the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

removable singularities and non uniqueness of solutions of a singular diffusion equation we prove that the solution u of the equation ut log u u 0 in x0 0 t 2 has removable singularities at x0 0 t if and only if for any 0 0 such that c1 x x0 u x t c2 x x0 holds for all 0 x x0 0 and a t b as a consequence we obtain a sufficient condition for removable singularities at 0 t for solutions of the above equation in 2 0 t and we prove the existence of infinitely many finite mass solutions for the equation in 2 0 t when 0 u0 l1 2 is radially symmetric and u0 lloc1 2

invariant geometric evolutions of surfaces and volumetric smoothing the study of geometric flows for smoothing multiscale representation and analysis of two and three dimensional objects has received much attention in the past few years in this paper we first survey the geometric smoothing of curves and surfaces via geometric heat type flows which are invariant under the groups of euclidean and affine motions second using the general theory of differential invariants we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group as an application we present the simplest affine invariant flow for convex surfaces in three dimensional space which like the affine invariant curve shortening flow will be of fundamental importance in the processing of three dimensional images

the singular set of mean curvature flow with generic singularities a mean curvature flow starting from a closed embedded hypersurface in mathbf r n 1 must develop singularities we show that if the flow has only generic singularities then the space time singular set is contained in finitely many compact embedded n 1 dimensional lipschitz submanifolds plus a set of dimension at most n 2 if the initial hypersurface is mean convex then all singularities are generic and the results apply in mathbf r 3 and mathbf r 4 we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time slice for 2 or 3 convex hypersurfaces in all dimensions the same arguments lead to the same conclusion the flow is completely smooth at almost all times and connected components of the singular set are contained in time slices a key technical point is a strong parabolic reifenberg property that we show in all dimensions and for all flows with only generic singularities we also show that the entire flow clears out very rapidly after a generic singularity these results are essentially optimal

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

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

dynamic solid solid transitions with phase characterized by an order parameter abstract we develop a thermodynamically consistent continuum theory for the study of solid solid phase transitions where deformation dominates heat and mass transfer we use an order parameter to characterize the notion of phase and identify phase interfaces with thin transition zones within which the strain and order parameter exhibit large gradients to model the growth of one phase at the expense of another we introduce configurational forces subject to their own balance that work against changes in the order parameter by studying traveling waves and the asymptotics of a transition layer we establish connections between our theory and more standard approaches that identify phase interfaces with surfaces of strain discontinuity

on the evolution of vector distance functions of closed curves inspired by the work by gomes et al we describe and analyze a vector distance function approach for the implicit evolution of closed curves of codimension larger than one the approach is set up in complete generality and then applied to the evolution of dynamic geometric active contours in mathbb r 4 codimension three case in order to carry this out one needs an explicit expression for the zero level set for which we propose a discrete connectivity method this leads us to make connections with the new theory of cubical homology we provide some explicit simulation results in order to illustrate the methodology

partial regularity at the first singular time for hypersurfaces evolving by mean curvature in this paper we consider smooth properly immersed hypersurfaces evolving by mean curvature in some open subset of mathbb r n 1 on a time interval 0 t 0 we prove that p integrability with p ge 2 for the second fundamental form of these hypersurfaces in some space time region b r y times 0 t 0 implies that the mathcal h n 2 p measure of the first singular set vanishes inside b r y for p 2 and n 2 this was established by han and sun our result furthermore generalizes previous work of xu ye and zhao and of le and sesum for p ge n 2 in which case the singular set was shown to be empty by a theorem of ilmanen our integrability condition is satisfied for p 2 and n 2 if the initial surface has finite genus thus the first singular set has zero mathcal h 2 measure in this case this is the conclusion of brakke s main regularity theorem for the special case of surfaces but derived without having to impose the area continuity and unit density hypothesis it follows from recent work of head and of huisken and sinestrari that for the flow of closed k convex hypersurfaces that is hypersurfaces whose sum of the smallest k principal curvatures is positive our integrability criterion holds with exponent p n 3 k alpha for all small alpha 0 as long as 1 le k le n 1 therefore the first singular set of such solutions is at most k 1 dimensional which is an optimal estimate in view of some explicit examples

motion of sets by curvature and derivative of capacity potential abstract we study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential we establish local well posedness and propose two possible weak formulations that exist after singularities

