In the talk we discuss two Dirichlet problems ("formally" in duality)where is a bounded open set in , ia bounded elliptic matrix, , are functions belonging to , , , .

In the first part we briefly recall some recent results:

• existence and summability properties of weak solutions ( ), if ;
• Calderon-Zygmund theory ( , infinite energy solutions), if ;
• uniqueness;
• the case , where ;
• the case .

Then we show:

• a new (simpler) existence proof, thanks to the presence of the zero order term, for ;
• a straight duality proof for ;
• continuous dependence of the solutions with respect to the weak convergence of the coefficients;
• regularizing effect of dominated coefficients ( or , );
• "weak" maximum principle: if [ ] and, of course, not =0 a.e., then [ ] and the set where [] is zero has zero measure (at most).

Work in progress: obstacle problem; nonlinear principal part.
Open problem: "strong" maximum principle.