Gradient estimates for a class of elliptic systems. (English) Zbl 0819.35019

The author proves partial regularity and everywhere regularity for solutions of a second order nonlinear elliptic system of the form \[ D_ \alpha (A_ i^ \alpha (x,u,Du))+ B_ i (x,u, Du)=0 \] in \(\Omega\) under additional structure conditions imposed on coefficients. Denote \((G^{\alpha \beta})\), \((g_{ij})\) positive matrices on \(\Omega\times \mathbb{R}^ N\) and set \(v= (G^{\alpha \beta} (x,z) g_{ij} (x,z) p_ \alpha^ i p^ \beta_ j )^{1/2}\). Then the main assumption is that there is a scalar function \(F\) on \(\Omega \times \mathbb{R}^ N\times \mathbb{R}^{Nn}\) satisfying \(A_ i^ \alpha (x,z,p)= \partial F(v)/ \partial p_ \alpha^ i\). The proof is based on a modification of Moser’s iteration scheme.
Reviewer: J.Stará (Praha)


35B65 Smoothness and regularity of solutions to PDEs
35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations
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