Elliptic equations with BMO nonlinearity in Reifenberg domains (Q952429)

The authors obtain the global \(L^p\) regularity of the gradient of the weak solutions of a second-order elliptic equation in divergence form with discontinuous nonlinearity in an irregular domain. The authors are interested in the well-posedness in \(W_0^{1,p}(\Omega)\) of the following nonlinear boundary-value problem: \[ \text{div}\,a(\nabla u,x)=\text{div}\,f\text{ in }\Omega\quad u=0\text{ on }\partial\Omega,\tag{1} \] where \(f\in L^p(\Omega; \mathbb{R}^n)\) is a given vector-valued function for some \(2\leq p<\infty\), \(a(x,\xi): \mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n\), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundarys and \(u:\Omega\to\mathbb{R}\).