dynamical renormalisation through classical equations of motion classical approximations for critical dynamics are taken as the basis of a new dynamical renormalisation group strategy in real space the approach is phenomenological and avoids proliferation of interactions as well as memory effects for the square and cubic glauber model the best determinations of static critical properties come up with values for the dynamic exponent which appear compatible with the most recent estimates by monte carlo or series expansion methods

renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

bounds for effective strains of geometrically linear elastic multiwell model we give a direct estimate for the quasiconvex hull q e k on linear strains generated by a finite set of linear strains k subset of m s 3 of symmetric matrices the problem is directly related to the microstructure modelled by the multiwell problem and its corresponding macroscopic effect we bound the quasiconvex envelope q e dist e x k near an exposed edge of the convex hull c k that does not have compatible connections our bounds depend on the weak type 1 1 estimate for certain singular integral operators and the eometric properties of the exposed edge

technical note the stochastic discrete dynamic lot size problem an open loop solution we investigate the stochastic dynamic lot size model taking into account probabilistic constraints on the state variable our approach is to obtain an open loop solution by solving a new equivalent deterministic dynamic lot size problem with new parameters we also discuss characteristics of the new equivalent deterministic problem and note in particular that the new formulation preserves concavity or convexity

image segmentation by variational methods mumford and shah functional and the discrete approximations in this paper we discuss the links between mumford and shah s variational problem for signal and image segmentation based on an energy functional of a continuous grey level function and the numerical algorithms proposed to solve it these numerical approaches are based on a discrete functional we recall that in one dimension this discrete functional is asymptotically equivalent to the continuous functional this can be summarized in a gamma convergence result we show that the same result holds in dimension two provided that the continuous energy is adapted to the anisotropy of the discrete approaches we display a few experimental results in dimensions one and two

stress regularity in quasi static perfect plasticity with a pressure dependent yield criterion abstract this work is devoted to establishing a regularity result for the stress tensor in quasi static planar isotropic linearly elastic perfectly plastic materials obeying a drucker prager or mohr coulomb yield criterion under suitable assumptions on the data it is proved that the stress tensor has a spatial gradient that is locally squared integrable as a corollary the usual measure theoretical flow rule is expressed in a strong form using the quasi continuous representative of the stress

a regularity theorem for minimizers of quasiconvex integrals we prove c1 partial regularity for minimizers of functionals with quasiconvex integrand f x u du depending on vector valued functions u the integrand is required to be twice continuously differentiable in du and no assumption on the growth of the derivatives of f is made a polynomial growth is required only on f itself

strong polyhedral approximation of simple jump sets abstract we prove a strong approximation result for functions u w 1 j where j is the union of finitely many lipschitz graphs satisfying some further technical assumptions we approximate j by a polyhedral set in such a manner that a regularisation term div j u i i 0 1 2 is convergent the boundedness of this regularisation functional itself introduced in t valkonen transport equation and image interpolation with sbd velocity fields j math pures appl 95 2011 459 494 doi 10 1016 j matpur 2010 10 010 ensures the convergence in total variation of the jump part div j u i of the distributional divergence

fine properties of functions with bounded deformation the paper is concerned with the fine properties of functions in the space of functions with bounded deformation we analyse the set of lebesgue points and the set where these functions have one sided approximate limits moreover following the analogy with we decompose the symmetric distributional derivative into an absolutely continuous part a jump part and a cantor part the main result of the paper is a structure theorem for functions showing that these parts of the derivative can be recovered from the corresponding ones of the one dimensional sections moreover we prove that functions are approximately differentiable in almost every point of their domain

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

a global method for relaxation in w1 p and in sbvp an integral representation formula for a class of functionals defined on and in the space of special functions of bounded variation is obtained without requiring the regularity conditions usually imposed in the literature the approach is based on the general results of 10 and on a poincare wirtinger type inequality introduced by de giorgi carriero leaci 25 applications to relaxation problems and dimension reduction problems in brittle thin films are presented

