set p δ 0 := x, and p δ N δ := y. By constructon for every = 1,..., N δ 1 we can fnd ponts p δ Cδ,x such that d(p δ, pδ +1 ) = δ, f = 1,..., N δ 2, and d(p δ N δ 1, y) = d(y, Cδ,x N δ 1 ). By the fact that has Lpschtz regular boundary t follows that there exsts a postve constant C, ndependent on δ and, and a curve γ jonng p δ wth pδ +1, such that (3.18) L(γ ) Cδ, where L(γ ) denotes the Eucldean length of γ. Jonng these curves, we obtan a curve γ δ : [0, 1] wth end-ponts x and y. In vew of Step 1, N δ δ s unformly bounded; by (3.18) t follows that also γ δ are unformly bounded n length. Therefore, f γ δ are parametrzed wth

17 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 17 constant velocty, they are unformly Lpschtz contnuous, and hence as δ 0 they converge unformly to a Lpschtz contnuous functon γ : [0, 1]. Fxed 0 = t 1 <... < t <... < t k+1 = 1 [0, 1], n vew of (3.18) we can select ponts p δ j 1,..., p δ j k+1 n p δ 0,..., pδ N δ } such that p δ j γ(t ) for every 1 k + 1. Therefore, we have (3.19) k =1 d ( γ(t ), γ(t +1 ) ) = lm δ 0 k =1 d(p δ j, p δ j +1 ) lm δ 0 d δ (x, y) = d 0 (x, y). Takng the supremum n (3.19) over all parttons of [0, 1], we obtan L d (γ) d 0 (x, y), and ths concludes the proof of the step. The concluson follows mmedately combnng Step 1 and Step 2. We are now n a poston to prove the followng theorem. Theorem 3.9. Let F be a postvely 1-homogeneous supremal functonal of the form (2.3) wth f satsfyng (2.4). Then the dstance d F defned n (1.8) s a geodesc dstance n D( 1 β, 1 α ). Proof. By defnton (see (3.12)) we have that d F (d F ). Let us prove that also the reverse nequalty holds. Let us fx x, y and a sequence δ n 0. Assume for the moment that the sets C δn,x defned n (3.15) are Lpschtz regular. Let us frst construct a sequence u n } W 1, (), wth F (u n ) 1, such that (3.20) (u n (y) u n (x)) d δn F (x, y) 0, where d δn F s defned as n (3.16), wth d replaced by d F and δ replaced by δ n. To ths am, let v n : R be the functon z d δn F (x, z). By constructon, t s easy to verfy that for every v n (p) v n (q) sup = 1. p,q C δn,x d F (p, q) Now denote by F the restrcton of F on C δn,x. More precsely F (u) = sup f(x, u(x)) C δn,x u W 1, (C δn,x ). By the defnton of the ntrnsc dstance (1.8) we have d F (x, y) > d F (x, y), for every x, y C δn,x, and thus v n (p) v n (q) sup 1. p,q C δn,x d F (p, q) In vew of the Lpschtz regularty of the sets C δn,x we may apply Theorem 3.5 to F and obtan that F (v n ) 1. Therefore there exsts a sequence u n, h } n W 1, (C δn,x ) such that (3.21) u n, h v n unformly on C δn,x and F (u n, h ) 1,

18 18 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI the second property beng guaranteed by the 1-homogenety of F. For every n and, let h(n, ) be the ndex such that (3.22) u n, h(n,) v n δ n /2n on C δn,x. Denote u n, := u n, h(n,). We are now n a poston to construct the approxmatng sequence u n }. Snce v n = const. = δ n on C δn,x \ and u n, are close to v n, n order to glue the functons u n, t s enough to slghtly translate and then truncate. Namely ( ) ( ) u n (y) := u n, (y) 2( 1)δ n /n ( 1)δ n (2 3)δ n /n ( ) δ n (2 1)δ n /n f y C δn,x. Usng that C δn,x = 0, for every and n, we have that F (u n ) = max F (u n, ) and hence, by (3.21), we have F (u n ) 1. Moreover for every y (u n (y) u n (x)) d δn F (x, y) = u n(y) v n (y) 2N δn δ n /n, whch tends to zero as n and then (3.20) s proved. Now, by defnton (1.8), we have d F (x, y) u n (y) u n (x) for any n. Therefore, usng (3.20) and Lemma 3.8 we obtan d F (x, y) lm n u n(y) u n (x) = lm n dδn F (x, y) = (d F ) (x, y). The proof that d F s geodesc s then concluded n the case where the sets C δn,x are Lpschtz regular. In the general case we need to slghtly modfy the argument. When we ntroduce the functonals F we have to replace the set C δn,x δn,x wth a Lpschtz regular set C C δn,x, δn,x wth the property that the Hausdorff dstance between C \ and C δn,x \ s smaller then ε n, for a sutable choce of ε n. More precsely we defne F (u) = sup C δn,x f(x, u(x)) u W 1, δn,x ( C ). Then we can apply Theorem 3.5 and obtan the functons u n, defned on enough, we have u n, h(n,) δ n δ n /n on C δn,x \. δn,x C. If ε n s small Ths permts to construct as above, by translaton and truncaton, a functon u n defned on n,x δn,x Cδ whch s constant on C \. Ths functon can be easly extended to a functon n W 1, () whch s locally constant on \ n,x Cδ and clearly stll satsfes (3.20). The concluson follows as above. Fnally the fact that d F D( 1 β, 1 α ) follows by extendng d F to (see Remark 1.1) and the growth condton (2.4) together wth the fact that x y = supu(x) u(y) : sup u 1}.

