Old and new results in regularity theory for diagonal elliptic systems via blowup techniques
The authors consider the diagonal elliptic system \[ \begin{alignedat}{2} & -\sum_{i,j=1}^d D_i(a_{ij}(x) D_j u_{\nu}(x)) + \kappa_{\nu} u_{\nu}(x) = H_{\nu}(x,u(x), \nabla u(x)) \qquad & \text{in } & \Omega,\\ & u_{\nu}(x) = 0 \qquad & \text{on } & \Gamma_D,\\ & \sum_{i,j=1}^d a_{ij}(x) D_j u_{\nu}(x) n_i(x) \qquad & \text{on } & \Gamma_N\end{alignedat} \] for \(\nu = 1, \ldots, N\) subject to mixed boundary conditions. Here \(\Gamma_{D}\) denotes the Dirichlet and \(\Gamma_{N}\) the Neumann part of the boundary. Further, \(\Omega \subset \mathbb{R}^d\) for \(d \geq 2\) is assumed to be a bounded Lipschitz domain, \(\kappa \in \mathbb{R}^N\) is a fixed vector with positive components and \(H: \Omega \times \mathbb{R}^N \times \mathbb{R}^{d \times N} \to \mathbb{R}^N\) is a nonlinearity that couples the equations. The system is assumed to be diagonal, i.e.\ \(a: \Omega \to \mathbb{R}^{d \times d}\), with bounded and measurable components. The equation is studied for nonlinearities \(H\) with critical quadratic growth in the gradient variable for which the existence of weak solutions may fail. It is shown that if \(H\) additionally satisfies estimates of the form \[ H_{\nu}(x,u,z) \leq K^* + K |z_{\nu}| |z| \] \[ \sum_{\nu = 1}^N q_{\nu} H_{\nu}(x,u,z) \geq -K^* - K \bigg| \sum_{\nu = 1}^{N} q_{\nu} z_{\nu} \bigg|, \] then the equation has a unique weak solution (which is defined in a slightly more general sense than usual) that is Hölder continuous for some fixed constant on open subsets of \(\Omega\) with positive distance to the Dirichlet part of the boundary. Further, \(W^{2,q}(\Omega)\)-estimates are obtained for \(C^{1,1}\) domains and coefficients \(a_{ij} \in C^{0,1}(\overline{\Omega})\). As key steps a Liouville theorem for solutions and a compactness result for weak solutions are obtained via blow-up techniques.
scientific article; zbMATH DE number 6484247
scientific article