Existence and \(W^{1,p}\) estimates of certain Maxwell type equations in Reifenberg domains (Q2061720)

The authors study existence and regularity questions regarding the weak solutions of the Dirichlet problem for a system related to Maxwell's equations \[ \begin{cases} \nabla \times \big(A(x)\nabla \times u)=\nabla\times f,\quad \nabla \cdot u=0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial\Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^3\) is a bounded domain with Reifenberg-flat boundary and the coeficient matrix is uniformly elliptic with small-BMO entries. Existence of weak solution and its \(W^{1,p}\)-regularity are proved by means of modified Vitali covering lemma, maximal function technique and the compactness method.