Electron. J. Diff. Eqns., Vol. 2001(2001), No. 02, pp. 1-7.

A stability result for p-harmonic systems with discontinuous coefficients

Bianca Stroffolini

Abstract:
The present paper is concerned with p-harmonic systems
$$ \mathop{\rm div} (\langle A(x) Du(x), Du(x) \rangle ^{{p-2}\over 2} A(x) Du(x))=\mathop{\rm div} ( \sqrt{A(x)} F(x)),$$
where $A(x)$ is a positive definite matrix whose entries have bounded mean oscillation (BMO), $p$ is a real number greater than 1 and $F\in L^{r\over {p-1}}$ is a given matrix field. We find a-priori estimates for a very weak solution of class $W^{1,r}$, provided $r$ is close to $2$, depending on the BMO norm of $\sqrt{A}$, and $p$ close to $r$. This result is achieved using the corresponding existence and uniqueness result for linear systems with BMO coefficients [St], combined with nonlinear commutators.

Submitted November 27, 2000. Published January 2, 2001.
Math Subject Classifications: 35J60, 47B47.
Key Words: Bounded mean oscillation, Linear and Nonlinear Commutators, Hodge Decomposition.

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Bianca Stroffolini
Dipartimento di Matematica e Applicazioni "R. Caccioppoli"
via Cintia, 80126 Napoli, Italy
e-mail: stroffol@matna2.dma.unina.it

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