Europe/Lisbon
Room P1, Mathematics Building — Online

Lucio Boccardo
Lucio Boccardo, Università di Roma La Sapienza

Recent developments on Dirichlet problems with singular convection/drift terms

In the talk we discuss two Dirichlet problems ("formally" in duality)(CP)(DP)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 (CP);
  • a straight duality proof for (DP);
  • 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.

Projecto FCT UIDB/04459/2020.

Projecto FCT UIDB/04459/2020.