# Article

 Title: On very weak solutions of a class of nonlinear elliptic systems (English) Author: Carozza, Menita Author: Passarelli di Napoli, Antonia Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 41 Issue: 3 Year: 2000 Pages: 493-508 . Category: math . Summary: In this paper we prove a regularity result for very weak solutions of equations of the type $- \operatorname{div} A(x,u,Du)=B(x, u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x, u, Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$\cases -\operatorname{div} A(x,u,Du)\,=B(x, u,Du) \quad & \text{in } \Omega , \ u-u_o\in W^{1,r}(\Omega,\Bbb R^m). \endcases$$ (English) Keyword: nonlinear elliptic systems Keyword: maximal operator theory MSC: 35B65 MSC: 35D05 MSC: 35J50 MSC: 35J55 MSC: 35J60 MSC: 35J99 MSC: 46E30 idZBL: Zbl 1119.35016 idMR: MR1795081 . Date available: 2009-01-08T19:04:34Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119185 . Reference: [AF] Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations.Arch. Rational Mech. Anal. 86 (1984), 125-145. Zbl 0565.49010, MR 0751305 Reference: [Do] Dolcini A.: A uniqueness result for very weak solutions of $p$-harmonic type equations.Boll. Un. Mat. Ital., Serie VII X-A (1996), 71-84. Zbl 0854.35047, MR 1386247 Reference: [FS] Fiorenza A., Sbordone C.: Existence and uniqueness result for solutions of nonlinear equations with right hand side in $L^1$.Studia Math. 127 (3) (1998), 223-231. MR 1489454 Reference: [G] Giaquinta M.: Multiple integrals in the Calculus of variations and nonlinear elliptic systems.Ann. of Math. Stud. 105, Princeton University Press, 1983. Zbl 0516.49003, MR 0717034 Reference: [GG] Giaquinta M., Giusti E.: On the regularity of the minima of variational integrals.Acta Math. 148 (1982), 31-46. Zbl 0494.49031, MR 0666107 Reference: [Gi] Giusti E.: Metodi diretti nel Calcolo delle Variazioni.U.M.I., 1984. Zbl 0942.49002 Reference: [GIS] Greco L., Iwaniec T., Sbordone C.: Inverting the $p$-harmonic operator.Manuscripta Math. 92 (1997), 249-258. Zbl 0869.35037, MR 1428651 Reference: [GLS] Giachetti D., Leonetti F., Schianchi R.: On the regularity of very weak minima.Proc. Royal Soc. Edinburgh 126A (1996), 287-296. Zbl 0851.49026, MR 1386864 Reference: [GS] Giachetti D., Schianchi R.: Boundary higher integrability for the gradient of distributional solutions of nonlinear systems.preprint. Zbl 0869.49020, MR 1439029 Reference: [GT] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order.Springer Verlag, 1982. Zbl 1042.35002 Reference: [H] Hedberg L.I.: On certain convolution inequalities.Proc. Amer. Math. Soc. 36 (1972), 505-510. MR 0312232 Reference: [I] Iwaniec T.: $p$-harmonic tensors and quasiregular mappings.Ann. of Math. 136 (1992), 589-624. Zbl 0785.30009, MR 1189867 Reference: [IS] Iwaniec T., Sbordone C.: Weak minima of variational integrals.J. Reine Angew. Math. 454 (1994), 143-161. Zbl 0802.35016, MR 1288682 Reference: [Le] Lewis J.: On very weak solutions of certain elliptic systems.Comm. Partial Differential Equations 18 (1993), 1515-1537. Zbl 0796.35061, MR 1239922 Reference: [M1] Moscariello G.: Weak minima and quasiminima of variational integrals.B.U.M.I. 7-11B (1997), 355-364. Zbl 0890.49003, MR 1459284 Reference: [M2] Moscariello G.: On weak minima of certain integral functionals.Ann. Polon. Math. LXIX.1 (1998), 37-48. Zbl 0920.49021, MR 1630200 Reference: [Mu] Muckenhoupt B.: Weighted norm inequalities for the Hardy Maximal Function.Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl 0236.26016, MR 0293384 Reference: [S] Sbordone C.: Quasiminima of degenerate functionals with non polynomial growth.Rend. Sen. Mat. Fis. Milano LIX (1989), 173-184. Zbl 0760.49023, MR 1159695 Reference: [T] Torchinsky A.: Real variable methods in harmonic analysis.Pure Appl. Math. 123, Academic Press, 1986. Zbl 1097.42002, MR 0869816 .

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