# Erik Duse: Hodge–Dirac operators and the generalised Beltrami equation in higher dimension

**Time: **
Wed 2023-04-19 11.00 - 12.00

**Location: **
Albano, Cramérrummet

**Participating: **
Erik Duse (KTH)

**Abstract.**

In this talk I will show how one can rewrite a second order scalar linear or nonlinear elliptic partial differential equation with measurable coefficients to a first order system involving Hodge–Dirac operators, which can be seen as higher dimensional analogue of the Beltrami equation in the plane. This involves relative de Rham cohomology and Hodge theory. Using the first order system I will show how one can solve the Dirichlet problem in a more or less explicit way and how the higher integrability of solutions of uniformly elliptic equations are directly linked to the \(L^p\)-norms of the higher dimensional Beurling–Ahlfors transform.