Rend. Lincei Mat. Appl. 20 (2009), 195–205

Partial Di¤erential Equations — A Calderon-Zygmund theory for infinite energy minima of some integral functionals,byLucio Boccardo, communicated on 8 May 2009.

‘‘. . . onore e lume vagliami ’l lungo studio e ’l grande amore che m’ha fatto cercar lo tuo volume.’’ (Inferno I, 82–84)1

Abstract. — A Calderon-Zygmund theory in Lebesgue and Marcinkiewicz spaces for infinite energy minima of some integral functionals is proved.

Key words: Calderon-Zygmund theory; Infinite energy minima.

AMS 2000 Mathematics Subject Classification: 49.10, 35J60.

Acknowledgements

This paper contains the unpublished part of the lecture held at the Conference ‘‘Recent Trends in Nonlinear Partial Di¤erential Equations (6.11.2008)’’. I would like to express my thanks to Accademia dei Lincei for the invitation to give a talk2 as a ‘‘tribute to Guido Stampacchia on the 30th anniversary of his death’’.

1. Introduction 1.1. Finite energy solutions. We recall the following regularity theorem by G. Stampacchia concerning solutions of linear Dirichlet problems in W, bounded subset of RN , N > 2, with right hand side a measurable function f ðxÞ. Consider a bounded elliptic matrix MðxÞ with ellipticity constant a > 0 and the related boundary value problem divðMðxÞDuÞ¼ f ðxÞ in W; ð1Þ u ¼ 0onqW:

1Always remembering (after 40 years) what I have learned from my teacher of ‘‘Istituzioni di Ana- lisi Superiore’’ and ‘‘Analisi Superiore’’ ([7] pg. 1, [8] pg. 1) 2... e piu´ d’onore ancora assai mi fenno, ch’e’ sı´ mi fecer de la loro schiera, sı´ ch’io fui sesto tra cotanto senno. (Inferno IV, 100–102) 196 l. boccardo

In [20] (see also3 [21] and [16]), the following result is proved about the solution 1; 2 4 u a W0 ðWÞ (recall that the coe‰cients of MðxÞ are not smooth ) under the as- sumption that f belongs to the Marcinkiewicz space M mðWÞ5: if m > N=2; then u a LlðWÞ; ð2Þ if 2N=ðN þ 2Þ < m < N=2; then u a M m ðWÞ;

mN where m ¼ðm Þ ¼ N2m (see [14] for new contributions in this field). The fun- damental tool for the proofs of (2) by Stampacchia is the use of the test function ½u TkðuÞ, where TkðuÞ is the truncation at the levels þk, k. Note that the proofs of (2) do not use the linearity of the di¤erential operator. Only the ellipticity is used, so that the results of (2) still hold for boundary value problems with nonlinear operators like divðaðx; u; DuÞÞ ¼ f ðxÞ in W; ð3Þ u ¼ 0onqW; under the ellipticity assumption aðx; s; xÞx b ajxj2, a > 0. If the matrix M is symmetric, the solution u of (1) is the minimum of the functional Z Z 1 1; 2 JðvÞ¼ MðxÞDvDv fv; v a W0 ðWÞ: 2 W W Thus the regularity theorem by Stampacchia can be stated in the following way: if f a M mðWÞ and m > N=2, the minimum u of J belongs to LlðWÞ;if 2N=ðN þ 2Þ < m < N=2, the minimum belongs to M m ðWÞ. Moreover, the proof of (2) can be easily adapted to the study of minima u of more general inte- gral functionals like Z Z 1; 2 jðx; v; DvÞ fv; v a W0 ðWÞ: W W The first result concerning the summability (in Lebesgue spaces) of u, solution of (1), is again due to G. Stampacchia: u belongs to Lm ðWÞ if f a LmðWÞ, 2N=ðN þ 2Þ a m < N=2. The proof uses (2), linear interpolation theory and the Marcinkiewicz–Zygmund Theorem. However, a summability result for weak (finite energy) solutions is proved in [11], [12] by a direct method which uses as test function a suitable power of u. In [13] the regularity results for minima of functionals are extend to the Lebesgue framework.

