Bounds for the singular set of solutions to non linear elliptic systems

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Abstract.

We give new estimates for the Hausdorff dimension of the singular set of solutions to elliptic systems \( {\mathrm div} a(x,u,Du) = b(x,u,Du)\;.\) If the vector fields a and b are Hölder continuous with respect to the variables (x,u) with exponent \(\alpha\), then, under suitable assumptions, the Hausdorff dimension of the singular set of any weak solution is at most \(n-2\alpha\). We consider natural growth assumptions on a(x,u,Du) with respect to u and critical ones on the right hand side b(x,u,Du), with respect to Du.