An extension of a theorem by C. Miranda in weighted spaces (Q2770414)

The authors consider the Dirichlet problem NEWLINE\[NEWLINE\begin{cases} Lu=f,\;f\in W^{-1,2} (\Omega)\\ u\in\overset \circ W^{1,2} (\Omega),\end{cases} \tag{1}NEWLINE\]NEWLINE where \(L\) is the second-order linear differential operator in divergence form NEWLINE\[NEWLINELu= -\sum^n_{i,j=1} (a_{ij}u_{x_i})_{x_j} +\sum^n_{i=1} b_iu_{x_i}+ cu\tag{2}NEWLINE\]NEWLINE with real and measurable coefficients. The authors extend the well-known theorem of C. Miranda concerning a priori estimates of weak solutions of (1)--(2) with discontinuous coefficients in a bounded domain of \(\mathbb{R}^n\).