19 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS Γ-convergence of supremal functonals The man result of ths secton s that the closure of 1-homogeneous supremal functonals wth respect to Γ-convergence s gven by the class of dfference quotent functonals assocated to a geodesc dstance. The followng proposton states the Γ-convergence of dfference quotents whenever the correspondng dstances unformly converge. Proposton 4.1. Let d n } be a sequence of dstances n D(α, β ). Assume that d n } converge to some dstance d D(α, β ). Then the functonals R dn defned n (2.1) Γ-converge to R d n W 1, () wth respect to the strong convergence n L. Proof. Let us prove that, for any sequence u n } n W 1, () convergng to some u unformly, we get (4.1) lm nf n Rdn (u n ) R d (u). For every x, y we have that u n (x) u n (y) lm nf n Rdn (u n ) lm nf = n d n (x, y) u(x) u(y). d (x, y) Takng the supremum n x and y we obtan (4.1). Now let u W 1, () and assume R d (u) < +. Let us defne u n W 1, () by [ ] (4.2) u n (x) = nf u(y) + R d (u)d n (x, y). y For every ε there exsts z n such that [ ] 0 u(x) u n (x) = u(x) nf u(y) + R d (u)d n (x, y) y u(x) u(z n ) R d (u)d n (x, z n ) + ε R d (u) (d (x, z n ) d n (x, z n )) + ε. Thus u n } converges unformly to u. On the other hand, by the defnton of u n t s easy to see that R dn (u n ) R d (u) for every n, and hence also the Γ-lmsup nequalty holds. Remark 4.2. Note that Proposton 4.1 holds true even f the sequence d n s a sequence of (not necessarly geodesc) dstances on, satsfyng d(x, y) M x y for every x, y for some postve constant M, and such that d n converge unformly to some functon d n. We mmedately deduce the followng Γ-convergence result for 1-homogeneous supremal functonals Theorem 4.3. Let F n : W 1, () R be a sequence of postvely 1-homogeneous supremal functonals defned by (4.3) F n (u) := sup f n (x, u(x)) where f n are Carathéodory functons satsfyng for every u W 1, (), (4.4) α ξ f n (x, ξ) β ξ for every ξ R n, for a.e. x. Then there exsts a subsequence (stll labeled by n) and a dfference quotent functonal R d, wth d D(α, β ), such that F n Γ-converges to R d n W 1, () wth respect to the strong convergence n L ().

