1,066 research outputs found
Stability of the Steiner symmetrization of convex sets
The isoperimetric inequality for Steiner symmetrization of any
codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets
Minimality via second variation for a nonlocal isoperimetric problem
We discuss the local minimality of certain configurations for a nonlocal
isoperimetric problem used to model microphase separation in diblock copolymer
melts. We show that critical configurations with positive second variation are
local minimizers of the nonlocal area functional and, in fact, satisfy a
quantitative isoperimetric inequality with respect to sets that are
-close. The link with local minimizers for the diffuse-interface
Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative
estimate, we get new results concerning periodic local minimizers of the area
functional and a proof, via second variation, of the sharp quantitative
isoperimetric inequality in the standard Euclidean case. As a further
application, we address the global and local minimality of certain lamellar
configurations.Comment: 35 page
Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization
Short time existence for a surface diffusion evolution equation with
curvature regularization is proved in the context of epitaxially strained
three-dimensional films. This is achieved by implementing a minimizing movement
scheme, which is hinged on the -gradient flow structure underpinning
the evolution law. Long-time behavior and Liapunov stability in the case of
initial data close to a flat configuration are also addressed.Comment: 44 page
A strong form of the Quantitative Isoperimetric inequality
We give a refinement of the quantitative isoperimetric inequality. We prove
that the isoperimetric gap controls not only the Fraenkel asymmetry but also
the oscillation of the boundary
Non trivial behavior of the linear response function in phase ordering kinetics
Drawing from exact, approximate and numerical results an overview of the
properties of the out of equilibrium response function in phase ordering
kinetics is presented. Focusing on the zero field cooled magnetization,
emphasis is on those features of this quantity which display non trivial
behavior when relaxation proceeds by coarsening. Prominent among these is the
dimensionality dependence of the scaling exponent which leads to
failure of the connection between static and dynamic properties at the lower
dimensionality , where . We also analyse the mean spherical
model as an explicit example of a stochastic unstable system, for which the
connection between statics and dynamics fails at all dimensionalities.Comment: 10 pages, 2 figures. Contribution to the International Conference
"Perspectives on Quantum Field Theory, Statistical Mechanics and Stochastics"
in honour of the 60th birthday of Francesco Guerr
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