442 research outputs found
Singularities of Nonlinear Elliptic Systems
Through Morrey's spaces (plus Zorko's spaces) and their potentials/capacities
as well as Hausdorff contents/dimensions, this paper estimates the singular
sets of nonlinear elliptic systems of the even-ordered Meyers-Elcrat type and a
class of quadratic functionals inducing harmonic maps.Comment: 18 pages Communications in Partial Differential Equation
Gauge theory of Faddeev-Skyrme functionals
We study geometric variational problems for a class of nonlinear sigma-models
in quantum field theory. Mathematically, one needs to minimize an energy
functional on homotopy classes of maps from closed 3-manifolds into compact
homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit
localized knot-like structure. Our main results include proving existence of
Hopfions as finite energy Sobolev maps in each (generalized) homotopy class
when the target space is a symmetric space. For more general spaces we obtain a
weaker result on existence of minimizers in each 2-homotopy class.
Our approach is based on representing maps into G/H by equivalence classes of
flat connections. The equivalence is given by gauge symmetry on pullbacks of
G-->G/H bundles. We work out a gauge calculus for connections under this
symmetry, and use it to eliminate non-compactness from the minimization problem
by fixing the gauge.Comment: 34 pages, no figure
Functional Liftings of Vectorial Variational Problems with Laplacian Regularization
We propose a functional lifting-based convex relaxation of variational
problems with Laplacian-based second-order regularization. The approach rests
on ideas from the calibration method as well as from sublabel-accurate
continuous multilabeling approaches, and makes these approaches amenable for
variational problems with vectorial data and higher-order regularization, as is
common in image processing applications. We motivate the approach in the
function space setting and prove that, in the special case of absolute
Laplacian regularization, it encompasses the discretization-first
sublabel-accurate continuous multilabeling approach as a special case. We
present a mathematical connection between the lifted and original functional
and discuss possible interpretations of minimizers in the lifted function
space. Finally, we exemplarily apply the proposed approach to 2D image
registration problems.Comment: 12 pages, 3 figures; accepted at the conference "Scale Space and
Variational Methods" in Hofgeismar, Germany 201
Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth
We establish global regularity for weak solutions to quasilinear divergence
form elliptic and parabolic equations over Lipschitz domains with controlled
growth conditions on low order terms. The leading coefficients belong to the
class of BMO functions with small mean oscillations with respect to .Comment: 24 pages, to be submitte
The mixed problem for the Laplacian in Lipschitz domains
We consider the mixed boundary value problem or Zaremba's problem for the
Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on
part of the boundary and Neumann data on the remainder of the boundary. We
assume that the boundary between the sets where we specify Dirichlet and
Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p
and the Dirichlet data is in the Sobolev space of functions having one
derivative in L^p for some p near 1. Under these conditions, there is a unique
solution to the mixed problem with the non-tangential maximal function of the
gradient of the solution in L^p of the boundary. We also obtain results with
data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since
the paper appeared long ago, this submission includes the complete paper,
followed by a short section that gives the correction to one step in the
proo
A note on boundedness of operators in Grand Grand Morrey spaces
In this note we introduce grand grand Morrey spaces, in the spirit of the
grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is
applicable to a variety of operators to reduce their boundedness in grand grand
Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of
this application, we obtain the boundedness of the Hardy-Littlewood maximal
operator and Calder\'on-Zygmund operators in the framework of grand grand
Morrey spaces.Comment: 8 page
Josephson nanocircuit in the presence of linear quantum noise
We derive the effective hamiltonian for a charge-Josephson qubit in a circuit
with no use of phenomenological arguments, showing how energy renormalizations
induced by the environment appear with no need of phenomenological
counterterms. This analysis may be important for multiqubit systems and
geometric quantum computation.Comment: Proceedings of LT23, to appear in Physica
On the Geometry and Entropy of Non-Hamiltonian Phase Space
We analyze the equilibrium statistical mechanics of canonical, non-canonical
and non-Hamiltonian equations of motion by throwing light into the peculiar
geometric structure of phase space. Some fundamental issues regarding time
translation and phase space measure are clarified. In particular, we emphasize
that a phase space measure should be defined by means of the Jacobian of the
transformation between different types of coordinates since such a determinant
is different from zero in the non-canonical case even if the phase space
compressibility is null. Instead, the Jacobian determinant associated with
phase space flows is unity whenever non-canonical coordinates lead to a
vanishing compressibility, so that its use in order to define a measure may not
be always correct. To better illustrate this point, we derive a mathematical
condition for defining non-Hamiltonian phase space flows with zero
compressibility. The Jacobian determinant associated with time evolution in
phase space is altogether useful for analyzing time translation invariance. The
proper definition of a phase space measure is particularly important when
defining the entropy functional in the canonical, non-canonical, and
non-Hamiltonian cases. We show how the use of relative entropies can circumvent
some subtle problems that are encountered when dealing with continuous
probability distributions and phase space measures. Finally, a maximum
(relative) entropy principle is formulated for non-canonical and
non-Hamiltonian phase space flows.Comment: revised introductio
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