REGULARITY UNDER SHARP ANISOTROPIC GENERAL GROWTH CONDITIONS

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1 REGULARITY UNDER SHARP ANISOTROPIC GENERAL GROWTH CONDITIONS GIOVANNI CUPINI PAOLO MARCELLINI ELVIRA MASCOLO Abstract. We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type 1.1) below, under sharp assumptions on the exponents p i in terms of p : the Sobolev conjugate exponent of p; i.e., p = np, 1 = P 1 n 1 n p p n p i. As a consequence, by mean of regularity results due to Lieberman [21], we obtain the local Lipschitz-continuity of minimizers under sharp assumptions on the exponents of anisotropic growth. 1. Introduction Integrals of the calculus of variations of the form Fu) = u xi x) pix) dx 1.1) for some bounded measurable functions p i x) may have not smooth, even unbounded, minimizers. This happens also in the case of constant exponents p i, i = 1,..., n, if they are spread out; i.e., if the ratio maxp i / minp i is not close enough to 1 in dependence on n. In fact integrals as in 1.1), with constant exponents p i, may have unbounded minimizers [18], [22], [23], see also [19]) for instance when n > 3 and 2 n 1) p 1 =... = p n 1 = 2, p n = q > n ) However a large literature already exists on regularity of solutions under suitable assumptions on the exponents when these exponents are not spread out; see the end of this section for details. Similar regularity questions can be posed for other integral-functionals, for instance of the form Du p log1 + Du ) + u xn q dx 1.3) for some exponents p, q 1 p < q), or [g Du )] p + [g u xn )] q dx, 1.4) where g = gt) is a convex function satisfying the so-called 2 -condition, namely there exists µ > 1 such that gλt) λ µ gt) for every λ > 1 and for every t sufficiently large see Section 2). An example of such a function, with a, b growth, is gt) = t [a+b+b a) sin log loge+t)]/2. The regularity results known in the literature seem not applicable to the integrals 1.3), 1.4) under sharp assumptions on the exponent p and q, as stated below Mathematics Subject Classification. Primary: 49N60; Secondary: 35J70. Key words and phrases. L regularity, gradient estimates, minimizers, anisotropic growth conditions, p, q growth conditions. 1

2 2 G. CUPINI P. MARCELLINI E. MASCOLO Recently Lieberman [21] proved that integrals of the calculus of variations as in 1.1) may have Lipschitz continuous local minimizers u, independently of any condition on the p i, if we assume a priori that u itself is bounded. This fact motivates the research proposed in this article. To this aim and for the sake of exposition we deal again with integrals as in 1.1) and we consider exponents p i, i = 1,..., n, and q greater than or equal to 1, such that pi p i x), a.e. x B r 1.5) q p i x), a.e. x B r, 1 i n, where B r is a ball of radius r > 0 contained in. Then let p be the harmonic average of the p i ; i.e., 1 p := 1 1 n p i and let p be the Sobolev conjugate exponent of p; i.e., p = np n p if p < n, while p is any fixed real number greater than p, if p n. The following regularity result holds. Theorem 1.1. Let u be a local minimizer of 1.1) and let q < p. Then u is locally bounded in and the following estimate holds 1+θ p u u r L B r/2 n) x 0 )) c 1 + u xi x) pix) dx, B rx 0 ) 1 for some constant c > 0, where u r = B rx 0 ) B rx 0 ) u dx, p = min p i and θ = p q p) 1 i n pp q). Observe that if p 1 =... = p n 1 = 2 and p n = q 2 then the assumption q < p gives q < 2 n 1) / n 3); this inequality is exactly the opposite of condition 1.2), apart from the equality which is not achieved, since the borderline case q = p is not included in Theorem 1.1. Thus, our regularity result is essentially sharp. As a consequence of the previous theorem and of the quoted result by Lieberman [21] we get the following gradient estimate under a sharp assumption on the exponents of the anisotropic growth. Corollary 1.2. Let u be a local minimizer of the integral F in 1.1) with exponents p i x), for i = 1,..., n, locally Lipschitz continuous in. Let px) be the harmonic average of the p i x) and let p x) be the Sobolev conjugate exponent of px). If p x 0 ) > p i x 0 ) for some x 0 and for every i = 1,..., n, then u is Lipschitz continuous in a neighborhood of x 0. We emphasize that in fact in this paper we consider integrals more general than 1.1), 1.3) and 1.4). Precisely, we are able to consider general integrals with non-homogeneous densities of the form fx, u x1,..., u xn ) dx with f satisfying some non-standard p i -q growth conditions; precise assumptions and statements are in Section 2. We observe explicitly that, in the case of the functional in 1.4), the assumptions involve the exponents p and q, but they are independent of the function g. The mathematical literature on the regularity in this context is very rich; energy functionals with anisotropic, non-standard or general growth have been studied by many authors and in different settings of applicability. Among the many related papers we quote, in a not exhaustive way, Marcellini [24], [25], Lieberman [20], Bhattacharya-Leonetti [5], Moscariello-Nania [27], Mascolo-Papi [26], Fan-Zhao [13], [14], Dall Aglio-Mascolo-Papi [12] and, in the vectorial setting, Acerbi-Mingione

