191 research outputs found
Homogenization of variational problems in manifold valued BV-spaces
This paper extends the result of \cite{BM} on the homogenization of integral
functionals with linear growth defined for Sobolev maps taking values in a
given manifold. Through a -convergence analysis, we identify the
homogenized energy in the space of functions of bounded variation. It turns out
to be finite for -maps with values in the manifold. The bulk and Cantor
parts of the energy involve the tangential homogenized density introduced in
\cite{BM}, while the jump part involves an homogenized surface density given by
a geodesic type problem on the manifold.Comment: 32 page
Homogenization of variational problems in manifold valued Sobolev spaces
Homogenization of integral functionals is studied under the constraint that
admissible maps have to take their values into a given smooth manifold. The
notion of tangential homogenization is defined by analogy with the tangential
quasiconvexity introduced by Dacorogna, Fonseca, Maly and Trivisa \cite{DFMT}.
For energies with superlinear or linear growth, a -convergence result
is established in Sobolev spaces, the homogenization problem in the space of
functions of bounded variation being the object of \cite{BM}.Comment: 22 page
Gamma-convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves
We study the variational convergence of a family of two-dimensional
Ginzburg-Landau functionals arising in the study of superfluidity or thin-film
superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this
regime and for large enough applied rotations (for superfluids) or magnetic
fields (for superconductors), the minimizers acquire quantized point
singularities (vortices). We focus on situations in which an unbounded number
of vortices accumulate along a prescribed Jordan curve or a simple arc in the
domain. This is known to occur in a circular annulus under uniform rotation, or
in a simply connected domain with an appropriately chosen rotational vector
field. We prove that, suitably normalized, the energy functionals
Gamma-converge to a classical energy from potential theory. Applied to global
minimizers, our results describe the limiting distribution of vortices along
the curve in terms of Green equilibrium measures
A quantitative isoperimetric inequality for fractional perimeters
Recently Frank and Seiringer have shown an isoperimetric inequality for
nonlocal perimeter functionals arising from Sobolev seminorms of fractional
order. This isoperimetric inequality is improved here in a quantitative form
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