On very weak solutions of a class of nonlinear elliptic systems

Menita Carozza; Antonia Passarelli di Napoli

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 3, page 493-508
  • ISSN: 0010-2628

Abstract

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In this paper we prove a regularity result for very weak solutions of equations of the type - div A ( x , u , D u ) = B ( x , u , D u ) , where A , B grow in the gradient like t p - 1 and B ( x , u , D u ) is not in divergence form. Namely we prove that a very weak solution u ∈ 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 - div A ( x , u , D u ) = B ( x , u , D u ) in Ω , u - u o ∈ W 1 , r ( Ω , ℝ m ) .

How to cite

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Carozza, Menita, and Passarelli di Napoli, Antonia. "On very weak solutions of a class of nonlinear elliptic systems." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 493-508. <http://eudml.org/doc/248613>.

@article{Carozza2000,
abstract = {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 \[ \left\lbrace \begin\{array\}\{ll\}-\operatorname\{div\} A(x,u,Du)\,=B(x, u,Du) \quad & \text\{in \} \Omega , \ u-u\_o\in W^\{1,r\}(\Omega ,\mathbb \{R\}^m). \end\{array\}\right.\]},
author = {Carozza, Menita, Passarelli di Napoli, Antonia},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonlinear elliptic systems; maximal operator theory; nonlinear elliptic system; maximal operator},
language = {eng},
number = {3},
pages = {493-508},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On very weak solutions of a class of nonlinear elliptic systems},
url = {http://eudml.org/doc/248613},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Carozza, Menita
AU - Passarelli di Napoli, Antonia
TI - On very weak solutions of a class of nonlinear elliptic systems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 3
SP - 493
EP - 508
AB - 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 \[ \left\lbrace \begin{array}{ll}-\operatorname{div} A(x,u,Du)\,=B(x, u,Du) \quad & \text{in } \Omega , \ u-u_o\in W^{1,r}(\Omega ,\mathbb {R}^m). \end{array}\right.\]
LA - eng
KW - nonlinear elliptic systems; maximal operator theory; nonlinear elliptic system; maximal operator
UR - http://eudml.org/doc/248613
ER -

References

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  1. Acerbi E., Fusco N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125-145. (1984) Zbl0565.49010MR0751305
  2. 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. (1996) Zbl0854.35047MR1386247
  3. 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. (1998) MR1489454
  4. Giaquinta M., Multiple integrals in the Calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud. 105, Princeton University Press, 1983. Zbl0516.49003MR0717034
  5. Giaquinta M., Giusti E., On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46. (1982) Zbl0494.49031MR0666107
  6. Giusti E., Metodi diretti nel Calcolo delle Variazioni, U.M.I., 1984. Zbl0942.49002
  7. Greco L., Iwaniec T., Sbordone C., Inverting the p -harmonic operator, Manuscripta Math. 92 (1997), 249-258. (1997) Zbl0869.35037MR1428651
  8. Giachetti D., Leonetti F., Schianchi R., On the regularity of very weak minima, Proc. Royal Soc. Edinburgh 126A (1996), 287-296. (1996) Zbl0851.49026MR1386864
  9. Giachetti D., Schianchi R., Boundary higher integrability for the gradient of distributional solutions of nonlinear systems, preprint. Zbl0869.49020MR1439029
  10. Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1982. Zbl1042.35002
  11. Hedberg L.I., On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. (1972) MR0312232
  12. Iwaniec T., p -harmonic tensors and quasiregular mappings, Ann. of Math. 136 (1992), 589-624. (1992) Zbl0785.30009MR1189867
  13. Iwaniec T., Sbordone C., Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143-161. (1994) Zbl0802.35016MR1288682
  14. Lewis J., On very weak solutions of certain elliptic systems, Comm. Partial Differential Equations 18 (1993), 1515-1537. (1993) Zbl0796.35061MR1239922
  15. Moscariello G., Weak minima and quasiminima of variational integrals, B.U.M.I. 7-11B (1997), 355-364. (1997) Zbl0890.49003MR1459284
  16. Moscariello G., On weak minima of certain integral functionals, Ann. Polon. Math. LXIX.1 (1998), 37-48. (1998) Zbl0920.49021MR1630200
  17. Muckenhoupt B., Weighted norm inequalities for the Hardy Maximal Function, Trans. Amer. Math. Soc. 165 (1972), 207-226. (1972) Zbl0236.26016MR0293384
  18. Sbordone C., Quasiminima of degenerate functionals with non polynomial growth, Rend. Sen. Mat. Fis. Milano LIX (1989), 173-184. (1989) Zbl0760.49023MR1159695
  19. Torchinsky A., Real variable methods in harmonic analysis, Pure Appl. Math. 123, Academic Press, 1986. Zbl1097.42002MR0869816

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