3 SHARP ANISOTROPIC GROWTH CONDITIONS 3 [2], Coscia-Mingione [11], Cavaliere-D Ottavio-Leonetti-Longobardi [8], Canale-D Ottavio-Leonetti- Longobardi [7]. Specific regularity results addressed to the study of functionals with anisotropic growth under the sharp condition on the exponents p > q, have been first obtained by Boccardo- Marcellini-Sbordone [6], see also a generalization due to Stroffolini [29]. Fusco-Sbordone [16] consider the borderline case p = q and, later, in [17] they study more general anisotropic integrands f = fx, u, Du) satisfying a growth of the form ) u xi x) p i fx, u, Du) c + u xi x) p i, obtaining a boundedness result by mean of De Giorgi s methods. More general functionals are considered in Cianchi [10], in which the study of the boundedness of minimizers is carried out using the optimal Sobolev conjugate of convex functions. Because of the p i q growth, we use a different approach based upon a variant of the classical Moser s iteration method, which has its starting in an inequality of Euler s type, see Theorem 5.1. Moreover, for the anisotropic behavior of the integrand, we base our estimates on an embedding result for anisotropic Sobolev spaces due to Troisi [31] see also Acerbi-Fusco [1] and Fragalà- Gazzola-Kawhol [15]). Our paper is organized as follows. In the next section we present the precise statement of our regularity theorem and few more examples of applicability. In Section 3 preliminary properties of convex functions are proved. Section 4 is devoted to higher integrability results for minimizers, Section 5 to the Euler s inequality and Section 6 to the proof of Theorem Assumptions and statement of the main results Let us define the integral functional Fu) := fx, Dux)) dx, 2.1) where is an open bounded subset of R n, n 2, and u W 1,1, R). For the sake of simplicity, and with a slight abuse of notation, we assume f = fx, u x1,..., u xn ). A more general case is considered in the last section. Denoting R n + the set [0, + ) n, we assume H1) f : R n + R, fx, ξ) = fx, ξ 1,..., ξ n ), is a Carathéodory function, convex and of class C 1 with respect to ξ and increasing with respect to each ξ i, H2) there exist µ 1 and t 0 0, such that for every λ > 1 and for a.e. x and every ξ, ξ t 0. fx, λξ) λ µ fx, ξ) 2.2) A growth condition on f is assumed.

4 4 G. CUPINI P. MARCELLINI E. MASCOLO H3) there exist a > 0 and 1 p i q, 1 i n, such that [gξ i )] p i fx, ξ) a 1 + [gξ i )] q 2.3) for a.e. x and every ξ R n +. Here g : R + R + is of class C 1, convex, increasing, non-constant, g0) = 0 and gλt) λ µ gt) for every λ > 1 and every t t ) Without loss of generality, we assume t 0 large so that gt) > 0 and fx, ξ) > 0 for all t > t 0 and all ξ with ξ t 0. We denote W 1,F ) the space W 1,F ) = u W 1,1 ) : Fu) < + and we write W 1,F 0 ) in place of W 1,1 0 ) W 1,F ). A function u W 1,1 ) is a local minimizer of 2.1) if u W 1,F ) and Fu) Fu + ϕ), for all ϕ W 1,F ) with supp ϕ. Our aim is to prove the local boundedness of local minimizers of 2.1). To do this, we need a restriction on the exponents p i and q. We will use the following notations: we write p in place of minp i and, as in the introduction, we denote by p the harmonic average of p i, i.e., 1 p := 1 n n 1 p i and by p the Sobolev exponent of p np p := n p if p < n, 2.5) any µ > p if p n. Theorem 2.1. Assume H1), H2) and H3), and let q < p. Then a local minimizer u of 2.1) is locally bounded. Moreover, for every B r x 0 ) the following estimates hold true: 1) there exists c > 0, depending on the data, such that g u ) L B r/2 x 0 )) c 1 + [g u )] q dx B rx 0 ) 1+θ q 2) there exists c > 0, depending on the data, such that g u u r ) L B r/2 n) x 0 )) c 1 + fx, u x1,..., u xn ) dx where θ = p q p) pp q) and u 1 r := B rx 0 ) B rx 0 ) u dx. B rx 0 ) For the sake of simplicity we wrote the growth condition 2.3) in place of b [gξ i )] p i c fx, ξ) a 1 + [gξ i )] q,, 2.6) 1+θ p, 2.7) with b > 0, c R. This is not a loss of generality since u is a local minimizer of 2.1) if and only if u is a local minimizer of the functional having the energy density f replaced by a 1 f +a 2, with some constants a 1 > 0 and a 2 R. Taking this into account, it is not difficult to check that Theorem 2.1 applies to the functionals 1.1), 1.3) and 1.4) in Section 1. For instance, as far as 1.1) is concerned, we can take p i and q as in 1.5), µ = q, gt) = t, a = n2 q 1, b = 2 1 q, c = n. Moreover Theorem 2.1 applies also to functionals F with different energy densities. We give below some more examples.

5 SHARP ANISOTROPIC GROWTH CONDITIONS 5 We can consider constants γ > 0 and α 1 such that αγ 1, a measurable function β : [β 1, β 2 ], with β 1 1 and β 1 γ 1, and for instance the integrand fx, ξ) = ξ α + ξ n βx) ) γ. 2.8) In this case p = p i := γα, if 1 i n 1, p n := γ maxα, β 1 and q := γ maxα, β 2. An other example can be exhibit through measurable functions r i : [p i, q] and fx, ξ) = ξ i rix) ) γ, 2.9) with p := minp i 1 satisfying 1 γp γq < γp). Here, γp is the harmonic average of γp 1,..., γp n. The previous example can be easily generalized to include integrands of the type ) fx, ξ) = F [h ξ i )] r ix) ; 2.10) or, more in general, ) fx, ξ) = F f i x, ξ i ). 2.11) In particular in 2.11) we consider a convex function fx, ξ) of class C 1 with respect to ξ, functions f i x, ξ i ) increasing with respect to each ξ i and satisfying 2.2), F increasing and satisfying 2.4). Finally the following growth condition holds with g as in H3). We begin clarifying the role played by 2.4). [gt)] p i F f i x, t)) a 1 + [gt)] q, 3. Preliminary results Lemma 3.1. Consider h : R + R + of class C 1, convex and increasing, and fix t 0 > 0 and µ 1. The following two properties hold: 1) Suppose that for every λ > 1 and t t 0 we have for all λ > 1 and t t 0. Then hλt) λ µ ht) 3.1) hλt) λ µ ht) + ht 0 )) and h t)t µht) + ht 0 )) for all t ) 2) Suppose that ht) > 0 for every t > t 0 and Then h t)t µht) for all t t ) hλt) λ µ ht) + ht 0 )) for all t ) Moreover, if 3.1) or 3.3) hold, then for every t 1,..., t k ) R k + we have: k k ) k k 1 ht i ) h t i k ht µ 0 ) + ht i ). 3.5) The lemma deals with well known properties of the convex functions see [28]), however for the sake of completeness we provide a proof.

