383 research outputs found
The distance function from the boundary in a Minkowski space
Let the space be endowed with a Minkowski structure (that
is is the gauge function of a compact
convex set having the origin as an interior point, and with boundary of class
), and let be the (asymmetric) distance associated to .
Given an open domain of class , let
be the Minkowski
distance of a point from the boundary of . We prove that a
suitable extension of to (which plays the r\"ole of
a signed Minkowski distance to ) is of class in a
tubular neighborhood of , and that is of class
outside the cut locus of (that is the closure of the set
of points of non--differentiability of in ). In addition,
we prove that the cut locus of has Lebesgue measure zero, and
that can be decomposed, up to this set of vanishing measure, into
geodesics starting from and going into along the
normal direction (with respect to the Minkowski distance). We compute
explicitly the Jacobian determinant of the change of variables that associates
to every point outside the cut locus the pair , where denotes the (unique) projection of on
, and we apply these techniques to the analysis of PDEs of
Monge-Kantorovich type arising from problems in optimal transportation theory
and shape optimization.Comment: 34 page
Crystalline Evolutions in Chessboard-like Microstructures
We describe the macroscopic behavior of evolutions by crystalline curvature
of planar sets in a chessboard--like medium, modeled by a periodic forcing
term. We show that the underlying microstructure may produce both pinning and
confinement effects on the geometric motion.Comment: 17 pages, 10 figures. arXiv admin note: text overlap with
arXiv:1707.0334
Crystalline evolutions with rapidly oscillating forcing terms
We consider the evolution by crystalline curvature of a planar set in a
stratified medium, modeled by a periodic forcing term. We characterize the
limit evolution law as the period of the oscillations tends to zero. Even if
the model is very simple, the limit evolution problem is quite rich, and we
discuss some properties such as uniqueness, comparison principle and
pinning/depinning phenomena.Comment: 28 pages, 17 figure
Duality arguments for linear elasticity problems with incompatible deformation fields
We prove existence and uniqueness for solutions to equilibrium problems for
free-standing, traction-free, non homogeneous crystals in the presence of
plastic slips. Moreover we prove that this class of problems is closed under
G-convergence of the operators. In particular the homogenization procedure,
valid for elliptic systems in linear elasticity, depicts the macroscopic
features of a composite material in the presence of plastic deformation
DOMINO PROJECT GUIDELINES FOR EXPERIMENTAL PRACTICE
The aim of this handbook of experimental guidelines is to level out analyses run during the "Domino project" on practices for sustainable management of organic apple orchard and vineyard in field condition
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