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

continuum discontinuum modelling of shear bands a methodology to model shear bands as strong discontinuities within a continuum mechanics context is presented the loss of hyperbolicity of the ibvp is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction separation model acting at the discontinuity surface the extended finite element method xfem is employed for the spatial discretization of the governing equations this enables the shear bands to be arbitrarily positioned within the mesh examples that study the shear band progression within a rate independent material are presented copyright 2005 john wiley sons ltd

a hierarchy of plate models derived from nonlinear elasticity by gamma convergence we derive a hierarchy of plate models from three dimensional nonlinear elasticity by convergence what distinguishes the different limit models is the scaling of the elastic energy per unit volume h where h is the thickness of the plate this is in turn related to the strength of the applied force h membrane theory derived earlier by le dret and raoult corresponds to 0 nonlinear bending theory to 2 von karman theory to 3 4 and linearized vk theory to 3 intermediate values of lead to certain theories with constraints a key ingredient in the proof is a generalization to higher derivatives of our rigidity result 29 which states that for maps v 0 1 3 3 the l2 distance of v from a single rotation is bounded by a multiple of the l2 distance from the set so 3 of all rotations

critical differentiability of maps and canceling operators for a first order homogeneous linear differential operator mathbb a and 1 q infty we show that l n n 1 q differentiability of bv mathbb a text loc maps is equivalent to van schaftingen s elliptic and canceling ec condition for mathbb a for higher order ec operators we give a generalization of dorronsoro s theorem on critical l p taylor expansions for bv k maps here bv mathbb a text loc is the space of u in l 1 text loc such that mathbb a u is a radon measure these results can be formulated as sharp pointwise regularity results for elliptic systems mathbb a u mu where mu is a radon measure thereby covering the limiting case p 1 of a theorem of calder on and zygmund which was open so far

existence of free discontinuity problems in sbd ω abstract in this paper we consider the variational problems for the following two functionals in s b d f 1 u f 1 x u d x h n 1 j u j u g x u u d h n 1 h u d x f 2 u f 2 x u u d x h n 1 j u j u g x u u d h n 1 h u d x where f 1 m sym n n 0 be a caratheodory function satisfying a weak convexity property and growth assumptions of order s 1 f 2 and g satisfy convexity property and lower semicontinuity h l p r n p n satisfies the compatibility condition 0 here we do not assume that u x h x a e in h x x be a uniformly bounded family of closed subsets of r n

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

magnetization at corners in two dimensional ising models we investigate the corner spin magnetization of two dimensional ferromagnetic ising models in various wedge geometries results are obtained for triangular and square lattices by numerical studies on finite wedges using the star triangle transformation as well as analytic calculations using corner transfer matrices and a fermionic representation of the row to row transfer matrix the corner magnetizations vanish at the bulk critical temperature tc with an exponent c differing from the bulk exponent for isotropic systems with free edges we find that c is given simply by c 2 where is the angle at the corner for apex magnetizations of conical lattices we obtain the strikingly similar result a 4 these formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle eff determined by the relative strengths of the interactions

a lusin type theorem for gradients abstract we prove that for every borel vector field f there exists a function u of class c 1 whose gradient du agrees with f outside a set of arbitrary small measure

a finite element approximation of the griffith s model in fracture mechanics the griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of sbd spaces we suggest an approximation in the sense of convergence by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations the quasi static evolution for boundary value problems is also taken into account and some numerical results are shown

korn type inequalities in orlicz spaces abstract we exhibit balance conditions between a young function a and a young function b for a korn type inequality to hold between the l b norm of the gradient of vector valued functions and the l a norm of its symmetric part in particular we extend a standard form of the korn inequality in l p with 1 p and an orlicz version involving a young function a satisfying both the 2 and the 2 condition

which special functions of bounded deformation have bounded variation functions of bounded deformation bd arise naturally in the study of fracture and damage in a geometrically linear context they are related to functions of bounded variation bv but are less well understood we discuss here the relation to bv under additional regularity assumptions which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the cantor part to vanish on the positive side we prove that bd functions that are piecewise affine on a caccioppoli partition are in gsbv and we prove that sbd p functions are approximately continuous almost everywhere away from the jump set on the negative side we construct a function that is bd but not in bv and has distributional strain consisting only of a jump part and one that has a distributional strain consisting of only a cantor part

