global strong solution with bv derivatives to singular solid on solid model with exponential nonlinearity abstract in this work we consider the one dimensional very singular fourth order equation for solid on solid model in attachment detachment limit regime with exponential nonlinearity h t 1 h e e h 1 h e h h where total energy e h is the total variation of h using a logarithmic correction for total energy e h ln h d x and gradient flow structure with a suitable defined functional we prove the one dimensional evolution variational inequality solution preserves a positive gradient h x which has upper and lower bounds but in bv space we also obtain the global strong solution to the solid on solid model which allows an asymmetric singularity h x x to happen

gradient estimates for solutions to the conductivity problem in this paper we derive very precise gradient estimates for solutions to the conductivity problem in the case where two circular conductivity inclusions are very close but not touching the novelty of these estimates is that they give very specific information about the blow up of the gradient as the conductivities of the inclusions degenerate

homeomorphic conjugates of fuchsian groups by a fuchsian group we mean a discrete subgroup of m it may contain also orientation reversing elements usually a fuchsian group is thought to act on i but this is not a problem since the action on s can be uniquely extended to an action on 1 and later we might also consider a fuchsian group to act on 1 depending on the situation we are here concerned with the problem when is a group g ofhomeornorphisms of s topologically conjugate to a fuchsian group we propose that this happens if and only if g is a convergence group of s which means that it satisfies the convergence property this means that whenever gi g are distinct we can pass to a subsequence in such a way that for some a b c s

determination of the insulated inclusion in conductivity problem and related eshelby conjecture abstract in the study of composites it is important to determine the shape of inclusions there is an interesting case in conductivity problem that the inclusion is insulated in present paper we first obtain the representation formula of the solution to an exterior problem and then prove that for any uniform loading such solution can be extended into the inclusion as an affine function if and only if the insulated inclusion is an ellipse or an ellipsoid we also show that an analogous result holds for the elasticity problem which is related to eshelby conjecture the main results in this paper are motivated by ammari kang kim and lee 2013 ammari kang and lim 2005 kang and milton 2008 and liu 2008

discussion of second order topological sensitivity analysis by j rocha de faria et al abstract the aim of this discussion is to expose incorrect results in a previous ijss article

optimal boundary gradient estimates for lame systems with partially infinite coefficients abstract in this paper we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the lame systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary when the distance tends to zero the optimal blow up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions

stability and resolution analysis for a topological derivative based imaging functional the aim of this paper is to study a topological derivative based anomaly detection algorithm we compare its performance with other imaging approaches such as multiple signal classification backpropagation and kirchhoff migration we also investigate its stability with respect to medium and measurement noises as well as its resolution a simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional both in terms of resolution and stability

continuum theory of epitaxial crystal growth i we present various continuum limits to describe epitaxial thin film growth we consider a hierarchy of models which can take into account the diffusion of terrace adatoms attachment and detachment of edge adatoms vapor phase diffusion and the effect of multiple species the starting point is the burton cabrera frank type step flow model we have obtained partial differential equations in the form of a coupled system of diffusion equation for the adatom density and a hamilton jacobi equation for the film height function this is supplemented with appropriate boundary conditions at the continuum level to describe the growth at the peaks and valleys on the film the results here can be used in a macroscopic description of thin film growth

energy change due to the appearance of cavities in elastic solids abstract the paper presents an overview of the problem of assessing an increment of strain energy due to the appearance of small cavities in elastic solids the following approaches are discussed the compound asymptotic method by mazja et al the eshelby like method used in the classical works on the mechanics of composites the homogenization method and the topological derivative method proposed by sokolowski and ochowski the increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid all the methods lead to the same formula for the increment of energy it is expressed by a quadratic form with respect to strains referring to the virgin solid this quadratic form turns out to be unconditionally positive definite explicit formulae are derived for an elliptical hole and for a spherical cavity the results derived determine the characteristic function of the bubble method of the optimal shape design of elastic 2d and 3d structures

higher order topological derivatives in elasticity the topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain in this work we compute higher order topological derivatives for elasticity problems so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain a critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space we present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion we also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown

on hyperbolic groups with spheres as boundary let g be a torsion free hyperbolic group and let n 6 be an integer we prove that g is the fundamental group of a closed aspherical manifold if the boundary of g is homeomorphic to an n 1 dimensional sphere