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

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

stability of stationary interfaces with contact angle in a generalized mean curvature flow the dynamics of a moving hypersurface in a domain d n is studied it is assumed that the hypersurface moves depending on its curvature normal vector and position with the boundary that intersects d with a constant contact angle a stability criterion about a stationary hypersurface is established in the form of an eigenvalue problem which includes geometrical information of d and the stationary hypersurface 1 introduction in this paper we study the dynamics of an oriented hy persurface t i n n that evolves in time t 0 continuously here we call such an oriented hypersurface an interface we assume that the motion of t depends on its curvature normal vector and position x t then we are led to the equation v x f x x x x t t 0 1 1 where x a nd x are the mean curvature and the outward unit normal vector of t a t x t respectively and v x is the normal velocity in the out ward direction at x t we assume that the function f depends smoothly on x 1 n 1 n and satisfies

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

existence of solutions of the very fast diffusion equation in bounded and unbounded domain we prove the existence of a unique solution of the following neumann problem u t phi m u xx u 0 in a b 0 t u x 0 u0 x 0 in a b phi m u x a t g t and phi m u x b t f t forall 0 0 int a bu 0dx int 0 t f g dt m 0 0 le f g in l loc infty 0 infty 0 le u 0 in l infty a b and the case 1 1 m we also obtain a similar result in higher dimensions as a corollary we will give a new proof of a result of a rodriguez and j l vazquez on the existence of infinitely many finite mass solutions of the above equation in mathbb r times 0 t for any 1 m 0 we also obtain the exact decay rate of the solution at infinity

rate of convergence of the mean curvature flow we study the flow mt of a smooth strictly convex hypersurface by its mean curvature in n 1 the surface remains smooth and convex shrinking monotonically until it disappears at a critical time t and point x which is due to huisken this is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere sn of radius n in this paper we will study the rate of exponential convergence of a rescaled flow we will present here a method that tells us that the rate of the exponential decay is at least 2 n we can define the arrival time u of a smooth strictly convex n dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u x t if x mt for x int m0 huisken proved that for n 2 u x is c2 near x the case n 1 has been treated by kohn and serfaty 11 they proved c3 regularity of u as a consequence of the obtained rate of convergence of the mean curvature flow we prove that u is not necessarily c3 near x for n 2 we also show that the obtained rate of convergence 2 n which arises from linearizing a mean curvature flow is the optimal one at least for n 2 2007 wiley periodicals inc

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

thermal capacity estimates on the allen cahn equation we consider the allen cahn equation in a well known scaling regime which gives motion by mean curvature a well known transformation of this pde using its standing wave yields a pde the solution of which is approximately the distance function to an interface moving by mean curvature we give bounds on this last fact in terms of thermal capacity our techniques hinge upon the analysis of a certain semimartingale associated with a certain pde the pde for the approximate distance function and an analogue of some results by bafnuelos and oksendal relating lifetimes of diffusions to exterior capacities

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

a new multiscale representation for shapes and its application to blood vessel recovery in this paper we will first introduce a novel multiscale representation msr for shapes via level set motions and pdes based on the msr we will then design a surface inpainting algorithm to recover three dimensional geometry of blood vessels because of the nature of irregular morphology in vessels and organs both phantom and real inpainting scenarios were tested using our new algorithm successful vessel recoveries are demonstrated with numerical estimation of the degree of arteriosclerosis and vessel occlusion

existence of singular solutions of a degenerate equation in r2 let a 1 a 2 a m 2 2 f c 0 g i c 0 be such that 0 g i t 2 on 0 i 1 m for any open image in new window p 1 we prove the existence and uniqueness of solutions of the equation u t logu u 0 in open image in new window satisfying open image in new window and logu x t log x f t as x logu x t log x a i g i t as x a i 0 uniformly on every compact subset of 0 t for any i 1 m under a mild assumption on u 0 where open image in new window we also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of 2 with prescribed singularities at a finite number of points in the domain