3but for me see ‘‘Appunti del corso di Analisi Superiore—Universita` di Roma—a.a. 1969–70’’ 4(Calderon-Zygmund without derivatives) 5M mðWÞ, m > 0, is the space of measurable functions v on W such that bC b 0 : measfx a W : jvðxÞj b tg a Ctm; Et > 0 a calderon-zygmund theory for minima of integral functionals 197

1.2. Infinite energy solutions. If the datum f belongs to larger spaces (LmðWÞ,1a m < 2N=ðN þ 2Þ or M mðWÞ,1< m < 2N=ðN þ 2Þ) the regularity of the distributional (infinite energy) solutions u of (3) and of Du (nonlinear Calderon–Zygmund Theory) is proved in [13] in the Lebesgue framework and in [6] in the Marcinkiewicz framework (see also [17]). If the datum f belongs to larger spaces as above, it is not possible to use the definition of minimum, because the associated functional is not well defined on 1; 2 the ‘‘energy space’’ W0 ðWÞ. A possible way to handle minimization problems is then the use of T-minima, introduced in [2]. Minimization problems for integral functionals with nonregular data are also studied in [3], [5], [4], [19] (in these papers the function j of (4) can also depend on u) and [18], where existence of minima is proved also for function- als with measure data, using the definition of ‘‘weak minimum’’ introduced by Iwaniec and Sbordone [15]. Of course, it is possible to work with the same proofs if the standard frame- 1; p 1; 2 work is W0 ðWÞ instead of W0 ðWÞ; that is: if the assumption of coercivity is ajxjp a jðx; xÞ a bjxjp,1< p a N, instead of (5) (see below). 1.3. Assumptions. Let jðx; xÞ be a function defined in W RN .On jðx; xÞ we assume the standard hypotheses of the integrands in the Calculus ofZ Variations, 1; 2 whichZ lead to existence and uniqueness of minima in W0 ðWÞ of jðx; DvÞ W f ðxÞvðxÞ,if f a L2ðWÞ, that is: W the function jðx; xÞ is measurable with respect to x ð4Þ and strictly convex with respect to x there exist a; b > 0 such that ð5Þ ajxj2 a jðx; xÞ a bjxj2; Ex a RN ; a:e: in W: Recalling the definition of truncation T : R 7! R ( k t; jtj a k; TkðtÞ¼ t k jtj ; jtj > k; we give the definition and the existence theorem for T-minima. Definition 1.1 ([2]). Let f a L1ðWÞ. A measurable function u is a T-minimum for the functional Z Z ð6Þ JðvÞ¼ jðx; DvÞ f ðxÞvðxÞ W W if 8 1; 2 >TiðuÞ a W ðWÞ; Ei > 0: <> Z 0 Z Z ð7Þ jðx; DuÞ f ðxÞTi½u j a jðx; DjÞ; > u ai u ai :> fj jj g W fj jj g E 1; 2 B l E j a W0 ðWÞ L ðWÞ; i > 0: 198 l. boccardo

Proposition 1 ([2]). Under the assumptions (4) and (5) there exists a T- minimum u of the functional J defined in (6). Moreover the T-minimum u is unique and, in the case of di¤erentiability of jðx; xÞ with respect to x, the T-minimum is the entropy solution (see [1]) of the Euler-Lagrange equation for J. Remark 1; q E 1. In [2] it is also proved that u a W0 ðWÞ, q < N=ðN 1Þ.

