Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
nonlinear elliptic systems; maximal operator theory
nonlinear elliptic systems; maximal operator theory
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 $$ \cases -\operatorname{div} A(x,u,Du)\,=B(x, u,Du) \quad & \text{in } \Omega , \ u-u_o\in W^{1,r}(\Omega,\Bbb R^m). \endcases $$
References:
[AF] Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125-145. MR 0751305 | Zbl 0565.49010
[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. MR 1386247 | Zbl 0854.35047
[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
[G] Giaquinta M.: Multiple integrals in the Calculus of variations and nonlinear elliptic systems. Ann. of Math. Stud. 105, Princeton University Press, 1983. MR 0717034 | Zbl 0516.49003
[GG] Giaquinta M., Giusti E.: On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31-46. MR 0666107 | Zbl 0494.49031
[Gi] Giusti E.: Metodi diretti nel Calcolo delle Variazioni. U.M.I., 1984. Zbl 0942.49002
[GIS] Greco L., Iwaniec T., Sbordone C.: Inverting the $p$-harmonic operator. Manuscripta Math. 92 (1997), 249-258. MR 1428651 | Zbl 0869.35037
[GLS] Giachetti D., Leonetti F., Schianchi R.: On the regularity of very weak minima. Proc. Royal Soc. Edinburgh 126A (1996), 287-296. MR 1386864 | Zbl 0851.49026
[GS] Giachetti D., Schianchi R.: Boundary higher integrability for the gradient of distributional solutions of nonlinear systems. preprint. MR 1439029 | Zbl 0869.49020
[GT] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer Verlag, 1982. Zbl 1042.35002
[H] Hedberg L.I.: On certain convolution inequalities. Proc. Amer. Math. Soc. 36 (1972), 505-510. MR 0312232
[I] Iwaniec T.: $p$-harmonic tensors and quasiregular mappings. Ann. of Math. 136 (1992), 589-624. MR 1189867 | Zbl 0785.30009
[IS] Iwaniec T., Sbordone C.: Weak minima of variational integrals. J. Reine Angew. Math. 454 (1994), 143-161. MR 1288682 | Zbl 0802.35016
[Le] Lewis J.: On very weak solutions of certain elliptic systems. Comm. Partial Differential Equations 18 (1993), 1515-1537. MR 1239922 | Zbl 0796.35061
[M1] Moscariello G.: Weak minima and quasiminima of variational integrals. B.U.M.I. 7-11B (1997), 355-364. MR 1459284 | Zbl 0890.49003
[M2] Moscariello G.: On weak minima of certain integral functionals. Ann. Polon. Math. LXIX.1 (1998), 37-48. MR 1630200 | Zbl 0920.49021
[Mu] Muckenhoupt B.: Weighted norm inequalities for the Hardy Maximal Function. Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 | Zbl 0236.26016
[S] Sbordone C.: Quasiminima of degenerate functionals with non polynomial growth. Rend. Sen. Mat. Fis. Milano LIX (1989), 173-184. MR 1159695 | Zbl 0760.49023
[T] Torchinsky A.: Real variable methods in harmonic analysis. Pure Appl. Math. 123, Academic Press, 1986. MR 0869816 | Zbl 1097.42002
Search
Partner of