evaporation of droplets in the two dimensional ginzburg landau equation abstract we consider the problem of coarsening in two dimensions for the real scalar ginzburg landau equation this equation has exactly two stable stationary solutions the constant functions 1 and 1 we assume most of the initial condition is in the 1 phase with islands of 1 phase we use invariant manifold techniques to prove that the boundary of a circular island moves according to allen cahn curvature motion law we give a criterion for non interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces

region tracking on level sets methods since the work by osher and sethian 1988 on level sets algorithms for numerical shape evolutions this technique has been used for a large number of applications in numerous fields in medical imaging this numerical technique has been successfully used for example in segmentation and cortex unfolding algorithms the migration from a lagrangian implementation to a eulerian one via implicit representations or level sets brought some of the main advantages of the technique i e topology independence and stability this migration means also that the evolution is parametrization free therefore the authors do not know exactly how each part of the shape is deforming and the point wise correspondence is lost in this note they present a technique to numerically track regions on surfaces that are being deformed using the level sets method the basic idea is to represent the region of interest as the intersection of two implicit surfaces and then track its deformation from the deformation of these surfaces this technique then solves one of the main shortcomings of the very useful level sets approach applications include lesion localization in medical images region tracking in functional mri fmri visualization and geometric surface mapping

a new approach to variational problems with multiple scales we introduce a new concept the young measure on micropatterns to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter e this allows one to extract in the limit e 0 the relevant information at the macroscopic scale as well as the coarsest microscopic scale say e and to eliminate all finer scales to achieve this we consider rescaled functions rx t x s e t viewed as maps of the macroscopic variable s with values in a suitable function space the limiting problem can then be formulated as a variational problem on the young measures generated by rex as an illustration we study a one dimensional model that describes the competition between formation of microstructure and highest gradient regularization we show that the unique minimizer of the limit problem is a young measure supported on sawtooth functions with a given period 2001 john wiley sons inc

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

weak solutions for a stochastic mean curvature flow of two dimensional graphs we study a stochastically perturbed mean curvature flow for graphs in mathbb r 3 over the two dimensional unit cube subject to periodic boundary conditions the stochastic perturbation is a one dimensional white noise acting uniformly in all points of the surface in normal direction we establish the existence of a weak martingale solution the proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach to overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods

large time behaviour of solutions of a singular diffusion equation in rn abstract let a 0 n 2 q n 2 p n q u 0 u 0 0 where 0 u 0 l loc p r n and l 1 r n l p r n satisfy a 1 1 k x q u 0 x a 1 1 k x q x r 0 k and a 1 1 k x q x 0 r 0 k x r 0 k 1 k 2 3 for some constants 0 r 0 k r 0 k 1 k 2 3 suppose u is the maximal solution of u t log u u 0 in r n 0 u x 0 u 0 x in r n and is the unique self similar solution of the equation with initial value a x q then the rescaled function v x t t q 2 q u t 1 2 q x t will converge uniformly on every compact subset of r n to x 1 as t

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

interface evolution in three dimensions with curvature dependent energy and surface diffusion interface controlled evolution phase transitions epitaxial growth of elastic films when the interfacial energy is a nonconvex function of orientation the anisotropic curvature flow equation becomes backward parabolic to overcome the instability thus generated a regularization of the equation that governs the evolution of the interface is needed in this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk surface interactions the theory is based on a superficial mass balance configurational forces and couples consistent with superficial force and moment balances a mechanical version of the second law that includes via the configurational moments work that accompanies changes in the curvature of the interface a constitutive theory whose main ingredient is a positive definite isotropic quadratic dependence of the interfacial energy on the curvature tensor two special cases are investigated i the interface is a boundary between bulk phases or grains and ii the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface

motion by curvature of planar curves with end points moving freely on a line this paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line we first prove the existence and uniqueness of self similar shrinking solution then we show that the curve shrinks to a point in a self similar manner if initially the curve is a graph

