Higher Integrability of the Gradient of Minimizers of Functionals with Nonstandard Growth Conditions

Transcription

1 Higher Integrability of the Gradient of Minimizers of Functionals with Nonstandard Growth Conditions NICOLA FUSCO Universitli di Salem0 AND CARL0 SBORDONE Universitli degli Studi, Naples Let us consider the functional Introduction where F: [O, a] [O, a] is an increasing function. These functionals have been extensively studied when F satisfies control assumptions of the type with 1 < p = q. In this paper we give a contribution to the study of the case in which p < q (see Theorem 3.2). Namely, assuming that F is a convex function satisfying F( t)/tp is increasing, F( t ) /t4 is decreasing, and q* < p 4, where q* = nq/(n + 41, we prove that if u E W,&'(sZ, RN) is a local minimizer of Z(u), then there exists r > 1 such that F(IDu1) E L;JsZ). As an example of a function to which our result applies, one can take F(t) = t P log(1 + t), P > 1, Communications on Pure and Applied Mathematics, Vol. XLIII, (1990) John Wiley & Sons, Inc. CCC /90/ $04.00

2 674 N. FUSCO AND C. SBORDONE for any dimension n, or if n = 2,3,4 (see Section 3). The higher integrability result is obtained by proving a suitable extension of the so-called Gehring's lemma. The proof given here is different from the ones given in [l], [7], [3], and follows the lines of the one suggested by [2] in the case F(t) = t P. We would like to point out that the results obtained here rely heavily on the fact that our integrand depends only on the modulus of the gradient. Moreover, the restrictions on p,q seem related to some counterexamples contained in [51, [lo]. 1. Preliminaries and Notation In the following we shall denote by a bounded open set of R", by Q a cube in R" with sides parallel to the coordinate axes, and by I(Q) its side; moreover, by aq we shall denote the cube with the same center as Q such that I(@) = al(q). If Qo is a fixed cube in R", f is a L,,(Qo) non-negative function, we shall denote by (where fa stands for (l/lql)) the local maximal function of f in Q,. Among the properties of Mf, we recall the following "weak type" inequality (see [l]). PROPOSITION 1.1. If Q, is a cube in R", f E I,'@,), f >= 0, then, for any t > 0, In the following, we shall denote by A a continuous function A : [O,m[ + [O, w[, such that (i) A(2t) ka(t) for all t > 0, (ii) there exists p > 1 such that A(t)/fp is increasing. We remark that if A is an increasing convex function, condition (i), which is known as the A, condition, is equivalent to the assumption that there exists q > 1 such that A(f)/tq is decreasing. On the other hand, condition (ii) is equivalent to the A2 condition for the conjugate function of A (see [9], $4).

3 MINIMIZERS OF FUNCTIONALS 675 We now state the following extension of the maximal theorem of Hardy- Littlewood given in [l]. PROPOSITION 1.2. If A satisfies (i) and (ii), f is a non-negative function such that A( f) E L (Q, ), then there mists a constant c = c(n, p, k) such that Proof If, for any t > 0, we set then, using Proposition 1.1, we have by Fubini s theorem and integration by parts A(Mf) dx = j+ma (t)a(t) dt jq0 0 = c( n) f( x) dx/2fc [ A ( t ) / t] dt Qo 0 + c(.>/ f(~)jo2 ~)[A(t)/t ] drdt Q.

4 676 N. FUSCO AND C. SBORDONE Using again the assumption on A it follows that Then, combining (1.2) and (1.3) we obtain (1.1). Remark 1.3. From the proof it is clear that Proposition 1.2 still holds if, instead of (ii), we assume that there exist p > 1, H > 0 such that A(At) =< HAP(t) for all t > 0, 0 s A 5 1. Let us now recall the following useful result of [12]. THEOREM 1.4. If f(x> is a non-negative L'(Q,) &fiction such that, for any cube Q c Qo, f( x) h 5 k ess inff( x), fq Q then there exists r > 1, L > 0 depending only on k and n such that We conclude this section, by proving the following lemma about cubes. LEMMA 1.5. If Q, Q" are two cubes contained in a Jixed cube Q, and (2 g ZQ', Q n Q' # 0, then there exists a cube Q 2 Q u Q', Q c Q, such that l<q> s 31(Q). Proof bil and Q' = n,"=,[a;, bf], then since Q n Q' # 0, [ai, bi] n [a;, bf] # 0 for all i = 1,. * a, n. If we set Let us suppose that Q, = [O,al". If Q = ni.=,[ai, 1 = max { max{ bi, bf} - min{ ai, a;}}, ljign