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

confidence based reasoning in stochastic constraint programming in this work we introduce a novel approach based on sampling for finding assignments that are likely to be solutions to stochastic constraint satisfaction problems and constraint optimisation problems our approach reduces the size of the original problem being analysed by solving this reduced problem with a given confidence probability we obtain assignments that satisfy the chance constraints in the original model within prescribed error tolerance thresholds to achieve this we blend concepts from stochastic constraint programming and statistics we discuss both exact and approximate variants of our method the framework we introduce can be immediately employed in concert with existing approaches for solving stochastic constraint programs a thorough computational study on a number of stochastic combinatorial optimisation problems demonstrates the effectiveness of our approach

rigidity and gamma convergence for solid solid phase transitions with so 2 invariance the singularly perturbed two well problem in the theory of solid solid phase transitions takes the form i e u 1 ew u e 2 u 2 where u r n r n is the deformation and w vanishes for all matrices in k so n a u so n b we focus on the case n 2 and derive by means of gamma convergence a sharp interface limit for i e the proof is based on a rigidity estimate for low energy functions our rigidity argument also gives an optimal two well liouville estimate if u has a small bv norm compared to the diameter of the domain then in the l 1 sense either the distance of u from so 2 a or the one from so 2 b is controlled by the distance of vu from k this implies that the oscillation of vu in weak l1 is controlled by the l1 norm of the distance of u to k

the jump set under geometric regularisation part 2 higher order approaches abstract in part 1 we developed a new technique based on lipschitz pushforwards for proving the jump set containment property h m 1 j u j f 0 of solutions u to total variation denoising we demonstrated that the technique also applies to huber regularised tv now in this part 2 we extend the technique to higher order regularisers we are not quite able to prove the property for total generalised variation tgv based on the symmetrised gradient for the second order term we show that the property holds under three conditions first the solution u is locally bounded second the second order variable is of locally bounded variation w bv loc r m instead of just bounded deformation w bd third w does not jump on j u parallel to it the second condition can be achieved for non symmetric tgv both the second and third condition can be achieved if we change the radon or l 1 norm of the symmetrised gradient ew into an l norm p 1 in which case korn s inequality holds on the positive side we verify the jump set containment property for second order infimal convolution tv ictv in dimension m 2 we also study the limiting behaviour of the singular part of du as the second parameter of tgv 2 goes to zero unsurprisingly it vanishes but in numerical discretisations the situation looks quite different finally our work additionally includes a result on tgv strict approximation in bv

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

characterization of generalized young measures generated by symmetric gradients this work establishes a characterization theorem for generalized young measures generated by symmetric derivatives of functions of bounded deformation bd in the spirit of the classical kinderlehrer pedregal theorem our result places such young measures in duality with symmetric quasiconvex functions with linear growth the local proof strategy combines blow up arguments with the singular structure theorem in bd the analogue of alberti s rank one theorem in bv which was recently proved by the authors as an application of our characterization theorem we show how an atomic part in a bd young measure can be split off in generating sequences

corner critical exponents from the mean field renormalisation group the previously introduced method of mean field renormalisation has provided a unified approach to bulk and surface critical behaviour the method is used for the computation of the critical exponent associated with the order parameter at corners applications to the square and triangular ising model are presented the angle dependence of the corner critical exponent is qualitatively reproduced the results for the bulk critical exponents are more accurate than in previously employed mean field renormalisation schemes

a geometrical approach to monotone functions in r n this paper is concerned with the fine properties of monotone func tions on r n we study the continuity and differentiability properties of these functions the approximability properties the structure of the distributional derivatives and of the weak jacobians moreover we exhibit an example of a monotone function u which is the gradient of a c 1 convex function and

a lower semicontinuity result for some integral functionals in the space sbd the purpose of this paper is to study the lower semicontinuity with respect to the strong l1 convergence of some integral functionals defined in the space sbd of special functions with bounded deformation precisely we prove that if u sbd uh sbd converges to u strongly in l1 rn and the measures ejuh converge weakly to a measure singular with respect to the lebesgue measure then f x eu dx liminfh f x euh dx provided the integrand f satisfies a weak convexity property and standard growth assumptions of order p 1

