Optimal regularity for quasilinear equations in stratified nilpotent Lie groups of step two

Luca Capogna, Purdue University

Abstract

We consider quasilinear equations$$\sum\limits\sbsp{i=1}{m}X\sb{i}A\sb{i}(p, Xu)=f(p)$$with quadratic growth in the gradient in a stratified nilpotent Lie group G. We prove that the gradient of the weak solutions is differentiable in the horizontal directions $X\sb{i}$ in the $L\sp2$ sense, and is Holder continuous with respect to the Carnot-Caratheodory metric. As an application, we prove a Liouville type theorem for the 1-quasiconformal mappings of the Heisenberg group, extending a famous result of Gehring.

Degree

Ph.D.

Advisors

Garofalo, Purdue University.

Subject Area

Mathematics

COinS