5 MINIMIZERS OF FUNCTIONALS 677 then 1 < a and there exists a cube Q = ll~~'-l[ci, + 11 containing Q U Qf. Now I 5 l(q) + l(q') and, since Q CL 2Q', Q n Q' # 0, we have l(q9 6 21(Q). 2. An Extension of Gehring's Lemma Now we give a suitable extension of Gehring's lemma for functions A satisfying (i) and (ii), following an idea of [2] relative to the case A(t) = t P. PROPOSITION 2.1. ZfA satisfies (i) and (ii) and f is an L:oc(R) function, f 2 0, such that, for any cube Q c R for which 2Q c c R, then there exist cl, c, > 0, r > 1, depending only on b,, b,, n, k, p such that, for any 2Q c c a, Proof Let us fur 2Q, c c R and denote by Mf(x) the local maximal function of f in Q,. Let us fur a cube Q' c Qo such that 4Q' c Q, and a point z E Q'. If Q c Q, is any cube containing z we have: if Q c 2Qf, then If Q ct 2Q', then, using Lemma 1.5, there exists a cube Q c Q, such that Q u Q' c Q, NQ) 5 31(Q), and so From this inequality and (2.31, taking the supremum over all cubes Q c Q,, z E Q, and then using assumption (i) we have

6 678 N. FUSCO AND C. SBORDONE Thus, integrating over Q' and using (2.1) and Proposition 1.2, we obtain I c infa(mf) + 1, - [Q' 1 which implies that, for any cube Q' c $Qo, Applying now Theorem 1.4, we deduce that there exist r > 1, c > 0 such that, (using again Proposition 1.2): So we have shown that, for any cube 2Q c C R, which implies (2.2) by a standard argument.

7 MINIMIZERS OF FUNCTIONAL Remark 2.2. We point out that in Proposition 2.1, in inequalities (2.1) and (2.2), cubes may easily be replaced by balls. Moreover, it is clear from the proof that if b, = 0, then also c2 = The Main Result In this section we shall suppose that the function A introduced in Section 1 satisfies: (i) A(2t) 5 M(t) for any t > 0, (ii) there exist 1 < p 5 q such that A(f)/fP is increasing, A(t)/tq is decreasing and p > 4* = nq/(n + q).' Remark 3.1. Assumption (ii) implies that A(t) satisfies the growth condition (iii) C'tP - C, sa(t) 5 C 3(P + l), but it may happen that the exponents p, q appearing in (ii) are not necessarily the best ones in order for (iii) to hold. For example, the convex function satisfies (ii) with p = 4 - \/z and q = and (iii) with p = 3 and q = 5. Moreover, if 4 - \/z < r < 4 + 6, A(t)/t' is neither strictly increasing nor decreasing. In the following we shall say that a function u E WA'(0; RN) is a minimizer for the functional (3.1) I( 0; v) = 1 A( (Dvl) dt: n if and only if, for every $ E W','(R; R) with compact support, I(supp $; u) s I(SUPP *; I. The first step in deriving the higher integrability result is the following version of the well-known Caccioppoli inequality (see [71, [131). THEOREM 3.2. If A satisfies (9, and there exists q > 1 such that A(t)/fq is decreasing, and u E W'*'(R; RN) is a minimizer of the functional (3.1), then, for 'If 1 < q < n/(n - l), we can replace q by n/(n - 1). With such a choice we still have A(f)/fq decreasing and q* = 1.

