Caccioppoli estimates and very weak solutions of elliptic equations

Iwaniec, Tadeusz ; Sbordone, Carlo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003), p. 189-205 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

Caccioppoli estimates are instrumental in virtually all analytic aspects of the theory of partial differential equations, linear and nonlinear. And there is always something new to add to these estimates. We emphasize the fundamental role of the natural domain of definition of a given differential operator and the associated weak solutions. However, we depart from this usual setting (energy estimates) and move into the realm of the so-called very weak solutions where important new applications lie. We carry out this task deliberately with a restricted generality in interest of readability, and we hope it pays off handsomely in mathematical insights.

Détail

Publié le : 2003-09-01

MR 2064266 | Zbl 1225.35069

Bibliographie

[1] Alvino, A. - Carbone, L. - Sbordone, C. - Trombetti, G., In ricordo di Renato Caccioppoli. Giannini Editore, Napoli 1999. | Zbl 0928.00071

[2] Astala, K. - Iwaniec, T. - Saksman, E., Beltrami operators in the plane. Duke Math. J., 107, n. 1, 2001, 27-56. | MR 1815249 | Zbl 1009.30015

[3] Baernstein, A. - Montgomery-Smith, S.J., Some conjectures about integral means of $\partial f$ and $\bar{\partial f}$. In: C. Kiselman (ed.), Complex analysis and differential equations (Uppsala, 1997). Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 64, 92-109. | MR 1758918 | Zbl 0966.30001

[4] Bañuelos, R. - Lindeman, A., A martingale study of the Beurling-Ahlfors transform in $\mathbb{R}^{n}$. J. Funct. Anal., 145, n. 1, 1997, 224-265. | MR 1442167 | Zbl 0876.60026

[5] Bañuelos, R. - Wang, G., Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J., 80, n. 3, 1995, 575-600. | MR 1370109 | Zbl 0853.60040

[6] Bénilan, P. - Boccardo, L. - Gallouët, T. - Gariepy, R. - Pierre, M. - Vázquez, J.L., An $L^{1}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 22, n. 2, 1995, 241-273. | MR 1354907 | Zbl 0866.35037

[7] Bers, L. - Nirenberg, L., On a representation theorem for linear elliptic equations with discontinuous coefficients and its applications. Convegno Int. Eq. Lin. Deriv. Parz., Trieste 1954, 111-140. | MR 76981 | Zbl 0067.32503

[8] Boccardo, L. - Gallouët, T., Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations, 17, n. 3-4, 1992, 641-655. | MR 1163440 | Zbl 0812.35043

[9] Bojarski, B., Homeomorphic solutions of Beltrami's systems. Dokl. Akad. Nauk SSSR, 102, 1955, 661-664. | MR 71620

[10] Bojarski, B., Remarks on the stability of reverse Hölder inequalities and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math., 10, 1985, 89-94. | MR 802470 | Zbl 0582.30016

[11] Bojarski, B. - Iwaniec, T., Analytic foundations of the theory of quasiconformal mappings in $\mathbb{R}^{n}$. Ann. Acad. Sci. Fenn. Ser. A., I 8, 1983, 257-324. | MR 731786 | Zbl 0548.30016

[12] Brezis, H. - Fusco, N. - Sbordone, C., Integrability for the Jacobian of orientation preserving mappings. J. Funct. Anal., 115, n. 2, 1993, 425-431. | MR 1234399 | Zbl 0847.26012

[13] Budney, L. - Iwaniec, T. - Stroffolini, B., Removability of singularities of $a$-harmonic functions. Differential Integral Equations, 12, n. 2, 1999, 261-274. | MR 1672754 | Zbl 1064.35505

[14] Caccioppoli, R., Limitazioni integrali per le soluzioni di un’equazione lineare ellitica a derivate parziali. Giorn. Mat. Battaglini, (4) 4(80), 1951, 186-212. | MR 46536 | Zbl 0043.31404

[15] Coifman, R. - Lions, P.L. - Meyer, Y. - Semmes, S., Compensated compactness and Hardy spaces. J. Math. Pures Appl., (9) 72, n. 3, 1993, 247-286. | MR 1225511 | Zbl 0864.42009

[16] Dal Maso, G. - Murat, F. - Orsina, L. - Prignet, A., Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 28, n. 4, 1999, 741-808. | MR 1760541 | Zbl 0958.35045

[17] Dolzmann, G. - Hungerbhler, N. - Müller, S., Uniqueness and maximal regularity for nonlinear elliptic systems of $n$-Laplace type with measure valued right hand side. J. Reine Angew. Math., 520, 2000, 1-35. | MR 1748270 | Zbl 0937.35065

[18] Elcrat, A. - Meyers, N., Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Math. J., 42, 1975, 121-136. | MR 417568 | Zbl 0347.35039

[19] Figiel, T. - Iwaniec, T. - Pełczyǹski, A., Computing norms and critical exponents of some operators in $L^{p}$-spaces. Studia Math., 79, 1984, 228-274. | MR 781720 | Zbl 0599.46035

