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Journal of Applied Mathematics and Physics, 2014, 2, 194-203 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.25024 How to cite this paper: Granucci, T. (2014) Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Function- als with General Growth Conditions. Journal of Applied Mathematics and Physics, 2, 194-203. http://dx.doi.org/10.4236/jamp.2014.25024 Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Functionals with General Growth Conditions Tiziano Granucci Scuola Superiore, Istituto P. Calamandrei, Firenze, Italy Email: tizianogranucci@libero.it Received September 2013 Abstract In this paper we proof a Harnack inequality and a regularity theorem for local-minima of scalar intagral functionals with general growth conditions. Keywords Harnack Inequality, Regularity, Hölder Continuity 1. Introduction In this paper we proof a Harnack inequality for local-minima of scalar intagral functionals of the calculus of variation of that type (1.1) where Ω is a bounded open subset of N , Φ:[0,+∞)→[0,+∞) is a N-function and Φ globally satisfies the Δ′- condition in [0,+∞), N f: is a Carathéodory function and there exist L,L0, L₁₂ ₂ and ∈∈ℝℝ for a. e. xΩ and for every (s,z)×N . The risearch of regularity results for elliptic and parabolic equations start from the basic and most important results of E. De Giorgi [5] and J. Nash [27]. In 1990s, beginning from the papers of G. Astarita and G. Marrucci [3] and J. P. Gosez [13] has been developed a remarkable production of regularity results for functionals with general growths. In [7], [8] and [25], M. Fuchs, G. Mingione, G. Seregin and F. Siepe have studied functionals of the type (1.2) showing results of partial and global regularity for the minimizer of such functional in the scalar and vectorial case. Moreover in [8] M. Fuchs and G. Mingione, have already studied functionals of this type (1.3) J,,, ufxuxuxdx ,,|| || zfxszL z ₂ |,1| J uuxlnuxdx ||,. J uudx T. Granucci 195 In papers [7,8,25] the regularity of the minimizer of the functionals (1.2) and (1.3) has been obtained starting from the weak Eulero-∈ Lagrange equations using the hypothesis: ΦC². We remember that in [7,8,25] there are important estimations on the L^{∞} norm of the gradient of the minima both in the scalar case and in the vec- torial one. In [24] E. Mascolo and G. Papi have determined an inequality of Harnack for the minimizer of the functional (1.3) under the condition ∈ ΦΔ 2∇ ∩2∈ . We observe that Φ Δ2∇ ∩2 implies (1.4) with real positive constants c,c,c,c₁₂₃₄ and 1 < p ≤ m. Therefore the functional (1.3) satisfies non-standard growth conditions. Classical regularity theorem for functionals with standard growth conditions (p = m) has been proved in [9] and [10] (for a didactic explanation refer to [2,11,12]). In [26], G. Moscariello and L. Nania has obtain a results of hölder continuity for the local-minima of functional of the type (1.1) under the hypothesis that (1.4) holds with 1 < p ≤ m < ((Np)/(N-p)). In [17], G. M. Lieberman proved an Harnack inequality for the local- ∈ minima of the functional (1.1) with ΦC² suth that verifies the following relation with 0 < c5 < c6. We are interested in functionals with quasi-linear growths and we will proof a regularity result which extend the ones obtained in [17,24,26] to a wider N-functional class. In particular we get that the local- minima of the following functionals: (1.5) are holder continuous functions. In [14] and [15] we start to study the regularity