2. A Calderon-Zygmund theory in Lebesgue spaces for infinite energy minima of some integral functionals

For Dirichlet problems with measure (or L1) data, existence of distributional so- lutions is proved in [9]; while in [10] it is proved that the assumption f a LmðWÞ, 1 < m < 2N=ðN þ 2Þ, yields more summability for the solutions and their gra- dients. We use the following definitions, for k a Rþ,

Ak ¼fx a W : k a juðxÞjg; Bk ¼fx a W : k a juðxÞj < k þ 1g: Theorem 2.1 (Calderon-Zygmund theory for functionals 1). Under the as- sumptions (4) and (5),iff a LmðWÞ, 1 < m < 2N=ðN þ 2Þ, then there exists a pos- itive constant Cf such that the T-minimum u of the functional J of (6) satisfies the estimates jj ujj L m ðWÞ a Cf and jj Dujj L m ðWÞ a Cf . Proof. Let k > 0 be fixed. In the definition of T-minimum we use as test func- tion j ¼ TkðuÞ and i ¼ 1. We then obtain Z Z Z

jðx; DuÞ a jðx; DTkðuÞÞ þ f ðxÞT1½u TkðuÞ; fjuTk ðuÞja1g fjuTk ðuÞja1g W which implies, since jðx; 0Þ¼0, Z Z Z Z 2 ð8Þ a jDuj a jðx; DuÞ a fT1½u TkðuÞ a j f j Bk Bk W Ak m 0 Define y ¼ 2 , so that y2 ¼ð2y 1Þm ¼ m . Then a consequence of (8) is that Z Z Z jDuj2 jDuj2 j f j 9 a a a a ; ð Þ 2ð1yÞ 2ð1yÞ 2ð1yÞ Bk ð1 þjujÞ Bk ð1 þ kÞ Ak ð1 þ kÞ which implies, summing on k ranging from 0 to M 1, Z Z k¼XM1 2 k¼XM1 jDuj j f j 10 a a ð Þ 2ð1yÞ 2ð1yÞ k¼0 Bk ð1 þjujÞ k¼0 Ak ð1 þ kÞ Z kX¼M j f j a : 2ð1yÞ k¼0 Ak ð1 þ kÞ a calderon-zygmund theory for minima of integral functionals 199 S S h¼l k¼M1 Observe that Ak ¼ h¼k Bh, and that fjuj a Mg¼ k¼0 Bk, so that a conse- quence of (10) is Z Z 2 XM hX¼l jDT ðuÞj j f j a M a : 2ð1yÞ 2ð1yÞ W ð1 þjTM ðuÞjÞ k¼0 h¼k Bh ð1 þ kÞ Exchanging the summation order in the right hand side, we obtain Z Z kX¼M hX¼l kX¼M hX¼l j f j 1 f 2ð1yÞ ¼ 2ð1yÞ j j B 1 k 1 k B k¼0 h¼k h ð þ Þ k¼0 ð þ Þ h¼k h Z Z hX¼l k¼XTM ðhÞ 1 hX¼l 1 ð11Þ f a f 1 T h 2y1 ¼ j j 2ð1yÞ j jð þ M ð ÞÞ Bh ð1 þ kÞ 2y 1 Bh h¼0 Z k¼0 h¼0 1 2y1 a j f j½1 þjTM ðuÞj ; 2y 1 W since one has Z Z k¼XTM ðhÞ k¼XTM ðhÞ k T h 1 þ1 dx 1þ M ð Þ dx a ¼ 2ð1yÞ x2ð1yÞ x2ð1yÞ k¼0 ð1 þ kÞ k¼0 k 0 ½1 þ T ðhÞ2y1 ¼ M : 2y 1 Thus we have Z Z jDT ðuÞj2 1 12 a M a f 1 T u 2y1; ð Þ 2ð1yÞ j j½ þj M ð Þj W ð1 þjTM ðuÞjÞ 2y 1 W and (thanks to the Sobolev inequality), y y T u 2 T u 2 C jj ð 1 þj M ð ÞjÞ jj L ðWÞ a jj½ð1 þj M ð ÞjÞ 1jjL ðWÞ þ W Z 1=2 2y1 a C1 j f j½1 þjTM ðuÞj þ CW: W

