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Partial Differential Equations 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 Acknowledgements This paper contains the unpublished part of the lecture held at the Conference ‘‘Recent Trends in Nonlinear Partial Di¤erential Equations (6.11.2008)’’.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 ÀdivðMðxÞDuÞ¼ 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 Ãà if 2N=ðN þ 2Þ < m < N=2; then u a M m ðWÞ; Ãà à à mN where m ¼ðm Þ ¼ NÀ2m (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 À TkðuÞ, where TkðuÞ is the truncation at the levels þk, Àk. 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 Àdivðaðx; 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 À 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 Ãà 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 Ãà 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 CtÀm; 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 Zwhich 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 À j a jðx; DjÞ; > u ai u ai :> fj Àjj g W fj Àjj 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 À 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 Ãà ðWÞ a Cf and jj Dujj L m à 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 À TkðuÞ; fjuÀTk ðuÞja1g fjuÀTk ðuÞja1g W which implies, since jðx; 0Þ¼0, Z Z Z Z 2 ð8Þ a jDuj a jðx; DuÞ a fT1½u À TkðuÞ a j f j Bk Bk W Ak m Ãà à 0 Ãà Define y ¼ 2 à , so that y2 ¼ð2y À 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 2ð1ÀyÞ 2ð1ÀyÞ 2ð1ÀyÞ Bk ð1 þjujÞ Bk ð1 þ kÞ Ak ð1 þ kÞ which implies, summing on k ranging from 0 to M À 1, Z Z k¼XMÀ1 2 k¼XMÀ1 jDuj j f j 10 a a ð Þ 2ð1ÀyÞ 2ð1ÀyÞ k¼0 Bk ð1 þjujÞ k¼0 Ak ð1 þ kÞ Z kX¼M j f j a : 2ð1ÀyÞ k¼0 Ak ð1 þ kÞ a calderon-zygmund theory for minima of integral functionals 199 S S h¼l k¼MÀ1 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 : 2ð1ÀyÞ 2ð1ÀyÞ 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 2ð1ÀyÞ ¼ 2ð1ÀyÞ 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 2yÀ1 ¼ j j 2ð1ÀyÞ j jð þ M ð ÞÞ Bh ð1 þ kÞ 2y À 1 Bh h¼0 Z k¼0 h¼0 1 2yÀ1 a j f j½1 þjTM ðuÞj ; 2y À 1 W since one has Z Z k¼XTM ðhÞ k¼XTM ðhÞ k T h 1 þ1 dx 1þ M ð Þ dx a ¼ 2ð1ÀyÞ x2ð1ÀyÞ x2ð1ÀyÞ k¼0 ð1 þ kÞ k¼0 k 0 ½1 þ T ðhÞ2yÀ1 ¼ M : 2y À 1 Thus we have Z Z jDT ðuÞj2 1 12 a M a f 1 T u 2yÀ1; ð Þ 2ð1ÀyÞ j j½ þj M ð Þj W ð1 þjTM ðuÞjÞ 2y À 1 W and (thanks to the Sobolev inequality), y y T u 2 à T u 2 à C jj ð 1 þj M ð ÞjÞ jj L ðWÞ a jj½ð1 þj M ð ÞjÞ À 1jjL ðWÞ þ W Z 1=2 2yÀ1 a C1 j f j½1 þjTM ðuÞj þ CW: W Using the Ho¨lder inequality, we then have Z Z 2 Ã=2m 0 y2 à 2 Ã=2 ð2yÀ1Þm 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 .
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