16/09/2021
18/09/2021
Raul Serapioni
Regular and irregular solutions of degenerate elliptic equations: a glance at vintage mathematics.
Seminario di analisi matematica
The problem of Holder regularity of a variational solutions u = u(x) of a degenerate uniformly elliptic second order equations as (1) \sum_{i=1}^n Di(w(x)Diu(x)) = 0; x\in\Omega\subset R^n has been addressed since the beginning of the seventies. Now it is well known that if w belongs to the Muckenhaupt class A_2 then variational solutions of (1) are Holder continuous. On the other side the necessity of the assumption w\in A_2, or of similar structural assumptions on the weights, is far from being well understood. The simpler question of the necessity/sufficiency of quantitative assumptions on w and 1/w, even if better understood, is not yet completely settled.