on second order weakly hyperbolic equations and the ultradifferentiable classes abstract we consider second order weakly hyperbolic equations with time dependent coefficients in the ultradifferentiable classes our main purpose of the present paper is an investigation the relation between the classes of the functions to be well posed and the following properties of the coefficients the order of degeneration stabilization to a monotonic function and their smoothness in the ultradifferentiable classes

modulus of continuity of the coefficients and non quasianalytic solutions in the strictly hyperbolic cauchy problem abstract in the strictly hyperbolic cauchy problem we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well posedness in beurling roumieu classes of ultradifferentiable functions and functionals we find well posedness in nonquasianalytic classes assuming that the coefficients have modulus of continuity t 1 t such that 0 1 1 t d t this condition is sharp because in the case 0 1 1 t d t we provide examples of cauchy problems which are well posed only in quasianalytic classes

understanding optimising and extending data compression with anisotropic diffusion galic et al journal of mathematical imaging and vision 31 255 269 2008 have shown that compression based on edge enhancing anisotropic diffusion eed can outperform the quality of jpeg for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation however the reasons for the good performance of eed remained unclear and they could not outperform the more advanced jpeg 2000 the goals of the present paper are threefold firstly we investigate the compression qualities of various partial differential equations this sheds light on the favourable properties of eed in the context of image compression secondly we demonstrate that it is even possible to beat the quality of jpeg 2000 with eed if one uses specific subdivisions on rectangles and several important optimisations these amendments include improved entropy coding brightness and diffusivity optimisation and interpolation swapping thirdly we demonstrate how to extend our approach to 3 d and shape data experiments on classical test images and 3 d medical data illustrate the high potential of our approach

euler s elastica and curvature based inpainting image inpainting is a special image restoration problem for which image prior models play a crucial role euler s elastica was first introduced to computer vision by mumford algebraic geometry and its applications springer verlag new york 1994 pp 491 506 as a curve prior model by functionalizing the elastica energy masnou and morel proceedings of the 5th ieee international conference image processing 3 1998 pp 259 263 proposed an elastica based variational inpainting model the current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of bertalmio sapiro caselles and ballester siggraph 2000 acm press new york 2000 and chan and shen j visual comm image rep 12 2001 pp 436 449 a computational scheme based on numerical pdes is presented which allows the automatic handling of topologically complex inpainting domains

higher order weighted sobolev spaces on the real line for strongly degenerate weights application to variational problems in elasticity of beams abstract for one dimensional interval and integrable weight function w we define via completion a weighted sobolev space h w m p of arbitrary integer order m the weights in consideration may suffer strong degeneration so that in general functions u from h w m p do not have weak derivatives this contribution is focussed on studying the continuity properties of functions u at a chosen internal point x 0 to which we attribute a notion of criticality of order k and with respect to the weight w for non critical points x 0 we formulate a local embedding result that guarantees continuity of functions u or their derivatives conversely we employ duality theory to show that criticality of x 0 furnishes a smooth approximation of functions in h w m p admitting jump type discontinuities at x 0 the work concludes with demonstration of established results in the context of variational problem in elasticity theory of beams with degenerate width distribution

on the asymptotic behavior of the energy for the wave equations with time depending coefficients we consider the asymptotic behavior of the total energy of solutions to the cauchy problem for wave equations with time dependent propagation speed the main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient order of the differentiability behavior of the derivatives as t and stabilization of the amplitude described by an integral moreover the optimality of these properties are ensured by actual examples

log lipschitz regularity for sg hyperbolic systems we consider the cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space not smooth in t and growing at infinity with respect to x we discuss well posedness in weighted sobolev spaces showing that the non lipschitz regularity in t has an influence not only on the loss of derivatives of the solution but also on its behaviour for x we provide examples to prove that the latter phenomenon cannot be avoided

crossing preserving coherence enhancing diffusion on invertible orientation scores many image processing problems require the enhancement of crossing elongated structures these problems cannot easily be solved by commonly used coherence enhancing diffusion methods therefore we propose a method for coherence enhancing diffusion on the invertible orientation score of a 2d image in an orientation score the local orientation is represented by an additional third dimension ensuring that crossing elongated structures are separated from each other we consider orientation scores as functions on the euclidean motion group and use the group structure to apply left invariant diffusion equations on orientation scores we describe how we can calculate regularized left invariant derivatives and use the hessian to estimate three descriptive local features curvature deviation from horizontality and orientation confidence these local features are used to adapt a nonlinear coherence enhancing crossing preserving diffusion equation on the orientation score we propose two explicit finite difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence enhancing diffusion the use of curvature leads to improved enhancement of curves with high curvature furthermore the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings

