Potential theory and doubly nonlinear PDEs: estimates, existence, and removable sets, part 1

Abstract: Recent advances in pointwise potential bounds and integral weighted estimates are discussed for a class of quasilinear elliptic equations with measure or distributional data. The connection of those estimates to Sobolev capacities and trace inequalities is presented. Applications include sharp existence criteria and characterizations of removable singular sets for doubly nonlinear equations of the form $-\Delta_p u= u^q +\sigma$, or $-\Delta_p u= |\nabla u|^q +\sigma$. Here $q>0$ could be arbitrarily large, $\Delta_p$ is the $p$-Laplacian ($p>1$), and $\sigma$ is a measure or sometimes a general signed distribution.

analysis of PDEs

Audience: researchers in the topic

IMS lecture series on regularity theory for quasilinear equations