8 680 N. FUSCO AND C. SBORDONE any ball B R C c R, (3.2) where ur = jb,u, and c depends on& on A. We shall also need the following LEMMA 3.3. Zf A satisfies (ii), then, setting, for any t > 0, we see that K(t) is a concave function such that there exists a c for which (3.3) H(t) s K(t) 5 ch(t) for all t > 0. The concavity of K(t) follows observing that K'(t) = [A(t'/q)/t](n+4)/q which is decreasing by (ii). Note also that Thus, by (3, one easily gets We can now prove the following higher integrability result. THEOREM 3.4. Zf A satisfies (i), (ii) and u E W'.'(n; RN) is a minimizer for the functional (3.1), then there exist r > 1, c > 0 such that, for any BR c c R, (3.4) Proof From Theorem 3.2, we deduce that

9 MINIMIZERS OF FUNCTIONALS 681 and thus, by Holder's inequality If H(t), K(t) are defined as in Lemma 3.3, using that lemma and noting that K is concave, we have (3.5) If we set B(t) = A(t'lq*), we have B(2t) 5 kb(t), and so with f(x) = IDU~~*, we deduce by (3.5) that from which it follows that B and f satisfy the assumptions of Proposition 2.1.

10 682 N. FUSCO AND C. SBORDONE Thus there exists r > 1 such that i.e., for any B, c R, (3.6) Setting A(t) = j"ds, 0 s c > 0, it is easy to check that 1 (3.7) qa(r) ra(t) SA(t), t > 0, that A(t) is a convex function and also that is convex, since its derivative is (l/p)a(t 'Ip)/t which is increasing. From a result of [9], page 169, it follows that (3.8) Taking into account that p > q*, we see from (3.71, (3.8) that From (3.61, (3.9) we deduce (3.4). Remark 3.5. Let F(x, s, 2): R x RN X RnxN -P R be a function satisfying: with A as in Theorem 3.4. Then if u E W,2:(Q,RN) is a minimizer of /F(x, u(x), Du(x)) dx, using the general form of Proposition 2.1 (with b, > 0)

11 one easily obtains that, for any BR c c a, MINIMIZERS OF FUNCTIONALS 683 Remark 3.6. If p 5 n, then q in Theorem 3.4 can be any number greater than p and (3.4) implies, by the Sobolev-Poincart? imbedding theorem, that the minimizer u is locally Holder continuous. Acknowledgment. This work was performed as part of a National Research Project and was supported by M.P.I. Bibliography [l] Bojarski, B., and Iwaniec, T., Analytical foundations of quasiconfomal mappings in R", Ann. Acad. Sci. Fen. 8, 1983, pp [2] Fabes, E., personal communication. [3] Fusco, N., and Sbordone, C., Higher integrability from reverse Jensen inequalities with different supports, in Partial Differential Equatwns and the Calculus of Variations: Essays in Honour of Ennw DeGiorgi, Prog. Nonlin. Diff. Eq. Appl., Birkhauser Boston, 1989, pp [4] Gehring, F. W., The Lp-integrability of the partial derivatives of a quasi conformal mapping, Acta Math. 130, 1973, pp [5] Giaquinta, M., Growth conditions and regularity: A counterexample, Manuscripta Math. 59, 1987, pp Giaquinta, M., and Giusti, E., On the regularity of the minima of variational integrals, Acta Math. 148, 1982, pp [7] Giaquinta, M., and Modica, G., Regulnrity results for some classes of higher-order nonlinear elliptic systems, J. Reine Angew. Math , 1979, pp [8] Hardy, G. H.. Littlewood, J. E., and Polya, G., Inequalities, Cambridge University Press, [9] Krasnosel'skii, M. A., and Ruticki, Y. B., Convex Functwns and Orlicz Spaces, Noordhoff Ltd., New York, [lo] Marcellini, P., Une example de solution discontinue d'un pmbl>me uariationnel dans le cas scalaire, preprint, [ll] Marcellini, P., Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rat. Mech. Anal. 105, 1989, pp [12] Muckenhoupt, B., Weighted norms inequalities for the Hardy mima1 function, Trans. Amer. Math. Soc. 165, 1972, pp [13] Sbordone, C., On some integral inequalities and their applications to the calculus of variations, Boll. U.M.I. An. Funz. Appl., serie VI, Vol. V, 1986, pp Received November, 1989.