time dependent propagation speed vs strong damping for degenerate linear hyperbolic equations abstract we consider a degenerate abstract wave equation with a time dependent propagation speed we investigate the influence of a strong dissipation namely a friction term that depends on a power of the elastic operator we discover a threshold effect if the propagation speed is regular enough then the damping prevails and therefore the initial value problem is well posed in sobolev spaces solutions also exhibit a regularizing effect analogous to parabolic problems as expected the stronger is the damping the lower is the required regularity on the contrary if the propagation speed is not regular enough there are examples where the damping is ineffective and the dissipative equation behaves as the non dissipative one

unique existence of evolution equations of hyperbolic type with countably many singular of degenerate points soit h un espace de hilbert et soit un operateur auto adjoint a spectre borne par dessous soit t 0 un nombre reel arbitraire avec 1 t 0 1 on etudie la resolubilite unique de l equation d evolution abstraite singuliere ou degeneree de type hyperbolique u t t u t 0 sur t 0 1 u t 0 u 0 u t 0 u 1 ou est une fonction non negative sur 1 1 avec des points singuliers et degeneres denombrables s accummulant en 0

global solvability for kirchhoff equation in special classes of non analytic functions abstract in this paper we derive the following two properties the first one is a precise representation of wkb solution to the cauchy problem of a linear wave equation with a variable coefficient with respect to time and the second one is the global solvability for kirchhoff equation in some special classes of nonreal analytic functions which is proved by grace of the first property

a geometric ginzburg landau problem for surfaces embedded in a three dimensional euclidean space consider a functional consisting of two terms a version of the willmore energy and an anisotropic area penalising the first component of the normal vector the latter weighted with the factor 1 epsilon 2 the asymptotic behaviour of such functionals as epsilon tends to 0 is studied in this paper the results include a lower and an upper bound on the minimal energy subject to suitable constraints moreover for embedded spheres a compactness result is obtained under appropriate energy bounds

on the ergodic convergence rates of a first order primal dual algorithm we revisit the proofs of convergence for a first order primal dual algorithm for convex optimization which we have studied a few years ago in particular we prove rates of convergence for a more general version with simpler proofs and more complete results the new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms

linear wave equations with time dependent propagation speed and strong damping abstract we consider a second order linear equation with a time dependent coefficient c t in front of the elastic operator for these equations it is well known that a higher space regularity of initial data compensates a lower time regularity of c t in this paper we investigate the influence of a strong dissipation namely a friction term which depends on a power of the elastic operator what we discover is a threshold effect when the exponent of the elastic operator in the friction term is greater than 1 2 the damping prevails and the equation behaves as if the coefficient c t were constant when the exponent is less than 1 2 the time regularity of c t comes into play if c t is regular enough once again the damping prevails on the contrary when c t is not regular enough the damping might be ineffective and there are examples in which the dissipative equation behaves as the non dissipative one as expected the stronger is the damping the lower is the time regularity threshold we also provide counterexamples showing the optimality of our results

non uniqueness in hyperbolic cauchy problems where t e r and x e ri we say that it possesses the uniqueness property at a point to xo if for any solution u e c v of 1 v being a neighborhood of t0o xo such that u 0 on v nf t to it results that u 0 on a neighborhood of to xo the first uniqueness results for equation 1 were supplied by the classical theorems of holmgren 12 carleman 6 and calderon 5 more refined versions of these theorems are the following ones

curvature dependent energies a geometric and analytical approach we consider the total curvature of graphs of curves in high codimension euclidean space we introduce the corresponding relaxed energy functional and prove an explicit representation formula in the case of continuous cartesian curves i e of graphs cu of continuous functions u on an interval we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature in this case moreover the total curvature does not depend on the cantor part of the derivative of u we treat the wider class of graphs of one dimensional bv functions and we prove that the relaxed energy is given by the sum of length and total curvature of the new curve obtained by closing with vertical segments the holes in cu generated by jumps of u in the mathematical literature functionals depending on second order derivatives have recently been applied e g in image restoration processes in order to overcome some drawbacks typical of approaches based on first order functionals as the total variation one instance is the approach by chan marquinamulet 6 who proposed to consider regularizing terms given by second order functionals of the type

