392 research outputs found
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
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
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
Lorentz space estimates for vector fields with divergence and curl in Hardy spaces
In this note, we establish the estimate on the Lorentz space for
vector fields in bounded domains under the assumption that the normal or the
tangential component of the vector fields on the boundary vanishing. We prove
that the norm of the vector field can be controlled by the norms of
its divergence and curl in the atomic Hardy spaces and the norm of the
vector field itself.Comment: 11page
Singular diffusivity facets, shocks and more
There is a class of nonlinear evolution equations with singular diffusivity, so that diffusion effect is nonlocal. A simplest one-dimensional example is a diffusion equation of the form u_t = \delta(u_x)u_{xx} for u = u(x; t), where \delta denotes Dirac s delta function. This lecture is intended to provide an overview of analytic aspects of such equations, as well as various applications. Equations with singular diffusivity are applied to describe several phenomena in the applied sciences, and to provide several devices in technology, especially image processing. A typical example is a gradient flow of the total variation of a function, which arises in image processing, as well as in material science to describe the motion of grain boundaries. In the theory of crystal growth the motion of a crystal surface is often described by an anisotropic curvature flow equation with a driving force term. At low temperature the equation includes a singular diffusivity, since the interfacial energy is not smooth. Another example is a crystalline algorithm to calculate curvature flow equations in the plane numerically, which is formally written as an equation with singular diffusivity. Because of singular diffusivity, the notion of solution is not a priori clear, even for the above one-dimensional example. It turns out that there are two systematic approaches. One is variational, and applies to divergence type equations. However, there are many equations like curvature flow equations which are not exactly of divergence type. Fortu-nately, our approach based on comparision principles turns out to be succesful in several interesting problems. It also asserts that a solution can be considered as a limit of solution of an approximate equation. Since the equation has a strong diffusivity at a particular slope of a solution, a flat portion with this slope is formed. In crystal growth ploblems this flat portion is called a facet. The discontinuity of a solution (called a shock) for a scalar conservation law is also considered as a result of singular diffusivity in the vertical direction
On the role of kinetic and interfacial anisotropy in the crystal growth theory
A planar anisotropic curvature flow equation with constant driving force term is considered when the interfacial energy is crystalline. The driving force term is given so that a closed convex set grows if it is sufficiently large. If initial shape is convex, it is shown that a flat part called a facet (with admissible orientation) is instantaneously formed. Moreover, if the initial shape is convex and slightly bigger than the critical size, the shape becomes fully faceted in a finite time provided that the Frank diagram of interfacial energy density is a regular polygon centered at the origin. The proofs of these statements are based on approximation by crystalline algorithm whose foundation was established a decade ago. Our results indicate that the anisotropy of intefacial energy plays a key role when crystal is small in the theory of crystal growth. In particular, our theorems explain a reason why snow crystal forms a hexagonal prism when it is very small
Stokes Resolvent Estimates in Spaces of Bounded Functions
The Stokes equation on a domain is well understood in
the -setting for a large class of domains including bounded and exterior
domains with smooth boundaries provided . The situation is very
different for the case since in this case the Helmholtz projection
does not act as a bounded operator anymore. Nevertheless it was recently proved
by the first and the second author of this paper by a contradiction argument
that the Stokes operator generates an analytic semigroup on spaces of bounded
functions for a large class of domains. This paper presents a new approach as
well as new a priori -type estimates to the Stokes equation. They
imply in particular that the Stokes operator generates a -analytic
semigroup of angle on , or a non--analytic
semigroup on for a large class of domains. The
approach presented is inspired by the so called Masuda-Stewart technique for
elliptic operators. It is shown furthermore that the method presented applies
also to different type of boundary conditions as, e.g., Robin boundary
conditions.Comment: 22 pages, to appear in Ann. Sci. \'Ec. Norm. Sup\'er. (4
A duality based approach to the minimizing total variation flow in the space
We consider a gradient flow of the total variation in a negative Sobolev
space under the periodic boundary condition. If
, the flow is nothing but the classical total variation flow. If ,
this is the fourth order total variation flow. We consider a convex variational
problem which gives an implicit-time discrete scheme for the flow. By a duality
based method, we give a simple numerical scheme to calculate this minimizing
problem numerically and discuss convergence of a forward-backward splitting
scheme. Several numerical experiments are given
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