20 20 ADRIANA GARRONI, MARCELLO PONSIGLIONE, AND FRANCESCA PRINARI Proof. Clearly we have that F n Γ-converges to R d f and only f the relaxaton F n of F n Γ-converges to R and hence t s enough to prove the statement for the sequence F n. By Theorem 3.5 we know that the relaxaton of F n s gven by F n = R dn, where d n = d Fn denotes the dstance defned by (1.8) correspondng to F n. By Theorem 3.9 d n s a geodesc dstance n D( 1 β, 1 α ). By Proposton 1.2, there exsts a subsequence (stll denoted by d n) and a dstance d D(α, β ) such that d n } converges unformly to d. Therefore by Proposton 4.1 the functonals R dn Γ-converge to R d n W 1, () wth respect to the strong convergence n L and ths concludes the proof. Next Theorem establshes that the class of dfference quotents assocated to a geodesc dstance s the closure of 1-homogeneous supremal functonals wth respect to Γ-convergence. Theorem 4.4. Let R d be a dfference quotent functonal assocated to a dstance d D(α, β ). Then there exsts a sequence of 1-homogeneous supremal functonals F n of the type (4.3), wth f satsfyng (4.4), such that F n Γ-converge to R d n W 1, () wth respect to the strong convergence n L (). Proof. In vew of Corollary 3.6 and Proposton 4.1 the proof reduces to approxmate the dstance d by a sequence of ntrnsc dstances d n D (α, β ) wth respect to the unform convergence. Ths s a consequence of the fact that every geodesc dstance can be approxmated by dstances assocated to smooth Fnsler metrcs satsfyng the same bounds, as proved n [12, Theorem 4.1]. Remark 4.5 (The class of supremal functonals s not closed under Γ-convergence). Note that Theorem 4.4 mples that the class of 1-homogeneous supremal functonal s not closed wth respect to Γ-convergence. In fact, let = ( 1, 1) 2 and let d be the non ntrnsc dstance gven n Example 1.8. In Example 2.6 we proved that R d can not be wrtten n a supremal form. On the other hand, by Theorem 4.4 there exsts a sequence of supremal functonals F n Γ-convergng to R d n W 1, () wth respect to the strong convergence n L (). An explct sequence of functonals F n Γ-convergng to R d s the followng: let S = ( 1, 1) 0} and for every n N let S n := ( 1, 1) ( 1/n, 1/n). Let F n be the supremal functonals assocated to the functons f n defned by β ξ f x S n, for every ξ R n ; f n (x, ξ) := α ξ f x \ S n, for every ξ R n. It s easy to check that d Fn d unformly, and hence the functonals F n Γ-converge to R d n W 1, () wth respect to the strong convergence n L (). References [1] E. Acerb, G. Buttazzo, F. Prnar: The class of functonals whch can be represented by a supremum. J. Convex Anal. 9 (2002), [2] G. Aronsson, M.G. Crandall, P. Juutnen: A tour on the theory of absolute mnmzng functons. Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, (electronc). [3] E. N. Barron, R. R. Jensen: Relaxed mnmal control. SIAM J. Control and Optmzaton 33, no.4, (1995), [4] E. N. Barron, R. R. Jensen, C. Y. Wang: Lower Semcontnuty of L Functonals. Ann. Inst. H. Poncaré Anal. Non Lnéare (4) 18 (2001), [5] E. N. Barron, W. Lu: Calculus of varatons n L. Appl. Math. Optm 35 (1997), [6] A. Bran, A. Davn: Monge solutons for dscontnuous Hamltonans. ESAIM Control Optm. Calc. Var. 11, no. 2, (2005),

21 FROM 1-HOMOGENEOUS SUPREMAL FUNCTIONALS TO DIFFERENCE QUOTIENTS 21 [7] G. Buttazzo, L. De Pascale, I. Fragalà: Topologcal equvalence of some varatonal problems nvolvng dstances. Dscrete Contn. Dynam. Systems 7, no. 2, (2001), [8] G. Buttazzo, L. De Pascale, I. Fragalà: Erratum to Topologcal equvalence of some varatonal problems nvolvng dstances. Dscrete Contn. Dynam. Systems 7, no. 2, (2001), [9] F. Camll, A. Sconolf: Nonconvex degenerate Hamlton-Jacob equatons. Math. Z. 242, no. 1, (2002), [10] T. Champon, L. De Pascale: A prncple of comparson wth dstance functons for absolute mnmzes. Preprnt. [11] M.G. Crandall, L.C. Evans, R.F. Garepy: Optmal Lpschtz extensons and the nfnty Laplacan. Calc. Var. Partal Dfferental Equatons 13, no. 2, (2001), [12] A. Davn: Smooth approxmaton of weak Fnsler metrcs. Dfferental Integral Equatons 18 no. 5, (2005), [13] F. Prnar: Γ -convergence of L functonals (n preparaton). [14] G. De Cecco, G. Palmer: Integral dstance on a Lpschtz Remannan Manfold. Math. Z. 207, no. 2, (1991), [15] G. De Cecco,G. Palmer: Dstanza ntrnseca su una varetá fnslerana d Lpschtz. Rend. Accad. Naz. Sc. V, XVIII, XL, Mem. Mat., 1, (1993) [16] G. De Cecco, G. Palmer: Lp manfolds: from metrc to fnsleran structure. Math. Z.218, (1995) [17] M. Gor - F. Magg: On the lower semcontnuty of supremal functonals. ESAIM: COCV, 9 (2003), [18] E. J. McShane: Extenson of range of functons. Bull. Amer. Math. Soc, 40, no.2, (1934), [19] F. Prnar: Relaxaton and Γ-convergence of supremal functonals. Boll. Unone Mat. Ital. Sez. B Artc. Rc. Mat. (to appear). [20] A. Sconolf: Metrc character of Hamlton-Jacob equatons. Trans. Amer. Math. Soc. 355, no. 5, (2003), (Adrana Garron) Dpartmento d Matematca, Unverstá La Sapenza, P.le Aldo Moro 5, Roma, Italy E-mal address, Adrana Garron: garron@gove.mat.unroma1.t (Marcello Ponsglone) Max-Planck Insttut für Mathematk n den Naturwssenschaften Inselstr , D Lepzg, GERMANY E-mal address, Marcello Ponsglone: ponsgl@ms.mpg.de (Francesca Prnar) Dpartmento d Matematca, Unverstá d Lecce, Va Prov.le Lecce-Arnesano, Lecce, Italy E-mal address, Francesca Prnar: prnar@mal.dm.unp.t