of the local-minima introducing a maximal LΦ-L∞ inequality and estimating the measure of the level set A(k,R). Moreover in [15] and [16] we have shown that the following hypothesis can be used in order to give a new estimation of the measure of the li- vel set A(k,R): H-1) Φ globally satisfies the Δ′-condition in [0,+∞); H-2) there exists a constant H c0 ₂ (1.6) H-3) there exists a constant H c0 ₃ (1.7) Under these hypotheses we can show the following result. Theorem 1: If uW¹LΦ(Ω) is a quasi-minima of the functional (1.1) and if Φ confirm the hypotheses H-1, H-2 and H-3; then u is locally hölder continuous. In these pages we show that the hypotheses H-2 and H-3 are purely technical and they can be eliminated. We can subsequently weaken besides H-1. We will suppose that the following hypothesis hold. G-1) Let ϖ: be an increasing function such that (1.8) for every t and for every ɛ(0,1), where cG > 0 is a real constant. Moreover we suppose that We say that Φ∈G if (1.8) holds. The hypothesis G-1 implicates a type of quasi-sub-homogeneity condition on the N-function Φ. Remark 1: We observe that if Φ∈Δ₂∩∇₂ then by Lemma 3 (i) we have Then the functions Φ∈Δ₂∩∇₂ verify the hypothesis G-1. Remark 2: We observe that if Φ∈Δ′ on (0,+∞) then Φ verify the hypothesis G-1; in fact 0 pm tc tctcfort ₂₃₄ '' ' 56 / 0 ct ttcfort |,1 |1 p Juulnudx withp 1/ 0,1; H ttcfor every t ₂ 11/ 0,1. m H tctforeveryt ₃ G tc t 0 lim 0. x s 1/ . rr r ttt . tc t T. Granucci 196 Our principal results will be, a weak inequality of Harnack [Theorem 5] and the corollary of regularity that it follows of it [Corollary 2]. The proof of the Harnack inequality uses the techniques introduced in [6,17] and [24]. The only present novelty in the demonstrative technique is the use of an ɛ-Young inequality. This simple trick allows to recover the results introduced in [15-17,24,26] in a simple way and without using the properties of the functions Δ2∩2 (see Lemma of [15,24] and [26]). We finally observe that the hypotheses Δ2∩2 it is not, in general, equivalent to H-1; therefore the hypothesis G-1 seems to be slightly more general of those introduced in [15-17,24,26]. Definition 1: Let p be a real valued function defined on [0,+∞) and having the following properties: p(0) = 0, p(t) > 0 if t > 0, p is nondecreasing and right continuous on (0,+∞). Then the real valued function Φ defined on [0,+∞) by (1.9) is called an N-function. The function Φ: [0,+∞)→[0,+∞) defined by (1.9) satisfies the following properties: Definition 2: Let p be a real valued function defined on [0,+∞) and having the following properties: p(0) = 0, p(t) > 0 if t > 0, p is nondecreasing and right continuous on (0,+∞). We define and (1.10) The N-functions Φ and Ψ given by (1.9) and (1.10) are said to be complementary. Particularly for us it will be important the following Lemma. Lemma 1: Let Φ be an N-function, let Ψ be the complemantary N-function of Φ then we have (1.11) ∀s, t∈ℝ⁺. Moreover for every ɛ > 0 we get (1.12) Definition 3: A N-function Φ is of class ₂ globally in (0,+∞) if exists k > 1 such that (1.13) Definition 4: A N-function Φ is of class ₂^{m} globally in (0,+∞), with m>1, if for every λ>1 (1.14) The N-functions m ₂ are characterized by the following result Lemma 2: Let Φ be a N-function and let Φ'₋ be its left derivative. For m > 1 the following properties are equivalent: 1) Φ(λt) ≤ λmΦ(t), for every t ≥ 0, for every λ>1; 2) tΦ'₋(t) ≤ mΦ(t), for every t ≥ 0; 3) the function Φ(t)/tr is non-increasing on (0,+∞). 