6 6 G. CUPINI P. MARCELLINI E. MASCOLO Proof. Let us prove 1). The first inequality in 3.2) is trivial, since, by the monotonicity of h, we have hλt) hλt 0 ) λ µ ht 0 ) for every t < t 0. Let us prove the other inequality in 3.2). By assumption, for every σ > 0 and t > t 0 we have ht + σ) ht) σ = ht1 + σ t )) ht) 1 + σ ) µ ht) 1 σ t σ and for σ 0 we get h t)t µht) for all t > t 0 and, by continuity, for t t 0. Since h is increasing, if t t 0 we have h t)t h t 0 )t 0 µht 0 ), which implies the last inequality in 3.2). Now, let us prove 2). By 3.3), for every t > t 0 and λ > 1 we obtain λt t h s) hs) ds µ so that hλt) λ µ ht). From that, 3.4) follows. λt t 1 s ds, The first inequality in 3.5) is implied by the monotonicity of h, since ht j ) h k t i) for all j. To prove the second inequality, use the monotonicity of h again and 3.4), obtaining h k and the conclusion follows. ) ) ) t i h k max t i k µ h max t i + ht 0 ) 1 i k 1 i k Now, we consider the case of functions depending on more than one variable. Lemma 3.2. Let f : R n + R + satisfy H1), H2) and H3). Then there exists c 0 such that i) fx, λξ) cλ nµ 1 + fx, ξ) for every ξ R n + and every λ > 1, ii) fx, ξ + ζ) c 1 + fx, ξ) + fx, ζ) for every ξ, ζ R n +, iii) ξ i x, ξ)ξ i c 1 + fx, ξ) for every ξ R n +. Proof. Fix i = 1,..., n. By H1) for a.e. x and for every ξ R n +, with ξ i t 0, we have fx, ξ 1,..., ξ i 1, λξ i, ξ i+1,..., ξ n ) fx, λξ) λ µ fx, ξ) for every λ > ) Therefore, by Lemma 3.1 1), then, for every ξ R n +, fx, ξ 1,..., ξ i 1, λξ i, ξ i+1,..., ξ n ) λ µ fx, ξ) + fx, ξ 1,..., ξ i 1, t 0, ξ i+1,..., ξ n ). 3.7) Now, fix ξ = ξ 1,..., ξ n ) R n + and k N, 1 k n 1. For each set of indexes i 1,..., i k, with 1 i 1 <... < i k n, we define two vectors ai 1,..., i k ) and bi 1,..., i k ) in R n with j-th component ξj if j i ai 1,..., i k ) j = 1,..., i k 0 if j i 1,..., i k and, respectively, bi 1,..., i k ) j = 0 if j i1,..., i k 2t 0 if j i 1,..., i k.

7 SHARP ANISOTROPIC GROWTH CONDITIONS 7 An iterated use of 3.7) implies that for every λ > 1 and every ξ R n + fx, λξ 1,..., λξ n ) 2λ) nµ f n 1 + 2λ) nµ x, ξ 1 2,..., ξ n 2 k=1 1 i 1 <...<i k n ) + f x, t 0,..., t 0 ) f x, 1 2 ai 1,..., i k ) + 1 ) 2 bi 1,..., i k ). Notice that by the monotonicity of f with respect to each variable ξ j and the right inequality in 2.3) f x, ξ 1 2,..., ξ ) n + f x, t 0,..., t 0 ) fx, ξ) + fx, 2t 0,..., 2t 0 ) 2 3.9) c1 + fx, ξ). To estimate the last sum in 3.8) we use the convexity of f and the monotonicity properties of f 3.8) f x, 1 2 ai 1,..., i k ) + 1 ) 2 bi 1,..., i k ) 1 2 f x, ai 1,..., i k )) f x, bi 1,..., i k )) 1 2 fx, ξ) f x, 2t 0,..., 2t 0 ) 3.10) and apply 3.9). Thus, i) is proved. Claim ii) is a trivial consequence of i): fixed ξ, ζ R n +, by 2.2) fx, ξ + ζ) = f x, 2 ξ + ζ ) c 2 nµ 1 + f x, ξ + ζ ) 2 2 and the convexity of f gives the conclusion. It remains to prove iii). Fix ξ = ξ 1,..., ξ n ) R n +. By 3.6) and Lemma 3.1 1) ξ i x, ξ)ξ i µ fx, ξ) + fx, ξ 1,..., ξ i 1, t 0, ξ i+1,..., ξ n ). The last term can be estimated using the monotonicity of f with respect to each variable ξ j and ii). In fact, fx, ξ 1,..., ξ i 1, t 0, ξ i+1,..., ξ n ) fx, ξ 1 + t 0,..., ξ i + t 0,..., ξ n + t 0 ) c 1 + fx, ξ) + fx, t 0,..., t 0 ). The last inequality in 2.3) implies iii). 4. The space W 1,F ) and some higher integrability results Due to the assumptions on f in Section 2 the space W 1,F ) is a vector space. Lemma 4.1. Assume H1), H2) and H3). Then W 1,F ) is a vector space. Proof. By the right inequality of 2.3), the function u 0 is in W 1,F ). Let us assume that u and v are both in W 1,F ) and γ R. By Lemma 3.2 ii) we immediately have that u + v is in W 1,F ).