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

regularity properties of solutions to transmission problems we show that the gradients of solutions to certain elliptic and parabolic transmission problems with internal lipschitz boundary and constant co efficients at each side of the internal boundary are square integrable along the internal boundary

gradient estimates for solutions of the lame system with partially infinite coefficients we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

a fem based topological sensitivity approach for fast qualitative identification of buried cavities from elastodynamic overdetermined boundary data a time domain topological sensitivity ts approach is developed for elastic wave imaging of media of arbitrary geometry the ts which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal hole is expressed in terms of the time convolution of the free field and a supplementary adjoint field as a function of that specified location following previous studies performed under mostly static or time harmonic conditions the ts field is here considered as a natural and computationally efficient approach for defining a defect indicator function this study emphasizes the implementation and exploitation of ts fields using standard displacement based fem approaches a straightforward task once the correct sensitivity formulation is available a comprehensive set of numerical experiments on 3d and 2d elastodynamic and acoustic configurations is reported allowing to assess and highlight many features of the proposed ts based fast qualitative identification such as its ability to identify multiple defects and its robustness against data noise

from crystal steps to continuum laws behavior near large facets in one dimension abstract the passage from discrete schemes for surface line defects steps to nonlinear macroscopic laws for crystals is studied via formal asymptotics in one space dimension our goal is to illustrate by explicit computations the emergence from step motion laws of continuum scale power series expansions for the slope near the edges of large flat surface regions facets we consider surface diffusion kinetics via the burton cabrera and frank bcf model by which adsorbed atoms diffuse on terraces and attach detach at steps nearest neighbor step interactions are included the setting is a monotone train of n steps separating two semi infinite facets at fixed heights we show how boundary conditions for the continuum slope and flux and expansions in the height variable near facets may emerge from the algebraic structure of discrete schemes as n our technique relies on the use of self similar discrete slopes conversion of discrete schemes to sum equations and their reduction to nonlinear integral equations for the continuum scale slope approximate solutions to the continuum equations near facet edges are constructed formally by direct iterations for elastic dipole and multipole step interactions the continuum slope is found in agreement with a previous hypothesis of local equilibrium

braid groups and discrete diffeomorphisms of the punctured disk we show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand as an application of this method we also prove that there is no cohomological obstruction to lifting the standard embedding mathrm br 2g 2 hookrightarrow mathrm mod g 2 to a group homomorphism between diffeomorphism groups

second order topological sensitivity analysis abstract the topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain classically this derivative comes from the second term of the topological asymptotic expansion dealing only with infinitesimal holes however for practical applications we need to insert holes of finite size therefore we consider one more term in the expansion which is defined as the second order topological derivative in order to present these ideas in this work we apply the topological shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the poisson s equations taking the total potential energy as cost function and the state equation as constraint furthermore we also study the effects of different boundary conditions on the hole neumann and dirichlet both homogeneous finally we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion which has two main features it allows us to deal with hole of finite size and provides a better descent direction in optimization process

strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions abstract we consider the lame system of linear elasticity when the inclusion has the extreme elastic constants we show that the solutions to the lame system converge in appropriate h 1 norms when the shear modulus tends to infinity the other modulus the compressional modulus is fixed and when the bulk modulus and the shear modulus tend to zero using this result we show that the asymptotic expansion of the displacement vector in the presence of small inclusion is uniform with respect to lame parameters

