Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 02, pp. 1-7.

Title: A stability result for p-harmonic  systems with discontinuous coefficients

Author: Bianca Stroffolini (R. Caccioppoli, Napoli, Italy)
          
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.