0,[] t tpsds 00 0 0; andtif t 0,; is continuous on 0,; isstrictlyincreasingon 0,; is convex on 0 lim/0 lim/; x x tt andtt 0, //. if stthensstt pt s qs supt 0[], . t tqsds s ts t 1/1/ ,. ststst ¡ 2 0,. tkt t 0, . m ttt T. Granucci 197 The N-functions Φ∈∇₂r are characterized by the following result Lemma 3: Let Φ be a N-function and let Φ'₋ be its left derivative. For r>1 the following properties are equiv- alent: 1) Φ(λt) ≥ λrΦ(t), for every t ≥ 0, for every λ > 1; 2) tΦ'₋(t) ≥ rΦ(t), for every t ≥ 0; 3) the function Φ(t)/λr is non-decreasing on (0,+∞). Definition 5: We say that a N-function Φ belongs to the class Φ∈∇₂r if any of the three condition (i)', (ii)' or (iii)' is satisfied. Definition 6: We say that the N-function Φ satisfies the Δ′-condition if there exist positive constants—c and t₀—such that (1.15) for every t,s ≥ t₀. Definition 7: We say that the N-function Φ globally satisfies the Δ′-condition in [0,+∞) if (1.12) holds for every t, s ≥ 0. We remember that if Φ∈C² then Φ∈Δ′ if tΦ''(t)/Φ'(t) is a non-increasing function, for further details refer to Theorems 5.1 and 5.2 and to the Lemma 5.2 of [19]. Lemma 4: If the N-function Φ satisfies the Δ′-condition then it also satisfies the ₂-condition The N-functions satisfy the Δ′-condition. Moreover ₁ and ₂ satisfy the Δ′-condition globally in [0,+∞) and belong to the class ∇2 globally in [0,+∞). The function ₃ does not satisfy Δ′-condition for all t,s ≥ 0 and Φ3∇2. Ossevia- mo inoltre che la funzione ₄₂₂ but ₄ does not satisfy the Δ′-condition. For further details refer to [1,19,28]. Now we can introduce Orlicz spaces and Orlicz Sobolev Spaces, LΦ and W¹LΦ; in these definitions and throughout the article we assume that Φ is a N-function of class m ₂ for some m > 1 and that Ω⊂ℝN is a bounded open set with Lipschitz boundary. Definition 8: If u is a LN-measurable function on Ω and: ∫ΩΦ(|u|)dx <+ ∞ then u∈LΦ(Ω). Moreover (1.16) where ∂iu, for I = 1,...,N, are the weak derivatives of u. Theorem 2: LΦ(Ω) e W¹LΦ (Ω) are Banach spaces with the following norms (1.17) and (1.18) For greater details we refer to [1,19,28]. If u∈Wloc¹LΦ (Ω), k is a real number and R Q, we set Remark 3: For almost each k∈ℝ we get |A(k,R)| = |QR|-|B(k,R)|. Definition 9: If uWloc¹LΦ (Ω), we say that u∈ODGΦ⁺(Ω,H, R) if for every couple of concentric balls Qϱ⊂QR⋐QR₀⋐Ω, with R<R₀, and for every k∈ℝ we have (1.19) ts cts ₄ 1; p ttwith p ₁ 1 1; p ttlntwith p ₂ 11; ttlntt ₃ ²/11 . tt lnt ₄ ¹: 1,..., i WLuLuLforiN , 0:|/1| uinfkukdx ‖‖ 1, ,,1,....,, . iNi uu u ‖‖‖‖‖ ‖ ,: , R R AkRxQuxkukQ ,: . R R BkRx QuxkukQ () (,,) || / Ak AkR udxHukRdx ñ ñ T. Granucci 198 Definition 10: If u∈Wloc¹LΦ (Ω), we say that u ∈ODGΦ⁻ (Ω,H,R₀) if for every couple of concentric balls Qϱ⊂QR⋐QR₀⋐Ω, with R < R₀, and for every k∈ℝ we have (1.20) Definition 11: If u∈Wloc¹LΦ (Ω), we say that u∈ODG_{Φ}(Ω,H, R₀) if u∈ODG_{Φ}^{±}(Ω,H,R₀), that is Theorem 3: If u∈ODGΦ⁺(Ω,H,R₀) then u is locally bounded above on Ω. Furthermore, for each x₀∈Ω and R ≤ min(R₀,d(x₀,∂Ω),1) there exists an universal constant c5 = c5(N,m,H) such that Proof: The proof follows using the demonstration methods presented in [24]. Corollary 1: If u∈ODGΦ(Ω,H,R₀) then u is locally bounded on Ω. Furthermore, for each x₀∈Ω and R ≤ min(R₀,d(x₀,∂Ω),1) there exists an universal constant c6 = c6(N,m,H) such that Proof: The proof comes after Theorem 3 remembering that if u∈DGΦ⁻(Ω,H,R₀) then -u∈DGΦ⁺(Ω,H,R₀). Moreover the following lemma is valid: Lemma 5: If u∈DGΦ⁺(Ω,H,R₀) then u is locally bounded above on Ω. Furthermore, for each x₀∈Ω, R ≤ min(R₀,d(x₀,∂Ω),1) and