# A non-local variational problem arising from studies of nonlinear charge screening in graphene monolayers

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Cyrill Muratov

created by novaga on 24 Jun 2015

15 jul 2015
-- 16:00
[open in google calendar]

Dipartimento di Matematica, Sala Seminari

**Abstract.**

This talk is concerned with energy minimizers in an orbital-free density
functional theory that models the response of massless fermions in a
graphene monolayer to an out-of-plane external charge. The considered energy
functional generalizes the Thomas-Fermi energy for the charge carriers in
graphene layers by incorporating a von-Weizsaecker-like term that penalizes
gradients of the charge density. Contrary to the conventional theory,
however, the presence of the Dirac cone in the energy spectrum implies that
this term should involve a fractional Sobolev norm of the square root of the
charge density. We formulate a variational setting in which the proposed
energy functional admits minimizers in the presence of an out-of-plane point
charge. The associated Euler- Lagrange equation for the charge density is
also obtained, and uniqueness, regularity and decay of the minimizers are
proved under general conditions. In addition, a bifurcation from zero to
non-zero response at a finite threshold value of the external charge is
proved. This is joint work with J. Lu and V. Moroz.