[20] Fiorenza, A. - Sbordone, C., Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^{1}$. Studia Math., 127, n. 3, 1998, 223-231. | MR 1489454 | Zbl 0891.35039

[21] Franchi, B. - Serra Cassano, F., Gehring's lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals. Studia Math., 120, n. 1, 1996, 1-22. | MR 1398170 | Zbl 0865.46017

[22] Fusco, N. - Sbordone, C., Higher integrability of the gradient of minimizers of functionals with non-standard growth conditions. Comm. Pure Appl. Math., 43, n. 5, 1990, 673-683. | MR 1057235 | Zbl 0727.49021

[23] Gehring, F.W., The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math., 130, 1973, 265-277. | MR 402038 | Zbl 0258.30021

[24] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105, Princeton1983. | MR 717034 | Zbl 0516.49003

[25] Giaquinta, M. - Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math., 311/312, 1979, 145-169. | MR 549962 | Zbl 0409.35015

[26] Goldstein, V. M. - Vodop'Yanov, S. K., Quasiconformal mappings and spaces of functions with first generalized derivatives. Sibirsk. Mat. Z., 17, n. 3, 1976, 515-531. | MR 414869 | Zbl 0339.30018

[27] Greco, L., A remark on the equality $det \, Df = Det \, Df$. Differential and Integral Equations, 6, n. 5, 1993, 1089-1100. | MR 1230483 | Zbl 0784.49013

[28] Greco, L. - Iwaniec, T., New inequalities for the Jacobian. Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, n. 1, 1994, 17-35. | MR 1259100 | Zbl 0848.58051

[29] Greco, L. - Iwaniec, T. - Moscariello, G., Limits of the improved integrability of the volume forms. Indiana Univ. Math. J., 44, n. 2, 1995, 305-339. | MR 1355401 | Zbl 0855.42009

[30] Greco, L. - Iwaniec, T. - Sbordone, C., Inverting the $p$-harmonic operator. Manuscripta Math., 92, n. 2, 1997, 249-258. | MR 1428651 | Zbl 0869.35037

[31] Hajłasz, P. - Iwaniec, T. - Malý, J. - Onninen, J., Weakly differentiable mappings between manifolds. Preprint. | Zbl 1145.58006

[32] Istituto Italiano Per Gli Studi Filosofici: Seminari Di Scienze, , Il pensiero matematico del XX secolo e l’opera di Renato Caccioppoli. Atti della giornata di studi tenuta a Pisa il 10 aprile 1987 per iniziativa dell’Istituto Italiano per gli Studi Filosofici e della Scuola Normale di Pisa, 1988.

[33] Iwaniec, T., Extremal inequalities in Sobolev spaces and quasiconformal mappings. Z. Anal. Anwendungen, 1, n. 6, 1982, 1-16. | MR 719167 | Zbl 0577.46038

[34] Iwaniec, T., Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators. Studia Math., 75, n. 3, 1983, 293-312. | MR 722254 | Zbl 0552.35034

[35] Iwaniec, T., $p$-harmonic tensors and quasiregular mappings. Ann. of Math., (2) 136, n. 3, 1992, 589-624. | MR 1189867 | Zbl 0785.30009

[36] Iwaniec, T. - Koskela, P. - Onninen, J., Mappings of finite distortion: monotonicity and continuity. Invent. Math., 144, n. 3, 2001, 507-531. | MR 1833892 | Zbl 1006.30016

[37] Iwaniec, T. - Koskela, P. - Onninen, J., Mappings of finite distortion: Compactness. Ann. Acad. Sci. Fenn. Math., 27, 2002, 391-417. | MR 1922197 | Zbl 1017.30024

[38] Iwaniec, T. - Koskela, P. - Martin, G. - Sbordone, C., Mappings of finite distortion: $L^{n} \log^{a}L$. J. London Math. Soc., to appear. | Zbl 1047.30010

[39] Iwaniec, T. - Lutoborski, A., Integral estimates for null Lagrangians. Arch. Rational Mech. Anal., 125, n. 1, 1993, 25-79. | MR 1241286 | Zbl 0793.58002

[40] Iwaniec, T. - Martin, G., Geometric function theory and non-linear analysis. Oxford Mathematical Monographs, Oxford Univ. Press, 2001. | MR 1859913 | Zbl 1045.30011

[41] Iwaniec, T. - Martin, G., Quasiregular mappings in even dimensions. Acta Math., 170, n. 1, 1993, 29-81. | MR 1208562 | Zbl 0785.30008

[42] Iwaniec, T. - Migliaccio, L. - Moscariello, G. - Passarelli Di Napoli, A., A priori estimates for nonlinear elliptic complexes. Advances on Diff. Eq., 8 (5), 2003, 513-546. | MR 1972489 | Zbl pre02002540

[43] Iwaniec, T. - Migliaccio, L. - Nania, L. - Sbordone, C., Integrability and removability results for quasiregular mappings in high dimensions. Math. Scand., 75, n. 2, 1994, 263-279. | MR 1319735 | Zbl 0824.30009