approximation by γ convergence of a curvature depending functional in visual reconstruction where r2 is a bounded open set the image domain h1 denotes the one dimensional hausdorff measure and g l is the input image the functional has to be minimized over all closed sets c and all u c1 c the function u represents a denoised approximation of the input image g and c represents the set of boundaries of the segmentation the mumford and shah variational model can be extended to several visual reconstruction problems see march 29 computation of depth from stereo images shah 39 computation of optical flow nesi 34 shape from shading shah 40 the existence of minimizers of gms has been proved independently by dal maso morel and solimini 21 and de giorgi carriero and leaci 23 using the compactness and lower semicontinuity theorems of ambrosio 3 mumford and shah 33 studied the properties of minimizers u c of gms assuming that c is a finite union of simple c1 1 curves meeting and meeting each other only at their endpoints they proved that the vertices of c may only be i triple points where three curves meet with equal angles ii points on the boundary of where one curve meets perpendicularly iii crack tips where a curve ends and meets nothing in computer vision the constraints imposed on the segmentations highlighted by such results constitute a drawback of the variational model in particular corners and t junctions which are relevant features for pattern recognition are distorted since the length measure is not sensitive to corners and junctions to allow for such singularities in the segmentations it is necessary to consider curvature depending energies functionals

calibrations and spinors introduction i spinors in dimension 8m 1 1 spinors and pinors 1 2 constructing calibrations from spinors 1 3 triality and the octonians ii dimension 16 and es ii 1 a model for the clifford algebra cl 16 ii 2 the orbit structure of spin 16 on 8 16 ii 3 maximal isotropy subgroups of spin 16 ii 4 calibration on r 16 constructed by squaring spinors iii additional results in higher dimension iii 1 a differential system for each geometry iii 2 pure spinors and calibrations iii 3 spinors in dimensions 50 mod 8 references

curvature varifolds with boundary we introduce a new class of nonoriented sets in rk endowed with a generalized notion of second fundamental form and boundary proving several compactness and structure properties our work extends the definition and some results of j e hutchinson 13 and can be applied to variational problems involving surfaces with boundary

an introduction to potential theory in calibrated geometry in this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry these functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties moreover they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce a number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds this paper introduces and investigates questions of pseudo convexity in the context of a general calibrated manifold x phi analogues of totally real submanifolds are introduced and used to construct enormous families of strictly phi convex spaces with every topological type allowed by morse theory specific calibrations are used as examples throughout in a sequel the duality between phi pluri sub harmonic functions and phi positive currents is investigated this study involves boundaries generalized jensen measures and other geometric objects on a calibrated manifold

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

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

on the global solvability of kirchhoff equation for non analytic initial data abstract we give sufficient conditions for the global solvability of kirchhoff equation in terms of the spectral resolutions of the initial data u 0 x u t 0 x we assume no smallness conditions and only sobolev type regularity

modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic cauchy problem abstract we deal with the cauchy problem for a strictly hyperbolic second order operator with non regular coefficients in the time variable it is well known that the problem is well posed in l 2 in case of lipschitz continuous coefficients and that the log lipschitz continuity is the natural threshold for the well posedness in sobolev spaces which in this case holds with a loss of derivatives here we prove that any intermediate modulus of continuity between the lipschitz and the log lipschitz one leads to an energy estimate with arbitrary small loss of derivatives we also provide counterexamples to show that the following classification modulus of continuity loss of derivatives is sharp lipschitz no loss intermediate arbitrary small loss log lipschitz finite loss

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

the elastic ratio introducing curvature into ratio based image segmentation we present the first ratio based image segmentation method that allows imposing curvature regularity of the region boundary our approach is a generalization of the ratio framework pioneered by jermyn and ishikawa so as to allow penalty functions that take into account the local curvature of the curve the key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment among ratios whose discrete counterparts can be globally minimized with our approach we focus in particular on the elastic ratio 0l c i c s c s ds vl c 0l c c s qds that depends given an image i on the oriented boundary c of the segmented region candidate minimizing this ratio amounts to finding a curve neither small nor too curvy through which the brightness flux is maximal we prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions we also prove that the discrete minimizers provided by our graph based algorithm converge as the resolution increases to continuous minimizers in contrast to most existing segmentation methods with computable and meaningful i e nondegenerate global optima the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations furthermore our results allow drawing conclusions about global optima of a parameterization independent version of the snakes functional the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution

