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

If you have a question about this talk, please contact Harsha Hutridurga.

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 Partial Differential Equations seminar series.

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