Remarks on the Continuity of the Local Minimizer of Scalar Integral Functionals With Nonstandard General Growth Conditions Dedicated to Caterina and

In this paper we show a regularity theorem for local minima of scalar integral functionals of the Calculus of Variations with nonstandard general growth conditions. Let us consider functionals in the following form F [u,Ω] = ∫ Ω f (x, u (x) ,∇u (x)) , dx where f : Ω × R × RN → R is a Carathéodory function satisfying the inequalities Φ (|z|) − c1 ≤ f (x, s, z) ≤ c2 [ 1 + (Φ∗ (|z|)) + (Φ∗ (|s|)) ] for each z ∈ RN , s ∈ R and for LN-a. e. x ∈ Ω, where c1 and c2 are two positive real constants, with c1 < c2, Ω is an open subset of RN , N ≥ 2, Φ ∈ 2 ∩ ∇2 [Definition 6 and Definition 8], 1 ≤ r < m < N and the function Φ∗ is the Sobolev conjugate of Φ [Definition 12], β is a positive real number that we will opportunely fix [Hypothesis H1, f ].


Introduction
In this paper we show a regularity theorem for local minima of scalar integral functionals of the Calculus of Variations with nonstandard general growth conditions.
Let us consider functionals in the following form for each z ∈ R N , s ∈ R and for L N -a.e. x ∈ Ω, where c 1 and c 2 are two positive real constants, with c 1 < c 2 , Ω is an open subset of R N , N ≥ 2, Φ ∈ m 2 ∩∇ r 2 [Definition 6 and Definition 8], 1 ≤ r < m < N and the function Φ * is the Sobolev conjugate of Φ [Definition 12], β is a positive real number that we will opportunely fix [Hypothesis H 1, f ].The research of regularity results for weak solutions of elliptic and parabolic PDEs starts from the basic results of De Giorgi (1957) and Nash (1958).In 1990s a remarkable production of regularity results for functionals with general growths has been developed.
In this paper we proof a theorem on the regulaity of quasi-minima of the functional F [u, Ω] with the following hypotheses.
for each z ∈ R N , s ∈ R and for L N -a.e. x ∈ Ω, where c 1 and c 2 are two positive real constants, with then for every r and m, with 1 ≤ r ≤ m < N, there exists β 1,r,m,N , implicitly defined by Γ 1 (N, m, r, β) = 0 , such that if r(N−m) mN < β < β 1,r,m,N , then β is a solution of the system (1.4).Our hypoteses on β will be r H 2, f For every t ≥ 0 we have t r ≤ Φ (t) . (1.5) Remark 1 For the whole paper we will suppose that the system (1.4) has some solutions, in Appendix we will study in detail such system and we will find some conditions of existence, the relationship (1.5).

Remark 3
The hypothesis H 2, f is purely tecnical, it can be removed [Corollary 1].
Moreover we will assume that f satisfies on of the following hypotheses.
H 3,2, f For almost every x ∈ Ω and for every s ∈ R, f (x, s, •) is a convex function; moreover there exists a constant for almost every x ∈ Ω, for every pair s 1 , s 2 ∈ R, with |s 1 | ≤ |s 2 |, and for every ξ ∈ R N .
Our principal result is the following theorem.
Particulary Theorem1 and Corollary 1 can be applied in the following cases with p ≥ 1 and a > 0, if t ≥ t 0 where sin (ln (ln (t 0 ))) = −1 and 1 < p < q < N.

