Boundary regularity for elliptic systems under a natural growth condition
This paper is concerned with the existence of regular boundary points for the gradient of bounded vector-valued weak solutions of the Dirichlet problem to the second order elliptic system, \(-\)div \( a(\cdot, u, D u)=b(\cdot,u,D u)\), in a bounded domain \(\Omega\subset\mathbb R^n\), \((n=2,3,4)\), of class \(C^{1,\alpha}\), where \(1/2\leq \alpha<1\), with boundary value \(u_0\in C^{1,\alpha}(\bar\Omega;\mathbb{R}^N)\). The coefficients \(a\) are assumed to be Hölder continuous with exponent \(\alpha\) with respect to the first two variables, and of class \(C^1\) in the last variable, and to satisfy quadratic growth and ellipticity conditions. The existence of regular boundary points for \(Du\) is proved if \(b\) is a Carathéodory map that satisfies \(|b(x,v,z)|\leq L+L_2|z|^2\), for all \((x,v,z)\in \Omega\times \mathbb{R}^N\times\mathbb{R}^{nN}\) such that \(|v|\leq M\), where \(L\) is the upper bound of \(D_za\), \(M\) is the \(L^\infty\)-bound of the weak solution \(u\), \(L_2\) is such that \(2L_2M<\nu\), with \(\nu\) being the ellipticity constant. The strategy of the proof relies on the accomplishment of fractional Sobolev estimates for tangential and full derivatives of \(u\), and an interpolation technique.
scientific article; zbMATH DE number 6031767
scientific article