behaviour of solutions of a singular diffusion equation near the extinction time abstract we prove that if 2 and 0 u0 l1 r2 lp r2 for some constant p 1 is a radially symmetric function u0 0 and u is the unique solution of the equation u t log u u 0 in r 2 0 t u x 0 u 0 x in r2 which satisfies r 2 u x t d x r 2 u 0 d x 2 t 0 and rur x t u x t uniformly on a b as r x for any 0 t r 2 u 0 d x 2 then there exist unique constants 0 1 2 2 1 such that the rescaled function v y s u y t t t t t with s log t t will converge uniformly on every compact subset of r2 to the solution y of the ode r r r 0 in 0 with r 0 0 0 1 for some constant 0 as s

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

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

existence of solutions of the very fast diffusion equation abstract let m u um m if 1 m u log u if m 0 for any m 1 r t 0 0 u 0 l loc 1 r with 0 u 0 d x 0 u 0 d x and 0 f bv 0 such that 0 f d s for some constant 0 we prove the existence of solution of the equation ut m u xx in r 0 t 0 u x u0 in r for some constant t0 t such that 0 t u m 1 u x 0 s d s 0 u y t d y m 1 t t f d s 0 0 which satisfies a b u m 1 u x x s d s a b f d s as x for all 0 log u x as x uniformly on a b if m 0 for all 0 0 u x t d x and 0 u x t d x for all 0 log u x f t as x uniformly on a b if m 0 for all 0 0 u 0 l loc 1 r with 0 u 0 d x 0 u 0 d x we prove the existence of infinitely many finite mass solutions of the above equation with prescribed mass and line integral at time t and show that the solution will satisfy some uniform decay rate as x

amorphous surface growth via a level set approach abstract we construct an equation for amorphous surface growth due to some deposition process within the level set approach and derive a unique viscosity solution to this equation from a source particles deposit on the initial hypersurface 0 in any direction and diffuse on the hypersurface for each time t we define thereby a hypersurface t

dirichlet duality and the nonlinear dirichlet problem on riemannian manifolds in this paper we study the dirichlet problem for fully nonlin ear second order equations on a riemannian manifold as in our previous paper hl4 we define equations via closed subsets of the 2 jet bundle where each equation has a natural dual equation basic existence and uniqueness theorems are established in a wide variety of settings however the emphasis is on starting with a constant coefficient equation as a model which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model for example this covers all branches of the homogeneous complex monge ampere equation on an almost complex hermitian manifold x in general for an equation f on a manifold x and a smooth domain x we give geometric conditions which imply that the dirichlet problem on is uniquely solvable for all continuous boundary functions we begin by introducing a weakened form of comparison which has the advantage that local implies global we then introduce two fundamental concepts the first is the notion of a monotonicity cone m for f if x carries a global m subharmonic function then weak comparison implies full compar ison the second notion is that of boundary f convexity which is defined in terms of the asymptotics of f and is used to define bar riers in combining these notions the dirichlet problem becomes uniquely solvable as claimed this article also introduces the notion of local affine jet equiva lence for subequations it is used in treating the cases above but gives results for a much broader spectrum of equations on manifolds including inhomogeneous equations and the calabi yau equation on almost complex hermitian manifolds a considerable portion of the paper is concerned with specific examples they include a wide variety of equations which make sense on any riemannian manifold and many which hold univer sally on almost complex or quaternionic hermitian manifolds or topologically calibrated manifolds

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

mean curvature flow without singularities we study graphical mean curvature flow of complete solutions defined on subsets of euclidean space we obtain smooth long time existence the projections of the evolving graphs also solve mean curvature flow hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow with one additional dimension

self similar solutions of a 2 d multiple phase curvature flow this article studies self similar shrinking stationary and expanding solutions of a two dimensional motion by curvature equation modelling evolution of grain boundaries in polycrystals here the interfacial energy densities are assumed to depend only on the grains and the herring condition is used for triple junctions the intersections of three grain boundaries in particular in the isotropic case a total of six configurations are classified as the only self similar shrinking solutions