the evolution of a crystal surface analysis of a one dimensional step train connecting two facets in the adl regime abstract we study the evolution of a monotone step train separating two facets of a crystal surface the model is one dimensional and we consider only the attachment detachment limited regime starting with the well known odes for the velocities of the steps we consider the system of odes giving the evolution of the discrete slopes it is the l 2 steepest descent of a certain functional using this structure we prove that the solution exists for all time and is asymptotically self similar we also discuss the continuum limit of the discrete self similar solution characterizing it variationally identifying its regularity and discussing its qualitative behavior our approach suggests a pde for the slope as a function of height and time in the continuum setting however existence uniqueness and asymptotic self similarity remain open for the continuum version of the problem

gradient estimates for the perfect conductivity problem we establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect stiff conductors are closely spaced inside an open bounded domain and away from the boundary these results give the optimal blow up rates of the stress for conductors with arbitrary shape and in all dimensions

application of asymptotic analysis to the two scale modeling of small defects in mechanical structures this work aims at designing a numerical strategy towards assessing the nocivity of a small defect in terms of its size and position in a structure at low computational cost using only a mesh of the defect free reference structure the modification of the fields induced by the presence of a small defect is taken into account by using asymptotic corrections of displacements or stresses this approach helps determining the potential criticality of defects by considering trial micro defects with varying positions sizes and mechanical properties taking advantage of the fact that parametric studies on defect characteristics become feasible at virtually no extra computational cost the proposed treatment is validated and demonstrated on two numerical examples involving 2d elastic configurations

strain energy change to the insertion of inclusions associated to a thermo mechanical semi coupled system abstract the topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation according to the literature the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations in addition the topological asymptotic analysis associated to multi physics problems has been reported in the literature only on the level of mathematical analysis of singularly perturbed geometrical domains in this work we present the topological derivative in its closed form for the total potential mechanical energy associated to a thermo mechanical semi coupled system when a small circular inclusion is introduced at an arbitrary point of the domain in particular we consider the linear elasticity system modeled by the navier equation coupled with the steady state heat conduction problem modeled by the laplace equation the mechanical coupling term comes out from the thermal stress induced by the temperature field since this term is non local we introduce a non standard adjoint state which allows to obtain a closed form for the topological derivative finally we provide a full mathematical justification for the derived formulas and develop precise estimates for the remainders of the topological asymptotic expansion

identification of small inhomogeneities of extreme conductivity by boundary measurements a theorem on continuous dependence we consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity embedded in a spatially varying reference medium firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential the location of the inhomogeneities and their geometry secondly we use this representation formula to prove a lipschitz continuous dependence estimate for the corresponding inverse problem this estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution

spectral analysis of the neumann poincare operator and characterization of the stress concentration in anti plane elasticity when holes or hard elastic inclusions are closely located stress which is the gradient of the solution to the anti plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero it is important to precisely characterize the blow up of the gradient of such an equation in this paper we show that the blow up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1 2 of a neumann poincare type operator defined on the boundaries of the inclusions by comparing the singular function with the one corresponding to two disks osculating to the inclusions we quantitatively characterize the blow up of the gradient in terms of explicit functions in electrostatics our results apply to the electric field which is the gradient of the solution to the conductivity equation in the case where perfectly conducting or insulating inclusions are closely located

stability analysis for magnetic resonance elastography we consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues the measurements are made on the entirety of a smooth domain since tissues can be modeled as quasi incompressible fluids we examine the stokes system and consider only the recovery of shear modulus distributions our main result is to establish lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields these estimates imply convergence of a numerical scheme known as the landweber iteration scheme for reconstructing the shear modulus distributions

gradient estimates for solutions of the lame system with partially infinite coefficients in dimensions greater than two abstract we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

a new reconstruction method for the inverse potential problem abstract the inverse potential problem consists in reconstructing an unknown measure with support in a geometrical domain from a single boundary measurement in order to deal with this severely ill posed inverse problem we rewrite it as an optimization problem where a kohn vogelius type functional measuring the misfit between the solutions of two auxiliary problems is minimized one auxiliary problem contains information on the boundary measurement while the other one corresponds to the boundary excitation the solutions of the auxiliary problems coincide once the inverse problem is solved in order to minimize the kohn vogelius criterion its total variation with respect to a set of ball shaped perturbations on the measure is explicitly evaluated then a new method for solving the inverse potential problem based on the expression obtained is devised finally some numerical results are presented in order to show the effectiveness of the proposed reconstruction algorithm