structural optimization of thin elastic plates the three dimensional approach the natural way to find the most compliant design of an elastic plate is to consider the three dimensional elastic structures which minimize the work of the loading term and pass to the limit when the thickness of the design region tends to zero in this paper we study the asymptotics of such a compliance problem imposing that the volume fraction remains fixed no additional topological constraint is assumed on the admissible configurations we determine the limit problem in different equivalent formulations and we provide a system of necessary and sufficient optimality conditions these results were announced in bouchitte et al c r acad sci paris ser i 345 713 718 2007 furthermore we investigate the vanishing volume fraction limit which turns out to be consistent with the results in bouchitte and fragala arch rat mech anal 184 257 284 2007 siam j control optim 46 1664 1682 2007 finally some explicit computation of optimal plates are given

contrast driven elastica for image segmentation minimization of boundary curvature is a classic regularization technique for image segmentation in the presence of noisy image data techniques for minimizing curvature have historically been derived from gradient descent methods which could be trapped by a local minimum and therefore required a good initialization recently combinatorial optimization techniques have overcome this barrier by providing solutions that can achieve a global optimum however curvature regularization methods can fail when the true object has high curvature in these circumstances existing methods depend on a data term to overcome the high curvature of the object unfortunately the data term may be ambiguous in some images which causes these methods also to fail to overcome these problems we propose a contrast driven elastica model including curvature which can accommodate high curvature objects and an ambiguous data model we demonstrate that we can accurately segment extremely challenging synthetic and real images with ambiguous data discrimination poor boundary contrast and sharp corners we provide a quantitative evaluation of our segmentation approach when applied to a standard image segmentation data set

global minimum for a finsler elastica minimal path approach in this paper we propose a novel curvature penalized minimal path model via an orientation lifted finsler metric and the euler elastica curve the original minimal path model computes the globally minimal geodesic by solving an eikonal partial differential equation pde essentially this first order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models to solve this problem we present an eikonal pde based finsler elastica minimal path approach to address the curvature penalized geodesic energy minimization problem we were successful at adding the curvature penalization to the classical geodesic energy caselles et al in int j comput vis 22 1 61 79 1997 cohen and kimmel in int j comput vis 24 1 57 78 1997 the basic idea of this work is to interpret the euler elastica bending energy via a novel finsler elastica metric that embeds a curvature penalty this metric is non riemannian anisotropic and asymmetric and is defined over an orientation lifted space by adding to the image domain the orientation as an extra space dimension based on this orientation lifting the proposed minimal path model can benefit from both the curvature and orientation of the paths thanks to the fast marching method the global minimum of the curvature penalized geodesic energy can be computed efficiently we introduce two anisotropic image data driven speed functions that are computed by steerable filters based on these orientation dependent speed functions we can apply the proposed finsler elastica minimal path model to the applications of closed contour detection perceptual grouping and tubular structure extraction numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results

curvature dependent energies the elastic case abstract we continue our analysis on functionals depending on the curvature of graphs of curves in high codimension euclidean space we deal with the elastic case corresponding to a superlinear dependence on the pointwise curvature we introduce the corresponding relaxed energy functional and prove an explicit representation formula different phenomena w r t the plastic case i e to the relaxation of the total curvature functional are observed a p curvature functional is well defined on continuous curves with finite relaxed energy and the relaxed energy is given by the length plus the p curvature the wider class of graphs of one dimensional bv functions is treated

ground states of simple bodies that may undergo brittle fractures abstract equilibrium states of elastic brittle solids that may suffer nucleation of cracks in finite deformation setting are analyzed crack patterns are described in terms of appropriate radon measures namely curvature varifolds with boundary a new form of the energy is presented it includes terms associated with the curvature of margins and tips of possible cracks existence of minima of the energy is established in classes of pairs of deformation and families of varifolds configurational balances in weak form are determined with reference to generic curvature varifolds with boundary they include non standard terms associated with the curvatures involved in the energy pointwise balances of configurational actions are also evaluated in a special case new pointwise balances at the tips and along the margins of the crack pattern emerge

calibrations and t duality for a subclass of hitchin s generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold under a suitable integrability condition on the ambient geometry these generalised calibrated submanifolds minimise a functional occurring as d brane energy in type ii string theories further we investigate the behaviour of calibrated cycles under t duality and construct non trivial examples

