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SUMMARY:Higher gradient integrability for $\\sigma$-harmonic maps in dime
 nsion two - Dr. Mariapia Palombaro\, University of Sussex
DTSTART:20141006T140000Z
DTEND:20141006T150000Z
UID:TALK55007@talks.cam.ac.uk
CONTACT:Harsha Hutridurga
DESCRIPTION:I will present some recent results concerning the higher gradi
 ent integrability of $\\sigma$-harmonic functions $u$ with discontinuous 
 coefficients $\\sigma$ i.e.\, weak solutions of $\\nabla\\cdot(\\sigma\\n
 abla u) = 0$. When $\\sigma$ is assumed to be symmetric\, then the optima
 l integrability exponent of the gradient field is known\nthanks to the wor
 k of Astala and Leonetti and Nesi. I will discuss the case when only the e
 llipticity 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 solutio
 ns\, via the exhibition of optimal microgeometries.\n\n(Joint work with V.
  Nesi and M. Ponsiglione.)
LOCATION:CMS\, MR13
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