Using the Ho¨lder inequality, we then have Z Z 2 =2m 0 y2 2 =2 ð2y1Þm 0 ð1 þjTM ðuÞjÞ a C2jj f jj L mðWÞ ½1 þjTM ðuÞj þ C2: W W

1 2N Note that 2 < y < 1 since 1 < m < Nþ2 . Thus we proved the inequality Z 1=m m ½1 þjTM ðuÞj a Cf jj f jj L mðWÞ þ Cf ; W 200 l. boccardo which implies, as M ! l (thanks to Fatou Lemma), Z 1=m m ð13Þ ½1 þjuj a Cf jj f jj L mðWÞ þ Cf ; W that is the first part of the result. Now we use (12), the above estimate (13) and Ho¨lder inequality. Z 2 Z 1=m 0 jDTM ðuÞj m a C f 1 C f m u C : 2ð1yÞ 4jj jj L ðWÞ þ 4jj jj L ðWÞ j j ¼ 0 W ð1 þjTM ðuÞjÞ W Then the use of (once more) estimate (13) and Fatou Lemma (as M ! l) in the inequality Z Z m jDT ðuÞj jDT ðuÞjm ¼ M ð1 þjT ðuÞjÞm ð1yÞ M m ð1yÞ M W W ð1 þjTM ðuÞjÞ ð14Þ Z ð2m Þ=2 m =2 m a C0 ð1 þjujÞ W gives the second part of the result. r

Theorem 2.2 (A borderline case). Under the assumptions (4) and (5),if Z j f j logð1 þjf jÞ < l; W 1; 1 then the T-minimum u of the functional J of (6) belongs to W0 ðWÞ. Proof. Since one has Z k¼XTM ðhÞ 1 k¼XTM ðhÞ kþ1 dx a 1 þ ¼ 1 þ log½1 þ T ðhÞ; 1 k x M k¼0 ð þ Þ k¼1 k if we put y ¼ 1=2 in (11), inequality (12) becomes

Z 2 Z Z jDTM ðuÞj ð15Þ a a j f jþ j f j logð1 þjTM ðuÞjÞ: W ð1 þjTM ðuÞjÞ W W

Then (thanks to Sobolev and Young inequalities) we have Z ðN2Þ=2N N=ðN2Þ ð1 þjTM ðuÞjÞ W 1=2 1=2 ¼jjð1 þjTM ðuÞjÞ jj L2N=ðN2ÞðWÞ a jj ð 1 þjTM ðuÞjÞ 1jj L2N=ðN2ÞðWÞ þ C1 Z Z Z 1=2 1=2 1=2 logð1þjTM ðuÞjÞ a C1 þ C2 j f j þ C2 j f j logð1 þjf jÞ þ C2 e : W W W a calderon-zygmund theory for minima of integral functionals 201

Thus we proved the inequality Z N=ðN2Þ jTM ðuÞj a Cf ; W which implies (as M ! l) Z N=ðN2Þ juj a Cf : W

As a consequence of this estimate, of inequality (15) and of Fatou Lemma (as M ! l) we have Z jDuj2 a C0: W ð1 þjujÞ

Here we repeat (14) with m ¼ 1 and we prove the result concerning the sum- mability of the gradient. r

3. A Calderon-Zygmund theory in Marcinkiewicz spaces for infinite energy minima of some integral functionals

Theorem 3.1 (Calderon-Zygmund theory for functionals 2). Under the as- sumptions (4) and (5),iffa M mðWÞ, 1 < m < 2N=ðN þ 2Þ, then there exists a positive constant Cf such that the T-minimum u of the functional J of (6) satisfies the estimates C measfk a jujg a f k m and C measft a jDujg a f : tm