on weakly hyperbolic operators with non regular coefficients and finite order degeneration we prove that the cauchy problem for a class of weakly hyperbolic equations satisfying a condition of finite order degeneration and having non lipschitz continuous coefficients is well posed in gevrey spaces

a linear framework for region based image segmentation and inpainting involving curvature penalization we present the first method to handle curvature regularity in region based image segmentation and inpainting that is independent of initialization to this end we start from a new formulation of length based optimization schemes based on surface continuation constraints and discuss the connections to existing schemes the formulation is based on a cell complex and considers basic regions and boundary elements the corresponding optimization problem is cast as an integer linear program we then show how the method can be extended to include curvature regularity again cast as an integer linear program here we are considering pairs of boundary elements to reflect curvature moreover a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables we show that by solving the linear programming relaxation one gets reasonably close to the global optimum and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

a class of non analytic functions for the global solvability of kirchhoff equation abstract we consider the global solvability to the cauchy problem of kirchhoff equation with generalized classes of manfrin s class manfrin s class is a subclass of sobolev space but we shall extend this class as a subclass of the ultradifferentiable class and we succeed to prove the global solvability of kirchhoff equation with large data in wider classes from the previous works

energy estimates at infinity for hyperbolic equations with oscillating coefficients abstract we study the behaviour for t of the energy of the solutions to the cauchy problem for some strictly hyperbolic second order equations with coefficients very rapidly oscillating

a γ convergence result for the two gradient theory of phase transitions the generalization to gradient vector fields of the classical double well singularly perturbed functionals i e u 1w u e 2 u 2 dx where w 0 if and only if a or b and a b is a rank 1 matrix is considered under suitable constitutive and growth hypotheses on w it is shown that i e converge to k h n 1 s u if u w 1 1 r d i u u bv a b otherwise where k is the constant interfacial energy per unit area

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

hyperbolic like estimates for higher order equations abstract the main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time dependent coefficients in particular we estimate the energy in the hyperbolic zone of the extended phase space by means of a function f t which depends on the principal part and on the coefficients of the terms of order m 1 then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space we call this class of estimates hyperbolic like since the energy behavior is deeply depending on the hyperbolic structure of the equation in some cases these estimates produce a dissipative effect on the energy

local and global smoothing effects for some linear hyperbolic equations with a strong dissipation we consider an abstract second order linear equation with a strong dissipation namely a friction term which depends on a power of the elastic operator in the homogeneous case we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions in the nonhomogeneous case we study how the regularity of solutions depends on the regularity of forcing terms and we characterize the spaces where a bounded forcing term yields a bounded solution what we discover is a variety of different regimes with completely different behaviors depending on the exponent in the friction term we also provide counterexamples in order to show the optimality of our results mathematics subject classification 2010 msc2010 35l10 35l15 35l20

on the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions abstract we consider the energy estimates for the wave equation with time dependent propagation speed it is known that the asymptotic behavior of the energy is determined by the interactions of the properties of the propagation speed smoothness oscillation and the difference from an auxiliary function the main purpose of the article is to show that if the propagation speed behaves asymptotically as a monotone decreasing function then we can extend the preceding results to allow faster oscillating coefficients moreover we prove that the regularity of the initial data in the gevrey class can essentially contribute for the energy estimate

cartesian currents weak diffeomorphisms and existence theorems in nonlinear elasticity in this paper we introduce some new classes of functions among these a class of weak diffeomorphisms in these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals in particular we prove the existence of one to one orientation preserving maps that minimize suitable energies associated with hyperelastic materials the minimizers are also proved to satisfy equilibrium equations finally radial deformations are discussed in connection with cavitation

convergence of the allen cahn equation with a zero neumann boundary condition on non convex domains we study a singular limit problem of the allen cahn equation with a homogeneous neumann boundary condition on non convex domains with smooth boundaries under suitable assumptions for initial data the main result is the convergence of the time parametrized family of the diffused surface energy to brakke s mean curvature flow with a generalized right angle condition on the boundary of the domain

