motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

anisotropic curvature driven flow of convex sets abstract we study in this paper the mean curvature evolution and in particular the anisotropic mean curvature evolution of convex sets in r n without driving forces if the anisotropy is smooth we show that the evolution remains convex if the anisotropy is crystalline we build a convex evolution which satisfies an equation which is a weak form of the crystalline curvature motion equation

the anisotropic total variation flow we prove existence and uniqueness of solutions of the anisotropic total variation flow when the initial data is an l2 function and we give a characterization of such solutions that allows us to find explicit evolutions of sets in the presence of an anisotropy

existence and uniqueness for anisotropic and crystalline mean curvature flows an existence and uniqueness result up to fattening for crystalline mean curvature flows with forcing and arbitrary convex mobilities is proven this is achieved by introducing a new notion of solution to the corresponding level set formulation such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the approach accounts for the possible presence of a time dependent bounded forcing term with spatial lipschitz continuity as a by product of the analysis the problem of the convergence of the almgren taylor wang minimizing movements scheme to a unique up to fattening flat flow in the case of general possibly crystalline anisotropies is settled

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

existence and uniqueness for a crystalline mean curvature flow an existence and uniqueness result up to fattening for a class of crystalline mean curvature flows with natural mobility is proved the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the comparison principle is obtained by means of a suitable weak formulation of the flow while the existence of a global in time solution follows via a minimizing movement approach 2016 wiley periodicals inc

existence and regularity of mean curvature flow with transport term in higher dimensions given an initial c 1 hypersurface and a time dependent vector field in a sobolev space we prove a time global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field we show that the hypersurfaces are c 1 for a short time and even after some singularities occur almost everywhere c 1 away from the higher multiplicity region

crystalline mean curvature flow of convex sets we prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in open image in new window this theorem can handle the facet breaking bending phenomena and can be generalized to any anisotropic mean curvature flow the method provides also a generalized geometric evolution starting from any compact convex set existing up to the extinction time satisfying a comparison principle and defining a continuous semigroup in time we prove that when the initial set is convex our evolution coincides with the flat curvature flow in the sense of almgren taylor wang as a by product it turns out that the flat curvature flow starting from a compact convex set is unique

generalized motion by nonlocal curvature in the plane this paper considers the level set equation for a general planar anisotropic curvature flow equation when the interfacial energy is very singular so that the anisotropic curvature effect is nonlocal a new notion of solutions is introduced to establish an analytic foundation of the level set method including a comparison principle and stability results the main idea behind the proofs is to convert the level sets of solutions into graph like functions this new procedure is called slicing and it is not limited to nonlocal curvature flow equations our theory is useful for establishing the convergence of a crystalline algorithm as well as for justifying the crystalline flow as a limit of anisotropic curvature flow with smooth interfacial energy

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the perturbed test function method for viscosity solutions of nonlinear pde the method of viscosity solutions for nonlinear partial differential equations pdes justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions we describe in this paper a perturbed test function device which entails various modifications of the test functions by lower order correctors applications include homogenisation for quasilinear elliptic pdes and approximation of quasilinear parabolic pdes by systems of hamilton jacobi equations

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

polynomial and poisson dependence in free poisson algebras and free poisson fields abstract we prove that any two poisson dependent elements in a free poisson algebra and a free poisson field of characteristic zero are algebraically dependent thus answering positively a question from makar limanov and umirbaev 2007 8 we apply this result to give a new proof of the tameness of automorphisms for free poisson algebras of rank two see makar limanov and umirbaev 2011 9 makar limanov et al 2009 10

on a perimeter preserving crystalline flow abstract the behavior of admissible polygons moving under a crystalline curvature flow is discussed it is proved that the perimeter of the evolving polygon is preserved its area is increasing its crystalline curvature is uniformly bounded under this flow and the final shape of the solution polygon is the boundary of a wulff shape

on boundary detachment phenomena for the total variation flow with dynamic boundary conditions abstract we combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions we analyze the behavior of facets at the parts of the boundary where these conditions are imposed we devote particular attention to the radially symmetric data we observe that the boundary layer detachment actually can happen at concave parts of the boundary

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

evolving graphs by singular weighted curvature a new notion of solutions is introduced to study degenerate nonlinear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of the unknowns that the equation is no longer a partial differential equation by extending the theory of viscosity solutions a comparison principle is established for periodic continuous initial data a unique global continuous solution periodic in space is constructed the theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interfacial energy with corners when the curves are the graphs of functions even if the driving force term homogeneous in space exists the initial value problem is solvable for general nonadmissible continuous periodic initial data

on a crystalline variational problem part ii bv regularity and structure of minimizers on facets for a nonsmooth positively one homogeneous convex function n 0 it is possible to introduce the class n of smooth boundaries with respect to to define their mean curvature and to prove that for e n l e 9 based on these results we continue the analysis on the structure of e and on the regularity properties of we prove that a facet f of e is lipschitz up to negligible sets and that has bounded variation on f further properties of the jump set of are inspected in particular in three space dimensions we relate the sublevel sets of on f to the geometry of the wulff shape 1

on a crystalline variational problem part i first variation and global l regularity let n 0 be a given positively one homogeneous convex function and let 1 pursuing our interest in motion by crystalline mean curvature in three dimensions we introduce and study the class n of smooth boundaries in the relative geometry induced by the ambient banach space n it can be seen that even when is a polytope n cannot be reduced to the class of polyhedral boundaries locally resembling curved portions must be necessarily included and this fact as well as the nonsmoothness of is the source of several technical difficulties related to the geometry of lipschitz manifolds given a boundary e in the class n we rigorously compute the first variation of the corresponding anisotropic perimeter which leads to a variational problem on vector fields defined on e it turns out that the minimizers have a uniquely determined intrinsic tangential divergence on e we define such a divergence to be the mean curvature of e the function is expected to be the initial velocity of e whenever e is considered as the initial datum for the corresponding anisotropic mean curvature flow we prove that is bounded on e and that its sublevel sets are characterized through a variational inequality