373 research outputs found
Sobolev inequalities and regularity of the linearized complex Monge-Ampere and Hessian equations
Let be a smooth, strictly -plurisubharmonic function on a bounded
domain with . The purpose of this paper
is to study the regularity of solution to the linearized complex Monge-Amp\`ere
and Hessian equations when the complex -Hessian of is bounded
from above and below. We first establish some estimates of Green's functions
associated to the linearized equations. Then we prove a class of new Sobolev
inequalities. With these inequalities, we use Moser's iteration to investigate
the a priori estimates of Hessian equations and their linearized equations, as
well as the K\"ahler scalar curvature equation. In particular, we obtain the
Harnack inequality for the linearized complex Monge-Amp\`ere and Hessian
equations under an extra integrability condition on the coefficients. The
approach works in both real and complex case.Comment: 34 pages. Comments welcome
Scaling of entanglement at quantum phase transition for two-dimensional array of quantum dots
With Hubbard model, the entanglement scaling behavior in a two-dimensional
itinerant system is investigated. It has been found that, on the two sides of
the critical point denoting an inherent quantum phase transition (QPT), the
entanglement follows different scalings with the size just as an order
parameter does. This fact reveals the subtle role played by the entanglement in
QPT as a fungible physical resource
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