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

a non autonomous strongly damped wave equation existence and continuity of the pullback attractor abstract in this paper we consider the strongly damped wave equation with time dependent terms u t t u t u t e t u t f u in a bounded domain r n under some restrictions on e t t and growth restrictions on the nonlinear term f the function e t depends on a parameter e e t e 0 0 we will prove under suitable assumptions local and global well posedness using the uniform sectorial operators theory the existence and regularity of pullback attractors a e t t r uniform bounds for these pullback attractors characterization of these pullback attractors and their upper and lower semicontinuity at 0

towards a variational theory of phase transitions involving curvature an anisotropic area functional is often used as a model for the free energy of a crystal surface for models of faceting the anisotropy is typically such that the functional becomes nonconvex and then it may be appropriate to regularize it with an additional term involving curvature when the weight of the curvature term tends to 0 this gives rise to a singular perturbation problem the structure of this problem is comparable to the theory of phase transitions studied first by modica and mortola their ideas are also use ful in this context but they have to be combined with adequate geometric tools in particular a variant of the theory of curvature varifolds intro duced by hutchinson is used in this paper this allows an analysis of the asymptotic behaviour of the energy functionals

filling in by joint interpolation of vector fields and gray levels a variational approach for filling in regions of missing data in digital images is introduced the approach is based on joint interpolation of the image gray levels and gradient isophotes directions smoothly extending in an automatic fashion the isophote lines into the holes of missing data this interpolation is computed by solving the variational problem via its gradient descent flow which leads to a set of coupled second order partial differential equations one for the gray levels and one for the gradient orientations the process underlying this approach can be considered as an interpretation of the gestaltist s principle of good continuation no limitations are imposed on the topology of the holes and all regions of missing data can be simultaneously processed even if they are surrounded by completely different structures applications of this technique include the restoration of old photographs and removal of superimposed text like dates subtitles or publicity examples of these applications are given we conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow

a regularized equation for anisotropic motion by curvature for realistic interfacial energies the equations of anisotropic motion by curvature exhibit backward parabolic behavior over portions of their domain thereby inducing phenomena such as the formation of facets and wrinkles in this paper a physically consistent regularized equation that may be used to analyze such phenomena is derived

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

blow up of the energy at infinity for 2 by 2 systems abstract the goal of this paper is to study the long time behavior of the energy of solutions to 2 by 2 linear hyperbolic systems some blow up results at t are given for a large class of initial data of the cauchy problem we shall prove the optimality of these results for a special class of systems we present also an example of generalized energy conservation under a c m stabilization condition

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

second order energies on thin structures variational theory and non local effects abstract for a given positive measure on r n we consider integral functionals of the kind f u r n f x u 2 u d u c 0 r n and we study their relaxation with respect to the l p topology p being the growth exponent of f to obtain the relaxed energy f we develop a suitable theory of second order intrinsic operators related to a cosserat vector field and to a curvature tensor our main theorem shows that the functional f is in general a non local one this unexpected feature occurs even in very simple examples when is the one dimensional hausdorff measure over a closed lipschitz curve in the plane

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

total roto translational variation we consider curvature depending variational models for image regularization such as euler s elastica these models are known to provide strong priors for the continuity of edges and hence have important applications in shape and image processing we consider a lifted convex representation of these models in the roto translation space in this space curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields the line energies are then easily extended to any scalar function it yields a natural generalization of the total variation to the roto translation space as our main result we show that the proposed convex representation is tight for characteristic functions of smooth shapes we also discuss cases where this representation fails for numerical solution we propose a staggered grid discretization based on an averaged raviart thomas finite elements approximation this discretization is consistent up to minor details with the underlying continuous model the resulting non smooth convex optimization problem is solved using a first order primal dual algorithm we illustrate the results of our numerical algorithm on various problems from shape and image processing

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

energy estimates for wave equations with time dependent propagation speeds in the gevrey class abstract the total energy of the wave equation is conserved with respect to time if the propagation speed is a constant but this is not true in general for time dependent propagation speeds indeed it is considered in hirosawa 2007 3 that the following properties of the propagation speed are crucial for the estimates of the total energy oscillating speed difference from the mean and the smoothness in c m category the main purpose of this paper is to derive a benefit of a further smoothness of the propagation speed in the gevrey class for the energy estimates