8 8 G. CUPINI P. MARCELLINI E. MASCOLO Let us prove that γu W 1,F ). If γ 1 the conclusion follows by the monotonicity of f, see H1). If, instead, γ > 1 then the conclusion follows by Lemma 3.2 i), which implies that there exists c independent of x and u, such that fx, γ u x1,..., γ u xn ) γ nµ c 1 + fx, u x1,..., u xn ). To prove our result we use the following suitable anisotropic Sobolev space endowed with the norm W 1,p 1,...,p n) ) := u W 1,1 ) : u xi L p i ), for all i = 1,..., n, u W 1,p 1,...,pn) ) := u L 1 ) + u xi L p i). We write W 1,p 1,...,p n) 0 ) in place of W 1,1 0 ) W 1,p 1,...,p n) ). These spaces are studied in [31], see also [1]. We remind an embedding theorem for this class of spaces see [31]). Theorem 4.2. Let R n be a bounded open set and consider u W 1,p 1,...,p n) 0 ), p i 1 for all i = 1,..., n. Let maxp i < p, with p as in 2.5). Then u L p ). Moreover, there exists c depending on n, p 1,..., p n if p < n, and also on if p n, such that n u n L p ) c u xi L p i). The following embedding result, which holds for the cubes of R n, is proved in [1]. Theorem 4.3. Let Q R n be a cube with edges parallel to the coordinate axes and consider u W 1,p 1,...,p n) Q), p i 1 for all i = 1,..., n. Let maxp i < p, with p as in 2.5). Then u L p Q). Moreover, there exists c depending on n, p 1,..., p n if p < n, and also on Q if p n, such that u L p Q) u c L 1 Q) + u xi L p iq). A variant of the above lemma can be proved using Theorem 4.3 and a suitable Poincaré inequality proved in [3]. Proposition 4.4. Let u W 1,1 ) and let g : R + R + be of class C 1, convex, increasing, nonconstant, g0) = 0, gλt) λ µ gt), for some µ 1 and every λ > 1 and every t t 0. Suppose that g u xi ) L p i g u ) L p loc) for every i = 1,..., n, with p i 1. Let maxp i < p, with p as in 2.5), then loc ). Moreover, if Q is a cube with edges parallel to the coordinate axes, then g u ) L p Q) 1 c + g u ) L 1 Q) + g u xi ) L p iq). 4.1) Proof. We split the proof into steps. Step 1. We claim that g Du ) L 1 loc ). In fact, since Du n u x i, then by 3.5) g Du ) n gt µ 0 ) + g u xi ). 4.2) Step 2. Let us prove that g u ) L 1 loc ).

9 SHARP ANISOTROPIC GROWTH CONDITIONS 9 For every convex bounded open set Σ, by Lemma 3.2 ii) we get g u ) g u u Σ + u Σ ) c 1 + g u u Σ ) + g u Σ ) where u Σ = Σ 1 Σ u dx and c is a positive constant independent of u and Σ. By Lemma 3.1 1) ) ux) g ux) u Σ ) dx max [diamσ)] µ uσ, 1 g + gt 0 ) dx 4.3) diamσ) Σ and a Poincaré inequality proved in [3] implies Σ g ) ux) uσ dx diamσ) Σ ωn [diamσ)] n Σ 1 1 n g Dux) ) dx, 4.4) Σ where ω n is the volume of the unit ball in R n. The conclusion follows by Step 1. Step 3. Let a k be an increasing sequence, a k + as k goes to +, such that the sets u = a k have zero measure. Define the increasing sequence of functions g k defined as g k t) = gt) if t < a k and g k t) = ga k ) if t a k. We claim that g k u ) W 1,p 1,...,p n) loc ). In fact, let Σ be an open subset, Σ. Since g k is bounded then g k u ) is bounded, too. It remains to prove that [g k u )] xi L p i Σ). We notice that the following inequality holds: given two non-decreasing and non-negative functions h 1 and h 2, it holds true that h 1 t 1 )h 2 t 2 ) h 1 t 1 )h 2 t 1 ) + h 1 t 2 )h 2 t 2 ) for every t 1, t ) Hence, we have that 1 p i [g k u )] xi L p iσ) [g u )] p i u p i dx Σ u a k + [g u xi )] p i u xi p i dx Σ u a k and from Lemma 3.1 1) we get Thus, the claim is proved. [g k u )] xi L p iσ) c 1 + g k u ) L p iσ) + g u xi ) L p iσ) < ) Step 4. Now, we conclude. Let Q be a cube with edges parallel to the coordinate axes. Since g k u ) W 1,p 1,...,p n) Q) we can apply Theorem 4.3, so that, using also 4.6), there exists c 1 > 0 such that g k u ) L p Q) c gk u ) L 1 Q) 4.7) + c 1 g k u ) L p iq) + g u xi ) L p iq). Notice that if p i > 1 and being maxp i < p, then there exists α i 0, 1) such that p 1 i = 1 α i ) + α i /p. Hence for every ɛ > 0 and for every i there exists c ɛ,i > 0 such that g k u ) L p iq) g k u ) α i L p Q) g k u ) 1 α i L 1 Q) ɛ g k u ) L p Q) + c ɛ,i g k u ) L 1 Q). 1 p i