Proof. m We start as in Theorem 2.1, we use (9) with y ¼ 2 . If we sum these inequalities, with k ranging now between j b 1 and M, we obtain Z Z Z 2 k¼XM1 2 kX¼M jDuj jDuj j f j ð16Þ a ¼ a a : 2ð1yÞ 2ð1yÞ k2ð1yÞ jajuj 1 2 m 0 "# Z 1=m m jTM ðuÞj a C1 jajuj and, as M ! l, "# Z 1=m 1=m m j½measf j a jujg a juj a C1; jajuj that is the Marcinkiewicz estimate on u: C ð18Þ measf j a jujg a 0 : j m With respect to the gradient, from (8) and (18) we have, since our assumption on m m implies m 0 < 1, Z Z Z k¼XM1 kX¼M a jDT ðuÞj2 ¼ a jDuj2 a j f j M W k¼0 Bk k¼0 Ak Z kX¼M C~ a j f jþ f a C Mð2N2mmNÞ=ðN2mÞ k m =m 0 1 W k¼1 Here we follow a technique of [1]. The previous estimate also implies Z C t2 jfjuj < kg B fjDuj b tgj a jDuj2 a 1 k1m ð11=mÞ fjuj

jfjDuj b tgj a jfjDuj b t; juj < kgj þ jfjuj b kgj and (18) give C k1m ð11=mÞ 1 jfjDuj b tgj a 1 þ C : a t2 0 k m Note that 1 ðm 1ÞN 1 2N mðN þ 2Þ m 1 ¼ ; 1 m 1 ¼ a ð0; 1: m N 2m m N 2m

The minimization with respect to k gives (choose k ¼ tðN2mÞ=ðNmÞ) a calderon-zygmund theory for minima of integral functionals 203

C ð19Þ jfjDuj b tgj a f ; tm as desired. r

Lemma 3.2 (A borderline case). Under the assumptions (4) and (5),if 2N=ðNþ2Þ f a M ðWÞ, then there exists a positive constant Cf such that the T- minimum u of the functional J of (6) satisfies the estimates C ð20Þ measfk a jujg a f k2N=ðN2Þ Proof. m Let 1 < m < 2N=ðN þ 2Þ and y ¼ 2 . We start as in Theorem 3.1, we get (17), we use Ho¨lder and Young inequalities and we have "# "#"#0 Z 2=2 Z 1=m Z 1=m jT ðuÞjm a C j f jm jT ðuÞjm M m M jajuj jajuj jajuj "#"# Z 2=2 Z ðN2Þ=ðN2mÞ 1 m m a jTM ðuÞj þ C1 j f j : 2 jajuj jajuj

Now, thanks to Fatou Lemma (as M ! l), we have "#"# Z 2=2 Z ðN2Þ=ðN2mÞ m m juj a 2C1 j f j : jajuj jajuj Now we use Ho¨lder inequality in Marcinkiewicz framework and we have "# Z 2=2 m ½2NmðNþ2ÞðN2Þ=2NðN2mÞ juj a C2½measf j a jujg jajuj which implies

2m =2 ðN2Þ=N ½2NmðNþ2Þm =mN2 j ½measf j a jujg a C2½measf j a jujg that is C measf j a jujg a f ; j 2N=ðN2Þ that is the Marcinkiewicz estimate on u. r

Remark 2. We are not able to prove the Marcinkiewicz estimate in M 2ðWÞ on the gradient, under the same assumptions of the previous lemma. Note that, if we put together the above estimate and (8), we have Z C a jDuj2 a f ; Bk k 204 l. boccardo which implies (with the previous techniques) Z 2 a jDTM ðuÞj a C0 logðMÞ: W

Then, if we follow the second part of the proof of Theorem 3.1, we can show

logðtÞ measft a jDujg a C ; f t2 but we are not able to prove that

1 ð21Þ measft a jDujg a C : f t2

We point out that the similar bordeline case for the Dirichlet problems involving equations has been recently treated in [17], where the estimate (21) is proved.

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Received 16 February 2009, and in revised form 7 May 2009.

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