for every p > 1 there exists an universal constant c7 = c7(p,N,m,H) such that (1.21) for each R QQ and 0 < ϱ < R. Proof: The proof comes after Theorem 3 using the demonstration methods presented in [24]. Definition 12: Let u∈Wloc¹LΦ(Ω) then it is a local minima of (1.1) if for every ϕ∈W₀¹LΦ(Ω) we have Moreover we get: Theorem 4 (Caccioppoli inequalities): If ₂ and u∈Wloc¹LΦ(Ω) is a local minima of (1.1) then u∈ODGΦ(Ω,H,R₀). Using the previous results we obtain the following theorems: Theorem 5 (Weak Harnack inequality): Let Φ be a N-function. Let u be a positive function satisfying (1,17). If Φ∈G; then there exists p > 1 and a constant c > 0 such that (1.22) Theorem 6 (Main Theorem-Harnack inequality): Let Φ be a N-function. Let u be a positive local minimizer of (1.1). If Φ∈G; then there exists a constant c > 0 such that, for σ∈(0,1) we have (1.23) Proof (Proof of the Main Theorem): Using the (1.21) and (1.22) we have (1.23). Corollary 2: Let Φ be a N-function. If Φ∈G and u∈W¹LΦ(Ω) is a local minimizer of the functional (1.1); then u is locally hölder continuous. Proof: Using (1.20) and the technique introduced in [6,11,12] we get the proof. We finish observing that with small changes our demonstrative technique can also be applied to the quasi-mi- nima of the functional (1.1). Besides we can also apply this demonstration using equivalent N-functions. Unfor- tunately, ttln1t ₂ does not verify H1; for this Φ₂∈Δ′ on [t₀,+∞) with t₀>0 but Φ₂∉△′ globally on [0,+∞). We should think to solve this problem using the concept of equivalent N-function; the demonstrative technique allows it, but we do not know if it exists a N-function ₃ equivalent to ₂ which globally verifies Δ′ globally on [0,+∞). It is still an unsolved problem.I thank the colleague Dott. Elisa Albano who translated the article into English supporting and encouraging me so much. () (,,) || / Bk BkR udx HkuRdx ñ ñ 000 ,,,,,, . ODGHRODGH RODGH R /2 7 /| | QRR QR ess supuxcQu dx /2 7 || ||/.|| QRR QR esssupu xcQudx 1/ 1/ 7( ||) ||/ N pp QQR esssupucRu dx ñ ñ ,,. JusuppJusupp 1/ 1/ /2 1/ . p Np QR QR essinfucRu dx . QR QR esssupucessinfu T. Granucci 199 2. Proof of the Weak Harnack Inequality 2.1. Lemmata Let define then we have the following Caccioppoli inequalities (2.1) and (2.2) where 0 < σ < τ < 1 and k∈ℝ. Let us start remembering the following lemma: Lemma 6: Let g(t), h(t) be a non-negative and increasing functions on [0,+∞) then g(t)h(s) ≤ g(t)h(t) + g(s)h(s) for every s, t∈[0,+∞). Proof: If s ≤ t then g(t)h(s) ≤ g(t)h(t) ≤ g(t)h(t) + g(s)h(s). If t ≤ s then g(t)h(s) ≤ g(s)h(s) ≤ g(t)h(t) + g(s)h(s). Let us remember for the sake of completeness the following lemma: Lemma 7: Let ₂ and u∈W¹LΦ(Ω). Suppose that u is positive in QR and satisfies (2.2) then there exists a positive constants δ₀ such that if for some θ > 0 we have |B(θ,u,R)| ≤ δ₀|QR|, then (2.3) Proof: The proof follows using the demonstration methods presented in [24]. Refer to Lemma 4.1 of [24]. Our demonstration of the weak inequality of Harnack founds him on the following Lemma 8. We have shown the Lemma 8 using an opportune ɛ-Young inequality. Lemma 8: Let be Φ a N-function and Φ∈G. Let u∈W¹LΦ(Ω). Suppose that u satisfies (2.2). For every δ∈(0,1) and T > (1/2), there exists a positive constant μ(δ,T) such that if u is positive on Q2TR and there exists θ > 0 such that |B(θ,u,R)| ≤ δ|QR|, we have (2.4) Proof: Let δ∈(0,1). We first prove that if u is positive in QR and there exists θ > 0 such that |B(θ,u,R)| < δ|QR|, there exists a constant