Higher gradient integrability for σ-harmonic maps in dimension two

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• Mariapia Palombaro (University of Sussex)
• Monday 06 October 2014, 15:00-16:00
• CMS, MR13.

If you have a question about this talk, please contact Amit Einav.

I will present some recent results concerning the higher gradient inte- grability of σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti & Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise unconstrained and show that the optimal exponent is attained on the class of two-phase conductivities σ:Ω⊂R2 →(σ1,σ2)⊂M2×2. 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.)

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