length scaling in mfrg the mean field renormalisation group mfrg has so far given poor estimates for critical indices it is argued that this can be traced to an unsatisfactory definition of the length scaling factor a new definition that improves the estimates in most cases is proposed

integral representation for functionals defined on sbd p in dimension two we prove an integral representation result for functionals with growth conditions which give coercivity on the space sbd p omega for omega subset mathbb r 2 which is a bounded open lipschitz set and p in 1 infty the space sbdp of functions whose distributional strain is the sum of an lp part and a bounded measure supported on a set of finite mathcal h 1 dimensional measure appears naturally in the study of fracture and damage models our result is based on the construction of a local approximation by w1 p functions we also obtain a generalization of korn s inequality in the sbdp setting

lusin type approximation of bd functions the purpose of this paper is to establish a lusin type approximation of functions with bounded deformation by lipschitz or c 1 functions the main ingredients inthe proof of our result are the maximal function of the measure eu the poincare type result by kohn and the approximate symmetric differentiability of bd functions by ambrosio and others

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

an efficient computational method for a stochastic dynamic lot sizing problem under service level constraints abstract we provide an efficient computational approach to solve the mixed integer programming mip model developed by tarim and kingsman 8 for solving a stochastic lot sizing problem with service level constraints under the static dynamic uncertainty strategy the effectiveness of the proposed method hinges on three novelties i the proposed relaxation is computationally efficient and provides an optimal solution most of the time ii if the relaxation produces an infeasible solution then this solution yields a tight lower bound for the optimal cost and iii it can be modified easily to obtain a feasible solution which yields an upper bound in case of infeasibility the relaxation approach is implemented at each node of the search tree in a branch and bound procedure to efficiently search for an optimal solution extensive numerical tests show that our method dominates the mip solution approach and can handle real life size problems in trivial time

homogenization of the nonlinear maxwell model of viscoelasticity and of the prandtl reuss model of elastoplasticity this paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusion here we denote by the stress tensor by e the linearized strain tensor by b x the compliance tensor and by x the subdifferential of a convex function x this relation accounts for elasto viscoplasticity including a nonlinear version of the classical maxwell model of viscoelasticity and the prandtl reuss model of elastoplasticity the constitutive law is coupled with the equation of continuum dynamics and well posedness is proved for an initial and boundary value problem the function and the tensor b are then assumed to oscillate periodically with respect to x and as this period vanishes a two scale model of the asymptotic behaviour is derived via nguetseng s notion of two scale convergence a fully homogenized single scale model is also retrieved and its equivalence with the two scale problem is proved this formulation is non local in time and is at variance with that based on so called analogical models that rest on a mean field type hypothesis

an optimal solution for the stochastic version of the wagner whitin dynamic lot size model we present an algorithm for determining the optimal solution over the entire planning horizon for the dynamic lot size model where demand is stochastic and non stationary the optimal solution to the deterministic problem is the well known wagner whitin algorithm the present work contributes principally to knowledge building and provides a tool for researchers one potentially useful contribution to practice is the solution to an important special case where demand follows normal distributions other contributions to practice will likely flow from the development of improved heuristics and the improved basis to evaluate heuristic performance

functionals defined on piecewise rigid functions integral representation and vargamma γ convergence we analyse integral representation and vargamma convergence properties of functionals defined on piecewise rigid functions that is functions which are piecewise affine on a caccioppoli partition where the derivative in each component is constant and lies in a set without rank one connections such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior our results are based on localization techniques for vargamma convergence and a careful adaption of the global method for relaxation bouchitte et al in arch ration mech anal 165 187 242 2002 bouchitte et al in arch ration mech anal 145 51 98 1998 to this new setting under rather general assumptions they constitute a first step towards the investigation of lower semicontinuity relaxation and homogenization for free discontinuity problems in spaces of generalized functions of bounded deformation