Interior partial regularity for nonlinear elliptic systems with Dini continuous coefficients for the case: \(1 < m < 2\)
The paper concerns the regularity of the weak solutions to a nonlinear elliptic system of the form \[ -\mathrm{div}A(x,u,Du)=B(x,u,Du) \] considered on a bounded domain \(\Omega\) in \(\mathbb{R}^n\), \(n\geq2\). Here \(A(x,\xi,p)\) and \(B(x,\xi,p)\) are measurable functions defined on \(\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{nN}\), \(N>1\), \(u\) and \(B\) take values in \(\mathbb{R}^N\). \(A(x,\xi,p)\) is assumed to be uniformly strongly elliptic, to have continuous derivative in \(p\) with \(|\frac{\partial A}{\partial p}(x,\xi,p)|\leq L(1+|p|^2)^{(m-2)/2}\) for all \((x,\xi,p)\in\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{nN}\) and such that \(|A(x,\xi,p)-A(\tilde{x},\tilde{\xi},p)|\leq K(|\xi|)\mu(|x-\tilde{x}|^m+|\xi-\tilde{\xi}|^m)\lambda (1+|p|)^{m/2}\) where \(1<m<2\) and \(K\) is a nondecreasing function. Next, \(B\) is assumed to satisfy either natural growth condition: \(|B(x,\xi,p)|\leq a|p|^m+b\) or the controllable growth condition: \(|B(x,\xi,p)|\leq c(1+|u|^{r-1}+|p|^{m(1-1/r)})\), where \(r=nm/(n-m)\). Now, the main results state that any weak solution belongs to \(C^1(\Omega\setminus\Sigma,\mathbb{R}^N)\) where the set \(\Sigma\) is of zero Lebesgue measure.
scientific article; zbMATH DE number 5994350
scientific article