## A stability result for $$p$$-harmonic systems with discontinuous coefficients.(English)Zbl 0965.35044

The author considers the $$p$$-harmonic system $\text{div} \left( \langle A(x)Du(x),Du(x)\rangle^{(p-2)/2} A(x)Du(x) \right) = \text{div}\left( \sqrt{A(x)}F(x) \right) \tag $$*$$$ where $$A(x)$$ is a positive definite matrix whose entries have bounded mean oscillation (BMO) and $$F$$ is a given matrix field. The main result is the following a priori estimate. If $$|r-2|<\epsilon$$ for some $$\epsilon>0$$ depending on the BMO norm of $$\sqrt A$$, then there is a $$\delta>0$$ such that $\|\sqrt{A} Du \|_r^r \leq C \|F \|_{r/(p-1)}^{r/(p-1)}$ whenever $$|p-r|<\delta$$ and $$u$$ is a very weak solution of $$(*)$$. The proof uses the existence and uniqueness results of the author [Potential Analysis 15, No. 3, 285-299 (2001)] for linear systems with BMO coefficients combined with nonlinear commutator relations.

### MSC:

 35J60 Nonlinear elliptic equations 47B47 Commutators, derivations, elementary operators, etc. 35B45 A priori estimates in context of PDEs
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