Dini continuity of the first-order derivatives of solutions to the Dirichlet problem for linear second-order elliptic equations in a nonsmooth domain (Q1271992)

The author considers the Dirichlet problem for the equation \[ \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i}\Bigl(a_{ij}u_{x_j}+ a_iu\Bigr) +\sum_{i=1}^{n}b_i u_{x_i}+ cu= g+ \sum_{j=1}^{n} \frac{\partial f_j}{\partial x_j} ,\quad x \in G\subset \mathbb{R}^n. \] The boundary of \(G\) is assumed to be a Dini--Lyapunov surface containing a conic point at the origin. Under minimal smoothness conditions for the coefficients \(a_{ij}\) and \(a_i\) (Dini continuity), Dini estimates for the first-order derivatives of generalized solutions are obtained. Existence of generalized solutions is also proved.