10 10 G. CUPINI P. MARCELLINI E. MASCOLO Of course, if p i = 1 the above inequality is trivial. Choosing ɛ = 2nc 1 ) 1 the above inequalities and 4.7) imply that a constant c 2 > 0 exists such that g k u ) L p Q) c 2 g k u ) L 1 Q) + 2c g u xi ) L p iq). Using the monotone convergence theorem, inequality 4.1) follows. A consequence of the above result is the following corollary. Corollary 4.5. Assume H1), H2) and H3), with q < p. If u W 1,F ), then g u ) L p loc ). 5. The Euler s inequality Since H1) does not imply the C 1 -regularity of ξ f ξ 1,..., ξ n ), ξ R n, in place of the Euler s equation, we prove an inequality. Theorem 5.1. Assume that H1), H2) and H3) hold true and let u W 1,F ) be a local minimizer of 2.1). Then u xi >0 for all ϕ W 1,F ), supp ϕ. x, u x1,..., u xn ) sgn u xi ) ϕ xi dx ξ i x, u x1,..., u xn ) ϕ xi dx, ξ i u xi =0 5.1) Proof. Let ϕ W 1,F ) be a function with compact support and λ 1, 0). For every i 1,..., n define H i : 1, 0) R + R +, H i x, λ, s) := fx, u x1 x) + λϕ x1 x),......, u xi 1 x) + λϕ xi 1 x), s, u xi+1 x),..., u xn x) ). Notice that if i n 1 then H i x, λ, u xi x) + λϕ xi x) ) = H i+1 x, λ, u xi+1 x) ). By the minimality of u and the convexity of f with respect to each variable ξ j, we get Since H i s 0 1 Fu + λϕ) Fu) λ = 1 H n x, λ, u xn + λϕ xn ) H 1 x, λ, u x1 ) dx λ = 1 H i x, λ, u xi + λϕ xi ) H i x, λ, u xi ) dx λ = ξ i, by Lemma 3.2 iii) we obtain H i s x, λ, u x i + λϕ xi ) u x i + λϕ xi u xi dx. λ 5.2)

11 SHARP ANISOTROPIC GROWTH CONDITIONS 11 H i s x, λ, u x i x) + λϕ xi x) ) u x i x) + λϕ xi x) u xi x) λ H i s x, λ, u x i x) + ϕ xi x) ) u xi x) + ϕ xi x) ) c1 + fx, u x1 x) + λϕ x1 x),......, u xi 1 x) + λϕ xi 1 x), u xi x) + ϕ xi x), u xi+1 x),..., u xn x) ) and, using the monotonicity property in H1) and Lemma 3.2 ii), fx, u x1 x) + λϕ x1 x),......, u xi 1 x) + λϕ xi 1 x), u xi x) + ϕ xi x), u xi+1 x),..., u xn x) ) f x, u x1 x) + ϕ x1 x),..., u xn x) + ϕ xn x) ) c 1 + fx, u x1 x),..., u xn x) ) + fx, ϕ x1 x),..., ϕ xn x) ). Now, notice that the right hand side is in L 1 ), being u, ϕ W 1,F ). regularity C 1 of fx, ), H i lim λ 0 s x, λ, u x i x) + λϕ xi x) ) = x, u x1 x),..., u xn x) ). ξ i Thus, by the dominated convergence theorem and 5.2) we get u xi x) + λϕ xi x) u xi x) x, u x1 x),..., u xn x) ) lim dx 0. ξ i λ 0 λ The conclusion follows. 6. Proof of the boundedness of local minimizers Fixed i 1,..., n and β 1, let Φ : R R be the odd function defined as follows Φ i,β) t) := t 0 Moreover, by the [g s )] p iβ 1) ds. 6.1) In a first step, we deal with an approximating sequence of odd functions Φ i,β) k. Fixed k N, the function Φ i,β) k : R R is defined in R + as Φ i,β) k t) := Φ i,β) t) if 0 t k t Φ i,β) ) k) + Φ i,β) k) k Φ i,β) ) k) if t > k. From now on, we do not write explicitly the dependence on i and β. Notice that the restriction of Φ k to R + is C 1, increasing and convex. Moreover, its first order derivative is bounded and 6.2) Φ k t) Φ k t) t for all t R. 6.3) In the following lemma we define ϕ k, an admissible test function for the Euler s inequality 5.1). Lemma 6.1. Assume H1), H2) and H3), with q < p. Let u W 1,F ), fix a ball x 0 ) and let η C c x 0 )) be a cut-off function, satisfying the following assumptions 0 η 1, η 1 in B ρ x 0 ) for some ρ < R, Dη 2 R ρ. 6.4)

12 12 G. CUPINI P. MARCELLINI E. MASCOLO Fixed k N, define with α 1. Then ϕ k is in W 1,F 0 x 0 )). ϕ k x) := Φ k ux))[ηx)] α for every x x 0 ), 6.5) Proof. By Lemma 4.1, Lemma 3.2 and the definition of Φ k we get the thesis if we prove that A := f x, [Φu)] x1,..., [Φu)] xn ) dx < + B := C := D := u <k u <k u k u k f x, Φu) η x1,..., Φu) η xn ) dx < + f ) x, [gk)] piβ 1) u x1,..., [gk)] piβ 1) u xn dx < + f x, Φ k u) η x1,..., Φ k u) η xn ) dx < +. Let us deal with A. By the monotonicity of g, [Φu)x)] xj [gk)] piβ 1) u xj x), for a.e. x u < k. Then, by H1) and Lemma 3.2 ii) we get nµ A c max[gk)] piβ 1), f x, u x1,..., u xn ) dx, u <k u <k which is finite being u W 1,F ). The boundedness of C follows similarly. As far as B is concerned, from H1), the assumptions on η and the monotonicity of g we obtain ) 2α B f x, R ρ k[gk)]p iβ 1) 2α,..., R ρ k[gk)]p iβ 1) dx, which is finite because of the growth condition 2.3). Let us prove the boundedness of D. From 6.3) we obtain Φ k ux)) [gk)] p iβ 1) ux) for a.e. x in the integration domain. Thus, Φ k ux)) η xj x) 2[gk)]p iβ 1) ux). R ρ Using the assumptions on f and the right inequality in 2.3) we get nµ 2[gk)] p iβ 1) D c max, R ρ u k Since q < p, Corollary 4.5 implies that the last term in 6.6) is finite. [g u )] q dx The lemma below is a simple consequence of the Hölder inequality. We omit the proof.. 6.6) Lemma 6.2. Let be a bounded measurable set. Suppose that 1 p q, β 1 and v L qβ ). Then 1 p p v q+pβ 1) dx v + 1) q q dx v β + 1) q q dx. Now, we turn to the proof of our main result. Proof of Theorem 2.1. Let u be a local minimizer of 2.1) and consider x 0 and R 0 > 0, such that 0 := 0 x 0 ). In particular, by Corollary 4.5 g u ) L p 0 ). Fix also 0 < ρ < R R 0. We split the proof into steps.

