Seminar series
Date
Thu, 04 Jun 2015
Time
12:00 -
13:00
Location
L6
Speaker
Mariapia Palombaro
Organisation
University of Sussex
I will present some recent results concerning the higher gradient integrability 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⊂R27→ {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.)