Higher gradient integrability for $\sigma$-harmonic maps in dimension two

I will present some recent results concerning the higher gradient integrability of $\sigma$-harmonic functions $u$ with discontinuous coefficients $\sigma$ i.e., weak solutions of $\nabla\cdot(\sigma\nabla u) = 0$. When $\sigma$ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. I will discuss the case when only the ellipticity is fixed and $\sigma$ is otherwise unconstrained and show that the optimal exponent is attained on the class of two-phase conductivities $\sigma : \Omega\subset\mathbb{R}2 \mapsto \{\sigma_1,\sigma_2\}\subset\mathbb{M}{2\times2}$. The optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries.

(Joint work with V. Nesi and M. Ponsiglione.)

This talk is part of the Geometric Analysis and Partial Differential Equations seminar series.