13 SHARP ANISOTROPIC GROWTH CONDITIONS 13 Step 1. Assume that g u ) L qβ ) for some β 1. Fixed i 1,..., n we prove that if η is a cut-off function satisfying 6.4), then [g u )] β 1) g u xi ) u xi η µ p i dx c q p ) R ρ) µ g u ) L q 0 ) + 1 p i 6.7) q g β q u ) + 1 dx, for some c depending on n, µ, p, q, a, gt 0 ) and R 0, but independent of i, β, u, R and ρ. We begin using Theorem 5.1 with the test function ϕ i,β) k now on, we write ϕ k and Φ k in place of ϕ i,β) k Thus, using 6.3), µ u xj >0 2µ R ρ and Φ i,β) k := Φ i,β) k η µ with Φ i,β) k as in 6.2). From, respectively. We obtain ξ j x, u x1,..., u xn ) u xj Φ k u)ηµ dx ξ j x, u x1,..., u xn ) Φ k u) η µ 1 η xj dx. ξ j x, u x1,..., u xn ) u xj Φ k u)ηµ dx ξ j x, u x1,..., u xn ) Φ k u) u ηµ 1 dx. We estimate from below the left hand side using the convexity of fx, ), obtaining x, u x1,..., u xn ) u xj Φ k ξ u) ηµ dx j f x, u x1,..., u xn ) f x, 0,..., 0) Φ k u) ηµ dx. Now, let us estimate from above the right hand side in 6.8). For a.e. x and every s 0 define H j x, s) := fx, u x1 x),..., u xj 1 x), s, u xj+1 x),..., u xn x) ). Let L > 0 to be chosen later. Since f is convex we have that H j s x, ) is increasing, then by 4.5) and Lemma 3.2 iii), the following chain of inequalities holds true for a.e. x η 0 2µ u x, u x1,..., u xn ) ξ j ηr ρ) 1 H j L s c 1 L with c 1 depending only on n, µ, q, a, gt 0 ). x, uxj ) u xj + 1 ) H j 2µL u 2µL u x, L s ηr ρ) ηr ρ) ) 1 + f x, u x1,..., u xn ) + H j x,, 2µL u ηr ρ) 6.8) 6.9)

14 14 G. CUPINI P. MARCELLINI E. MASCOLO Now, denote with e j the vector 0,..., 0, 1, 0,..., 0). Using Lemma 3.2 ii) with j ξ := u x1,..., u xj 1, 0, u xj+1,..., u xn ), ζ := 2µL u η R ρ) e j, j and the monotonicity property in H1), we have that there exists c 2 such that ) H j x, c fx, u x1,..., u xn ) + f x, Thus, 2µL u ηr ρ) 2µ u x, u x1,..., u xn ) ξ j ηr ρ) c 3 L 1 + fx, u x 1,..., u xn ) + f x, 2µL u ηr ρ) e j ) 2µL u ηr ρ) e j with c 3 depending only on n, µ, q, a, gt 0 ). Choosing L > max2c 3, 2µ) 1 R 0, which implies 2µL > η R ρ), and using Lemma 3.1 1), the above inequality implies 2µ u x, u x1,..., u xn ) ξ j η R ρ) 1 2 fx, u x 1,..., u xn ) + c 4 η µ R ρ) µ 1 + fx, u e j ) ). 6.10) for some positive c 4. Collecting 6.8), 6.9) and 6.10), we obtain fx, u x1,..., u xn )Φ k u) ηµ dx 2 fx, 0,..., 0)Φ k u) dx + 2c 4 R ρ) µ 1 + By H3) and Lemma 3.1 1) applied with h = g p i fx, u x1,..., u xn ) fx, u e j )Φ k u) dx. [g u xj )] p j g u xi )] p i 1 µq gp i ) u xi ) u xi [gt 0 ) + 1] q. 6.11) 6.12) Moreover, by 2.3) 1 + fx, 0,..., 0) + fx, u e j ) c 5 [g u )] q )

15 Inequalities 6.11), 6.12) and 6.13) give SHARP ANISOTROPIC GROWTH CONDITIONS 15 g p i ) u xi ) u xi Φ k u) ηµ dx c 6 R ρ) µ [g u )] q + 1 Φ k u) dx. We recall that Φ k = Φ i,β) k and we explicitly notice that c 6 is independent of β, ρ and R. Using the monotone convergence theorem we let k go to + and by the definition of Φ we obtain [g u )] piβ 1) [g u xi )] pi 1 g u xi ) u xi η µ dx c 6 R ρ) µ [g u )] piβ 1) + [g u )] q+p iβ 1) dx. Now, by the Hölder inequality there exists c, depending on R 0, such that [g u )] piβ 1) dx [g u )] β pi + 1) dx c ) p i 6.14) q [g u )] β q + 1 dx. Moreover, by Lemma 6.2 applied to v = g u ), with p replaced by p i, we get the existence of a positive constant c, independent of β, such that [g u )] q+p iβ 1) dx c κ + 1 q p i ) p i q [g u )] β q + 1 dx, where κ := g u ) L q 0 ) is finite by Corollary 4.5 and the assumption q < p. So, it follows that [g u )] piβ 1) [g u xi )] pi 1 g u xi ) u xi η µ dx κ + 1 q p i ) p i 6.15) q c 7 R ρ) µ [g u )] β q + 1 dx. Now, by 2.4) and by the first step of the proof of Lemma 3.1 1) we get that for a.e. x u xi > t 0 the inequality g u xi ) 1 µ g u xi ) u xi holds true. Moreover, being µ 1 and p i q we get µ pi 1 µ q 1. Thus, 6.15) implies [g u )] piβ 1) [g u xi )] p i u xi p i η µ dx u xi >t 0 c 8 κ + 1 q p i R ρ) µ ) p i 6.16) q [g u )] β q + 1 dx, with c 8 independent of i. Filling the hole, that is adding to both sides [g u )] piβ 1) [g u xi )] p i u xi p i η µ dx, u xi t 0