[44] Iwaniec, T. - Sbordone, C., On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal., 119, n. 2, 1992, 129-143. | MR 1176362 | Zbl 0766.46016

[45] Iwaniec, T. - Sbordone, C., Weak minima of variational integrals. J. Reine Angew. Math., 454, 1994, 143-161. | MR 1288682 | Zbl 0802.35016

[46] Iwaniec, T. - Sbordone, C., Div-curl fields of finite distortion. C. R. Acad. Sci. Paris Sr. I Math., 327, n. 8, 1998, 729-734. | MR 1660033 | Zbl 0916.30015

[47] Iwaniec, T. - Sbordone, C., Quasiharmonic fields. Ann. Inst. H. Poincaré Anal. Non Linéaire, 18, n. 5, 2001, 519-572. | MR 1849688 | Zbl 1068.30011

[48] Iwaniec, T. - Scott, C. - Stroffolini, B., Nonlinear Hodge theory on manifolds with boundary. Ann. Mat. Pura Appl., 177 (4), 1999, 37-115. | MR 1747627 | Zbl 0963.58003

[49] Iwaniec, T. - Verde, A., A study of Jacobians in Hardy-Orlicz spaces. Proc. Roy. Soc. Edinburgh, Sect. A, 129, n. 3, 1999, 539-570. | MR 1693625 | Zbl 0954.46018

[50] Iwaniec, T. - Verchota, G.C. - Vogel, A.L., The Failure of Rank-One Connection. Arch. Rational Mech. Anal., 163, n. 2, 2002, 125-169. | MR 1911096 | Zbl 1007.35014

[51] Kauhanen, J. - Koskela, P. - Malý, J., Mappings of finite distortion: condition N. Michigan Math. J., 49, n. 1, 2001, 169-181. | MR 1827080 | Zbl 0997.30018

[52] Kauhanen, J. - Koskela, P. - Malý, J., Mappings of finite distortion: discreteness and openness. Arch. Rational Mech. Anal., 160, n. 2, 2001, 135-151. | MR 1864838 | Zbl 0998.30024

[53] Kilpeläinen, T. - Malý, J., Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19, n. 4, 1992, 591-613. | MR 1205885 | Zbl 0797.35052

[54] Migliaccio, L. - Moscariello, G., Higher integrability of div-curl products. Ricerche Mat., 49, n. 1, 2000, 151-161. | MR 1795037

[55] Moscariello, G., On the integrability of the Jacobian in Orlicz spaces. Math. Japon., 40, n. 2, 1994, 323-329. | MR 1297249 | Zbl 0805.46026

[56] Moscariello, G. - Sbordone, C., $A^{\infty}$ as a limit case of reverse-Hölder inequalities when the exponent tends to 1. Ricerche Mat., 44, n. 1, 1995, n. 1, 131-144. | MR 1470190 | Zbl 0920.26017

[57] Müller, S., Higher integrability of determinants and weak convergence in $L^{1}$. J. Reine Angew. Math., 412, 1990, 20-34. | MR 1078998 | Zbl 0713.49004

[58] Nazarov, F. - Volberg, A., Heating of the Beurling operator and the estimates of its norm. Amer. J. of Math., to appear.

[59] Passarelli Di Napoli, A. - Sbordone, C., Elliptic equations with right hand side in $L(\log L)^{a}$. Rend. Accad. Sci. Fis. Mat. Napoli, 62 (4), 1995, 301-314 (1996). | MR 1419292 | Zbl 1162.35354

[60] Piccinini, L. - Spagnolo, S., Una valutazione della regolarità delle soluzioni di sistemi ellittici variazionali in due variabili. Ann. Scuola Norm. Sup. Pisa, 27 (3), 1973, 417-429. | MR 369906 | Zbl 0279.49014

[61] Sbordone, C., New estimates for div-curl products and very weak solutions of PDEs. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (4), n. 3-4, 1997, 739-756 (1998). | MR 1655540 | Zbl 1073.35515

[62] Sbordone, C., Grand Sobolev spaces and their applications to variational problems. Le Matematiche (Catania), 51, n. 2, 1996, 335-347 (1997). | MR 1488076 | Zbl 0915.46030

[63] Sbordone, C., Maximal inequalities and applications to regularity problems in the calculus of variations. Calculus of variations, applications and computations (Pont-à-Mousson, 1994). Pitman Res. Notes Math. Ser., 326, Longman Sci. Tech., Harlow1995, 230-244. | MR 1419348 | Zbl 0835.35046

[64] Sbordone, C., Reduction of nonlinear PDEs to linear equations. Milan Journal of Math., to appear. | MR 2120914 | Zbl 1121.35058

[65] Sbordone, C., Nonlinear commutators and applications to the regularity properties of the Jacobian. Atti Sem. Mat. Fis. Univ. Modena, 43, n. 2, 1995, 363-369. | MR 1366066 | Zbl 0851.46023

[66] Stredulinsky, E.W., Higher integrability from reverse Hölder inequalities. Indiana Univ. Math. J., 29, n. 3, 1980, n. 3, 407-413. | MR 570689 | Zbl 0442.35064