Definitions
Definition 2 A continuous and convex function Φ: (2.1)For exemple the function Φ p,β (t) = t p ln β (1 + t) for p > 1 and β ≥ 0 or p = 1 and β > 0 is a N-function.Actually, only the growth at infinity really matters in the definition of N-function.Indeed, given a continuous and convex function there exist a N-function Φ and t 0 > 0 such that for every t > t 0 there holds The function A is called principal part of the N-function Φ.For exemple there exists a N-function Φ such that Φ (t) = t ln(t) near infinity or there exists a N-function Φ such that Φ (t) = t ln(t) near infinity.
Definition 3 If Φ 1 and Φ 2 are two N-functions we say that Φ 1 dominates Φ 2 near infinity if there exist two positive constants κ and t 0 such that Φ 2 (t) ≤ Φ 1 (κt) for all t ≥ t 0 .
Definition 4 If Φ 1 and Φ 2 are two N-functions we say that Φ 1 and Φ 2 are equivalent near infinity (Φ 1 ∼ Φ 2 ) if and only if there exist some positive constants κ 1 , κ 2 and t 0 such that for all t ≥ t 0 .
Φ 2 (t) < +∞ then Φ 1 and Φ 2 are equivalent near infinity.Let us introduce two important classes of N-functions.
The class 1 2 contains only linear functions.
The N-functions Φ ∈ m 2 are characterized by the following result.Lemma 1 Let Φ be a N-function and let Φ− be its left derivative.For m ≥ 1 the following properties are equivalent: , for every t ≥ 0, for every λ > 1; (ii) t Φ− (t) ≤ mΦ (t), for every t ≥ 0; (iii) the function Φ(t)  t m is nonincreasing on (0, +∞).
The N-functions Φ ∈ ∇ r 2 are characterized by the following result.Lemma 2 Let Φ be a N-function and let Φ− be its left derivative.For r ≥ 1 the following properties are equivalent: for every t > 1.
Definition 10 If Ω ⊂ R N is a bounded open set and Φ ∈ 2 then we define where ∂ i u are the weak derivatives of u for i = 1, ..., N.
Definition 11 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 increasing and right continuous on (0, +∞).We define and The N-functions Φ and Φ are complementary N-functions.
Particularly from the relationship (2.1) of the Definition 2 we get the following Young inequality ab ≤ Φ (a) + Φ (b) .
( 2 .8) We now recall the notion of Sobolev conjugate of a N-function.For a sake of simplicity, we will only consider the case of a function in m 2 .Definition 12 Assume that Φ ∈ m 2 , with 1 ≤ m < N. We define the Sobolev conjugate of Φ as the function Φ * whose inverse is defined by ( 2 .9) for almost every k ∈ R, so that when necessary we can assume without loss of generality that all the vcalues k under consideration will satisfy this relation.
Our proof is based on the following Caccioppoli inequalities.
then there exist two real positive constants C 1,Cac , C 2,Cac depending only on c 1 , c 2 , N, p and β such that for every x 0 ∈ Ω, for every cube Q (x 0 ) ⊂⊂ Q R (x 0 ) ⊂⊂ Ω and for every k ∈ R we have Proof.It follows by Theorem 4.1 of Dall'Aglio, Mascolo, and Papi (1997).
Remark 14 Let us take v = u + 1 and h = k + 1, then using (3.1) and (3.2) we get Using an abuse of notation we will always identify u with v and h with k.
We can now introduce the adequate De Giorgi classes related to the functional (1.1).
and for all k ≥ k 0 ≥ 0 we have (3.3) (3.4) . Lemma 4 Let θ > 0 and let {x i } i∈N be a sequence of real positive numbers such that and lim then u is locally bounded in Ω and there exists a R 0 > 0 such that for every x 0 ∈ Ω and 0 < R ≤ min (R 0 , dist (x 0 , ∂Ω)) we have where C (R) is a real positive constant depending on R, N, m, r and β.
Now we introduce our first result that improves the precedent theorem shown in general in (Dall'Aglio, Mascolo, & Papi, 1997, 1998).
where α , and C 22 is a real positive number depending on N, β, α 1 , α 2 and γ. where using the Sobolev Inequality we get where Using the Caccioppoli inequality we obtain where From (3.14) and (3.15) it follows be a constant that we shall fix later, and define for i ≥ 1; (3.17) where Φ * ,β (t) = (Φ * (t)) β , then, using (3.16), we get Now we give two alternative estimates of Since u is bounded we get where (3.20) Using (3.18), (3.19) and (3.20) it follows and where we can apply Lemma 4; then lim The condition imposed on d will be satisfied taking where and ϑ is the positive solution of ϑ (ϑ where where . Since t r ≤ g (t) and β > (N−m)r mN then βmN (N−m)r > 1 and we get where then, using Caccioppoli inequality (3.1) for the levels k = k i = M (2R) − 2 −i−1 osc (u, 2R) and h = k i−1 , proceeding as in the previous Theorem 5 we get Summing over i from 1 to ν we have and Lemma 6 Let ϕ be a positive function, and assume that there exist a constant q and a number τ, 0 < τ < 1 such that for every R < R 0 ϕ (τR) ≤ τ δ ϕ (R) + BR κ (4.9) with 0 < κ < δ and ϕ (t) ≤ qϕ τ k R (4.10) where C is a constant depending only on q, τ, κ and δ.
Now we can prove our main Theorem.

Appendix: Hypotheses on β
Let us consider the system then the system (5.1) can be written in this way In the following figures we draw the graphs of E (y, w), e 1 (y), e 2 (y) and e 3 (y).