16 16 G. CUPINI P. MARCELLINI E. MASCOLO and noticing that, due to the convexity of g, 2.4), the first step of the proof of Lemma 3.1 1) and 6.14) imply [g u )] piβ 1) [g u xi ) u xi ] p i η µ dx u xi t 0 u xi t 0 [g u )] p iβ 1) [g t 0 )t 0 ] p i η µ dx µ q [g u )] p iβ 1) [gt 0 )] p i η µ dx µ q [gt 0 ) + 1] q [g u )] p iβ 1) η µ dx c 9 ) p i q [g u )] β q + 1 dx, we obtain that [g u )] β 1) g u xi ) u xi pi η µ dx c 10 κ + 1 q p i R ρ) µ ) p i q [g u )] β q + 1 dx. Since η µp i η µ and p i p we get 6.7). Step 2. In this step we prove that if g u ) L qβ ) for some β 1, then there exists c, independent of β, R and ρ, such that [ ] η µ [g u )] β i + 1) x p dx i c βγ q p R ρ) γ g u ) L q 0 ) + 1 p i 6.17) [g u )] β + 1) q q dx, with γ = maxµ, q. We begin noticing that [ ] η µ [g u ) β i + 1) x p dx i [η 2 q 1 µ ] xi [g u )] β pi + 1) dx + 2 q 1 β q [g u )] β 1 g u ) u xi η µ p i dx = I1 + I ) By 6.4) and the Hölder inequality we have that I 1 c p i R ρ) q [g u )] β + 1) q q dx. 6.19) Let us consider I 2. Use 4.5) with h 1 = g p i, h 2 = id p i, t 1 = ux) and t 2 = u xi x), obtaining [g ux) )] p i u xi x) p i [g u xi x) )] p i u xi x) p i + [g ux) )] p i ux) p i.

17 Therefore SHARP ANISOTROPIC GROWTH CONDITIONS 17 I 2 2 q 1 β q + 2 q 1 β q [g u )] β 1) g u xi ) u xi η µ p i [g u )] β 1 g u ) u η µ p i The first term in the right hand is estimated by 6.7). To estimate the second term, use Lemma 3.1 1), which implies g u ) u c g u ) + 1, obtaining [g u )] β 1 g u ) u η µ p i dx c [g u )] piβ 1) [g u )] p i + 1) dx ) c [g u )] β pi ) p i q + 1 dx c [g u )] β q + 1 dx. Thus, I 2 c βq q p ) R ρ) µ g u ) L q 0 ) + 1 p i q [g u )] β q + 1 dx. 6.20) The inequalities 6.18), 6.19) and 6.20) imply 6.17). Step 3. Now, we prove the boundedness of u and the estimate 1). If g u ) L qβ ) for some β 1, then Step 2 implies that x η µ x)[g ux) )] β + 1 is in W 1,p 1,...,p n) 0 ). Multiplying 6.17) on i and being p i p, we get n ) 1 D i η µ [g u )] β p i p i + 1) dx β c 11 R ρ B ρ nγ p g u ) L q 0 ) + 1 n q p p with c 11 independent of β, R and ρ. By Theorem 4.2 we get ) p 1 p [g u )] β + 1 dx γ β p c 12 g u ) L R ρ q 0 ) + 1 q p p dx. dx n [g u )] β + 1) q q dx, [g u )] β + 1) q dx and, defining Gx) := max1, g ux) ), we obtain 1 p [Gx)] βp dx B ρ 6.21) γ 1 β p 2c 12 g u ) L R ρ q 0 ) + 1 q p p [Gx)] βq q dx. ) For all h N define β h = p h 1, ρh = R 0 /2 + R 0 /2 h+1 and R h = R 0 /2 + R 0 /2 h. By 6.21), q replacing β, R and ρ with β h, R h and ρ h, respectively, we have that G L β hq h ) implies 1 q