λ(θ) such that (2.5) We consider the function wR define by wR(y) = 0 if vR(y) ≥ k, wR(y) = k − vR if k > vR(y) > h, wR(y) = k − h if vR(y) ≤ h where vR(y) = ((u(Ry))/R), y∈Q₁. Let us consider ki = (θ/(2iR)) with I ≤ ν, since wR = 0 in iR Q\Bk,v ,1₁ and by Sobolev inequality we have and (2.6) where Δi = B(ki,vR,1)\B(ki-1,vR,1). Using the Young inequality ab ≤ Ψ(a) + Φ(b), where Φ is the complementary function of Φ, we have /, R vyuRyRyQ ₁ ),(,, ),( || / AkvRRAk vRR vdxHvk dx ),(,,),( || / BkvRRBk vRR vdxHkv dx /2 /2 . QR inf u ,. QTR inf uT /2 QR inf u \,,11 iR R QBkv Q ₁ /1 1/ () )1/ 1 ,,1 ,,1(] NN NNN i iiRiRQR kk BkvBkvwdx ₁ 1/ 1 ,,1,,1|' N iiiRSNiRiR R kkBkvCBkvww dx /('/ iRRi RR ww dxmwmw dx T. Granucci 200 and then (2.7) Since from the inequality (see inequality (6), page 230 of [1]) we have then Moreover, since Φ globally satisfies the Δ′-condition in [0,+∞), it follows since using Caccioppoli’s inequality (2.2) we have Summing the last inequality on i from 0 to ν we have and Fix ɛ = (1/(1 + ν)1/2), then and (2.9) From (2.9) we have Since ϖ(s)↓0 for s↓0 then we can choose ν such that where δ₀ is the constant in Lemma 7, then there exists λ(δ₀) such that /'//'(||/ iRRi RR mwmwdxm wmwdx '|.|// iRRi RR wwdxmwmwdx ) '/ ('// R RR RRR wmw w mwww /) ttt /||, iRRiR R wwdxm wwdx 1/ 1 ,,1,,1/|| . N iiiRSNiRiRR k kBkvC Bkvmwwdx 1/ 1 1 ,,1,,1/| | N i iiRSNiRi iiGiR kk BkvCBkvmkkmccwdx ₁ || || iR iR wdx vdx 11/ | ,,1 /. N iRSNiSN BkvCmC mcQ ₂₂ 11/ 1,,1 /1 N i RSNSN BkvCmQC mcQ ₁₂ ₂ 11/ ,,1/ 1. N i RSNSN BkvCmQC mcQ ₁₂₂ 11/ 1/ 21/ 2 ,,1/11/1 N i RSNSN BkvCmQC mcQ ₁₂ ₁ 11/ 11/1/ 21/2 ,,1 1/11/1 N N iR SN BkvC mcQ ₂₁ /1 /1 1/2( 1) 2)/( (,,11/1 1/1. NN NN iR SN BkvC mcQ ₂₁ () () /1 /1 1/21/21/ 0 1/11/1/ 2(1 NN N NNN SN Cm c ₂ /2 . QR inf u T. Granucci 201 Let now T > (1/2) and assume |B(θ,u,R)| ≤ δ|QR| and u positive in Q2R. Since we have then there exists a constant depending on δ and T such that (2.4) holds. Using the technique introduces in [11] we get the following lemma. Lemma 9: Let u∈DGΦ⁻ with k₀ = 0 and let u be positive in Q₂. Let δ∈(0,1) and t > 0. If then where c = c(δ) being as in Lemma 8 with δ = 2-N-1. Proof: For s = 0 the claim is true by Lemma 8. Now we use the inductive process. We assume the claim true for some s and we prove it for s + 1. Let us define Ai = {x∈Q₁:u(x) > cit}; by hipothesis, if A₀ = {x∈Q₁:u(x) > t} then We have two alternative. 1) We assume |A₀| > 2-s|Q₁|, the by inductive hypothesis 2) Otherwise 2-s-1|Q₁| < |A₀| < 2-s|Q₁|. Let us assume g = χ_{A₀} and apply the Calderon-Zygmund argument to g in Q₁ with parameter (1/2) then we find a sequence of dyadic cubes {Qj} such that 1 1/2 ; j j g in QQ if Qj is one of the 2N subcubes of Pi arising during the Calderon-Zygmund process, then From (2) and (3) we get 000 1/2; ii i ii i A AP APP moreover We apply Lemma 8 and we obtain Let us consider then PiA₁ and by inductive hypotesis 2 ,,211/2 R TR A uTRQT Q ⁿ 2 ,,21 1/2 TR BuTRTQ ⁿ :2 s x Qux tQ ₁₁ 1/ 2 s Q infuc t 1 0 1/ 2. s AQ ₁ 1 1/2 s s Q infuc tct 1 |1/ 21/2;| N jj QQgdx ||1/1 /2. iPi P gdx 1 00| | 1/ 21/ 2|. N ij ji A PAQ QP Pi inf uct : A xQuxct ₁₁ 1 0 21/2 s QA A ₁₁ 1 1/ 2 s Q infuc t T. Granucci 202 2.2. Proof of the Weak Harnack Inequality Now we can proof the inequality (1.19) using the technique introduced by Di Benedetto-Trudinger in [6]. Proof (Proof of Theorem 5); Given any t > 0 choose an integer s such that i.e. then by Lemma 9 we get therefore Let us define then where α = ln((1/2)/ln(c). 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