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

sounding of finite solid bodies by way of topological derivative this paper is concerned with an application of the concept of topological derivative to elastic wave imaging of finite solid bodies containing cavities building on the approach originally proposed in the elastostatic theory of shape optimization the topological derivative which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity free reference body is used as a void indicator through an assembly of sampling points where it attains negative values the computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary value problems for the reference body which are here formulated as boundary integral equations for a comprehensive treatment of the subject formulas for topological sensitivity are obtained using three alternative methodologies namely i direct differentiation approach ii adjoint field method and iii limiting form of the shape sensitivity analysis the competing techniques are further shown to lead to distinct computational procedures methodologies i and ii are implemented within a bem based platform and validated against an analytical solution a set of numerical results is included to illustrate the utility of topological derivative for 3d elastic wave sounding of solid bodies an approach that may perform best when used as a pre conditioning tool for more accurate gradient based imaging algorithms despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques

electrical impedance tomography by elastic deformation this paper presents a new algorithm for conductivity imaging our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations using asymptotics of the fields in the presence of small volume inclusions we relate the pointwise values of the energy density to the measured data through a nonlinear pde our algorithm is based on this pde and takes full advantage of the enriched data we give numerical examples that illustrate the performance and the accuracy of our approach

higher order topological sensitivity for 2 d potential problems application to fast identification of inclusions abstract this article concerns an extension of the topological derivative concept for 2 d potential problems involving penetrable inclusions whereby a cost function j is expanded in powers of the characteristic size e of a small inclusion the o e4 approximation of j is established for a small inclusion of given location shape and conductivity embedded in a 2 d region of arbitrary shape and conductivity and then generalized to several such inclusions simpler and more explicit versions of this result are obtained for a centrally symmetric inclusion and a circular inclusion numerical tests are performed on a sample configuration for i the o e4 expansion of potential energy and ii the identification of a hidden inclusion for the latter problem a simple approximate global search procedure based on minimizing the o e4 approximation of j over a dense search grid is proposed and demonstrated

characteristic classes of surface bundles let m be a c manifold if one wants to classify all differentiable fibre bundles over a given manifold which have m as fibres then one must study the topology of the corresponding structure group diffm the group of all diffeomorphisms of m equipped with the c topology it is easy to see that the connected component diffo m of the identity of diff m is an open subgroup therefore the natural topology on the quotient group m diff m diffo m the diffeotopy group of m is the discrete topology hence we have a fibration

realization of the mapping class group by homeomorphisms in this paper we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface more precisely let pr mathcal h textit omeo m to mathcal m mathcal c m denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface m of genus g 5 we show that there is no homomorphism mathcal e mathcal m mathcal c m to mathcal h textit omeo m such that pr circ mathcal e is the identity this answers a question by thurston see 11

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

a semismooth newton method for topology optimization abstract this paper deals with elliptic optimal control problems for which the control function is constrained to assume values in 0 1 based on an appropriate formulation of the optimality system a semismooth newton method is proposed for the solution convergence results are proved and some numerical tests illustrate the efficiency of the method

the homology of the mapping class group 1 2 h btg a h mg q in this paper we will show that m g btg and b i l sg get more and more complicated as the genus g tends to infinity more precisely we will prove theorem 1 1 let q z2 z4 z6 denote the polynomial algebra of generators z2n in dimension in n 1 2 3 there are classes y2 y4 y2n with y2n in the 2nth cohomology group h 2n b i sg z such that the homomorphism of algebras sending z2n toy2n