critical counterexamples for linear wave equations with time dependent propagation speed abstract we investigate an abstract wave equation with a time dependent propagation speed and we consider both the non dissipative case and the case with a strong damping that depends on a power of the elastic operator previous results show that depending on the values of the parameters and on the time regularity of the propagation speed this equation exhibits either well posedness in sobolev spaces or well posedness in gevrey spaces or ill posedness with severe derivative loss in this paper we examine some critical cases that were left open by the previous literature and we show that they fall into the pathological regime the construction of the counterexamples requires a redesign from scratch of the basic ingredients and a suitable application of baire category theorem in place of the usual iteration scheme

cm theory of damped wave equations with stabilisation abstract the aim of this note is to extend the energy decay estimates from j wirth wave equations with time dependent dissipation i non effective dissipation j differential equations 222 2006 487 514 to a broader class of time dependent dissipation including very fast oscillations this is achieved using stabilisation conditions on the coefficient in the spirit of f hirosawa on the asymptotic behavior of the energy for wave equations with time depending coefficients math ann 339 4 2007 819 839

long time asymptotics for 2 by 2 hyperbolic systems the goal of this paper is to study the behavior of the energy for 2 by 2 strictly hyperbolic systems on the one hand we are interested in generalized energy conservation which excludes blow up and decay of the energy for t on the other hand we present scattering results which take account of terms of order zero

integral representation of convex functionals on a space of measures abstract in duality pairs such as m b c 0 and w 1 p w 0 1 p a convex integral functional on the space of functions has a polar which admits an integral representation this representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part the duality is useful in plasticity theory in the sobolev case the study of non parametric integrands is new a description of the sub differential is obtained

clusters minimizing area plus length of singular curves soap bubble clusters which seek the least area way to enclose and separate given volumes of air often have been used as models for metallic structures which likewise seek to minimize interfacial areas between regions for metals however there may well be additional energy associated with the singular curves affecting both the global and local structure cf mt bo see fig 1 we give the first existence and regularity results for clusters of prescribed volumes in r 3 minimizing area plus length of singular curves almgren s general theory of soap bubble clusters a2 does not apply because it provides no control over the singular curves although some of his ideas and lemmas prove useful

structure and rigidity of functions in bv2loc r2 with gradients taking only 3 values consider a function u bv2loc r2 such that u takes values in a fixed set of 3 vectors almost everywhere this condition implies that u is piecewise affine away from a closed set of vanishing 1 dimensional hausdorff measure furthermore there is some rigidity in the sense that away from the exceptional set small perturbations of u will result only in controllable changes of the structure

generalised energy conservation law for wave equations with variable propagation speed abstract we investigate the long time behaviour of the l 2 energy of solutions to wave equations with variable speed of propagation the novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property

abstract singular evolution equations of hyperbolic type abstract this paper is concerned with the study of a large class of linear abstract evolution equations with singular coefficients under kato s hyperbolicity assumptions existence and uniqueness results for strong and strict solutions are obtained in suitable weighted spaces

quasianalytic and nonquasianalytic solutions for a class of weakly hyperbolic cauchy problems abstract in the weakly hyperbolic cauchy problem we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well posedness in beurling roumieu classes of ultradifferentiable functions and functionals we find a sharp condition on the modulus of continuity assuring the well posedness in nonquasianalytic classes

unilateral problems in nonlinear three dimensional elasticity in 1977 john ball obtained a series of very important existence results in hyperelasticity the objective of the present paper is to extend his results in two directions first by considering unilateral boundary conditions corresponding to contact without friction secondly by allowing locking constraints in the set of admissible deformations

anisotropic fast marching on cartesian grids using lattice basis reduction we introduce a modification of the fast marching algorithm which solves the anisotropic eikonal equation associated to an arbitrary continuous riemannian metric cal m on a two or three dimensional domain the algorithm has a complexity cal o n ln n n ln kappa cal m where n is the discrete domain cardinality the logarithmic dependency in the maximum anisotropy ratio kappa cal m of the riemannian metric allows us to handle extreme anisotropies for a limited numerical cost we prove the convergence of the algorithm and illustrate its efficiency by numerical experiments the algorithm relies on the computation at each grid point z of a special system of coordinates a reduced basis of the lattice bbb z m with respect to the symmetric positive definite matrix cal m z encoding the desired anisotropy at this point

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding

on t dependent hyperbolic systems part 2 abstract we consider hyperbolic equations with time dependent coefficients and develop an abstract framework to derive the asymptotic behaviour of the representation of solutions for large times we are dealing with generic situations where the large time asymptotics is of hyperbolic type our approach is based on diagonalisation procedures combined with asymptotic integration arguments