49,977 research outputs found
Anisotropic total variation flow of non-divergence type on a higher dimensional torus
We extend the theory of viscosity solutions to a class of very singular
nonlinear parabolic problems of non-divergence form in a periodic domain of an
arbitrary dimension with diffusion given by an anisotropic total variation
energy. We give a proof of a comparison principle, an outline of a proof of the
stability under approximation by regularized parabolic problems, and an
existence theorem for general continuous initial data, which extend the results
recently obtained by the authors.Comment: 27 page
Periodic total variation flow of non-divergence type in Rn
We introduce a new notion of viscosity solutions for a class of very singular
nonlinear parabolic problems of non-divergence form in a periodic domain of
arbitrary dimension, whose diffusion on flat parts with zero slope is so strong
that it becomes a nonlocal quantity. The problems include the classical total
variation flow and a motion of a surface by a crystalline mean curvature. We
establish a comparison principle, the stability under approximation by
regularized parabolic problems, and an existence theorem for general continuous
initial data.Comment: 36 pages, 2 figure
A level set crystalline mean curvature flow of surfaces
We introduce a new notion of viscosity solutions for the level set
formulation of the motion by crystalline mean curvature in three dimensions.
The solutions satisfy the comparison principle, stability with respect to an
approximation by regularized problems, and we also show the uniqueness and
existence of a level set flow for bounded crystals.Comment: 55 pages, 4 figure
Almost classical solutions to the total variation flow
The paper examines one-dimensional total variation flow equation with
Dirichlet boundary conditions. Thanks to a new concept of "almost classical"
solutions we are able to determine evolution of facets -- flat regions of
solutions. A key element of our approach is the natural regularity determined
by nonlinear elliptic operator, for which is an irregular function. Such
a point of view allows us to construct solutions. We apply this idea to
implement our approach to numerical simulations for typical initial data. Due
to the nature of Dirichlet data any monotone function is an equilibrium. We
prove that each solution reaches such steady state in a finite time.Comment: 3 figure
A Mean-Field Theory for Coarsening Faceted Surfaces
A mean-field theory is developed for the scale-invariant length distributions
observed during the coarsening of one-dimensional faceted surfaces. This theory
closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in
two-phase systems [1-3], but the mechanism of coarsening in faceted surfaces
requires the addition of convolution terms recalling the work of Smoluchowski
[4] and Schumann [5] on coalescence. The model is solved by the exponential
distribution, but agreement with experiment is limited by the assumption that
neighboring facet lengths are uncorrelated. However, the method concisely
describes the essential processes operating in the scaling state, illuminates a
clear path for future refinement, and offers a framework for the investigation
of faceted surfaces evolving under arbitrary dynamics.
[1] I. Lifshitz, V. Slezov, Soviet Physics JETP 38 (1959) 331-339.
[2] I. Lifshitz, V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35-50.
[3] C. Wagner, Elektrochemie 65 (1961) 581-591.
[4] M. von Smoluchowski, Physikalische Zeitschrift 17 (1916) 557-571.
[5] T. Schumann, J. Roy. Met. Soc. 66 (1940) 195-207
Time-Constrained Temporal Logic Control of Multi-Affine Systems
In this paper, we consider the problem of controlling a dynamical system such
that its trajectories satisfy a temporal logic property in a given amount of
time. We focus on multi-affine systems and specifications given as
syntactically co-safe linear temporal logic formulas over rectangular regions
in the state space. The proposed algorithm is based on the estimation of time
bounds for facet reachability problems and solving a time optimal reachability
problem on the product between a weighted transition system and an automaton
that enforces the satisfaction of the specification. A random optimization
algorithm is used to iteratively improve the solution
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