WP3: Degenerate, nonlinear and nonlocal PDE

The instruments of singular integrals and Fourier analysis developed in WP2 will be used to develop PDE theory. We will start studying totally degenerate equations in Lie groups. For this type of equations, even the Schauder estimate at the boundary is not know, and classical approach can not be applied. Schauder estimates at the boundary will allow to study existence of solutions to nonlinear equations. We will focus on existence and regularity for p-Laplacian, infinity Laplacian or curvature flow, both in elliptic and subelliptic setting. Another emerging direction of research, which requires to develop new ad hoc instruments is the theory nonlocal differential equations, related to fractional Laplacians. These operations can account long-range interactions, but problems here are open even in the Riemannian case.