18 18 G. CUPINI P. MARCELLINI E. MASCOLO G L β h+1q h+1 ). Precisely, G L β h+1 q h+1 ) 2c 2 h+1 12 R 0 p q ) h 1 γ p g u ) L q 0 ) + 1 q p p 1 β h G LβhqBRh ) 6.22) holds true for every h. Corollary 4.5 and the inequality q < p imply G L q 0 ). An iterated use of 6.22) implies the existence of a constant c 13 such that G L 0 /2x 0 )) c 13 g u ) L q 0 ) + 1 p q p) pp q) G L q 0 x 0 )). Therefore, by the very definition of G, p g u ) L 0 /2x 0 )) c 14 g u ) q p) pp L q 0 x 0 )) + 1 q) +1. From H3), which implies that gt) + as t +, the above inequality implies that u is in L 0 /2x 0 )). Step 4. Here we prove estimate 2.7). Fix B r x 0 ). Notice that if Q s x 0 ) denotes the cube with edges parallel to the coordinate axes, centered at x 0 and with side length 2s, then B r/ n x 0 ) Q r/ n x 0 ) B r x 0 ). Let u W 1,F ) be a local minimizer of F and define u r := B rx 0 ) u dx. Since u u r is a local minimizer, too, then by 2.6) and the Hölder inequality p q p) g u u r ) L B r/2 n) x 0 )) c 1 + g u u r ) L p pp q) +1 B r/ n x 0 )). 6.23) By 4.1) in Proposition 4.4 g u u r ) L p B r/ n x 0 )) g u u r ) L p Q r/ n x 0 )) c 1 + g u u r ) L 1 B rx 0 )) + g u xi ) L p ibrx0 )). and by the Poincaré inequality proved in [3], see 4.2), 4.3) and 4.4)) g u u r ) L 1 B rx 0 )) c 1 + g u xi ) L 1 B rx 0 )). Therefore, using 6.24) we get g u u r ) L p B r/ n x 0 )) c ) g u xi ) L p ibrx0 )). 6.25) Now, 2.3) implies g u xi ) L p ibrx0 )) c fx, u x1,..., u xn ) p L 1 B rx 0 )). 6.26) The final estimate 2.7) follows collecting 6.23), 6.24), 6.25) and 6.26).

19 SHARP ANISOTROPIC GROWTH CONDITIONS 19 Remark 6.3. It is not difficult to see that similar results to those stated in Theorem 2.1 can be proved for functionals 2.1) with more general Lagrangians f. For instance, few and straightforward changes in the proof of Theorem 2.1 allow to obtain the local boundedness of local minimizers of 2.1), together with estimates similar to 2.6) and 2.7), under the following set of assumptions: f : R n R + is a Carathéodory function, convex and of class C 1 with respect to ξ, satisfying the growth assumption fx, ξ 1,..., ξ n ) fx, ξ) M 1 + fx, ξ 1,..., ξ n ), M > 0, with f : R n + R +, f = fx, z 1,..., z n ), satisfying H1), H2) and H3), and such that, for some Λ > 0, x, ξ) ξ i Λ f x, ξ 1,..., ξ n ) for all ξ R n. z i References [1] E. Acerbi and N. Fusco, Partial regularity under anisotropic p, q) growth conditions, J. Differential Equations, ) [2] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., ) [3] T. Bhattacharya and F. Leonetti, A new Poincaré inequality and its application to the regularity of minimizers of integral functionals with nonstandard growth, Nonlinear Anal., ) [4] T. Bhattacharya and F. Leonetti, Some remarks on the regularity of minimizers of integrals with anisotropic growth, Comment. Math. Univ. Carolin., ) [5] T. Bhattacharya and F. Leonetti, W 2,2 regularity for weak solutions of elliptic systems with nonstandard growth, J. Math. Anal. Appl., ) [6] L. Boccardo, P. Marcellini and C. Sbordone, L -regularity for variational problems with sharp nonstandard growth conditions, Boll. Un. Mat. Ital. A, ) [7] A. Canale, A. D Ottavio, F. Leonetti and M. Longobardi, Differentiability for bounded minimizers of some anisotropic integrals, J. Math. Anal. Appl., ) [8] P. Cavaliere, A. D Ottavio, F. Leonetti and M. Longobardi, Differentiability for minimizers of anisotropic integrals,comment. Math. Univ. Carolinae, ) [9] A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations, ) [10] A. Cianchi, Local boundedness of minimizers of anisotropic functionals, Ann. Inst. Henri Poincaré, Analyse non linéaire, ) [11] A. Coscia and G. Mingione, Hölder continuity of the gradient of px)-harmonic mappings, C. R. Acad. Sci. Paris Sér. I Math., ) [12] A. Dall Aglio, E. Mascolo and G. Papi Local boundedness for minima of functionals with nonstandard growth conditions,rend. Mat. Appl ) [13] X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., ) [14] X. Fan and D. Zhao, Regularity of quasi-minimizers of integral functionals with discontinuous px)-growth conditions, Nonlinear Anal., ) [15] I. Fragalà, F. Gazzola and B. Kawhol, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. Henri Poincaré, Analyse non linéaire, ) [16] N. Fusco and C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math., ) [17] N. Fusco and C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equations ) [18] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math., ) [19] Hong Min-Chun, Some remarks on the minimizers of variational integrals with nonstandard growth conditions, Boll. Un. Mat. Ital. A, ) [20] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural tseva for elliptic equations, Comm. Partial Differential Equations, ) [21] G.M. Lieberman, Gradient estimates for anisotropic elliptic equations, Adv. Differential Equations, )

20 20 G. CUPINI P. MARCELLINI E. MASCOLO [22] P. Marcellini, Un example de solution discontinue d un problème variationnel dans le cas scalaire, Preprint 11, Istituto Matematico U.Dini, Università di Firenze, [23] P. Marcellini, Regularity of minimizers of integrals in the calculus of variations with non standard growth conditions, Arch. Rational Mech. Anal., ) [24] P. Marcellini, Regularity and existence of solutions of elliptic equations with p q-growth conditions, J. Differential Equations, ) [25] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, ) [26] E. Mascolo and G. Papi, Local boundedness of minimizers of integrals of the Calculus of Variations, Ann. Mat. Pura Appl., ) [27] G. Moscariello and L. Nania, Hölder continuity of minimizers of functionals with nonstandard growth conditions, Ricerche Mat., ) [28] M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc., New York, [29] B. Stroffolini, Global boundedness of solutions of anisotropic variational problems, Boll. Un. Mat. Ital. A, ) [30] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., ) [31] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche di Mat., ) Dipartimento di Matematica U. Dini, Università di Firenze, Viale Morgagni 67/A, Firenze, Italy address: cupini@math.unifi.it address: marcellini@math.unifi.it address: mascolo@math.unifi.it