a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

approximation of biholomorphic mappings by automorphisms of cn in this paper we prove several results on approximation of biholomorphic mappings between domains in c n 2 by holomorphic automorphisms of c recall that the group of holomorphic automorphisms of c denoted aut c consists of those holomorphic mappings c c that have a holomorphic inverse 4 1 c c with the topology of uniform convergence on compact sets aut c is a topological group while the group aut c 1 of automorphisms of the complex plane consists only of the linear mappings az b a b e c a 4 0 the group aut c is very large and complicated when n 2 let us choose a linear coordinate system on c and write the coordinates as z z w where z z1 zn i c c n 1 and w z c then autc n contains mappings of the form

totally real immersions of surfaces first published in transactions of the american mathematical society in volume 362 and issue 1 published by the american mathematical society

embedding holomorphic discs through discrete sets the most interesting case is n 2 for n 3 this was proved by a different method in 16 for f c n and in 9 for all convex domains o c c n in the case when f2 c n 3 one can in addition prescribe a discrete set c c and require that f j z j for all j 16 for n 2 the methods of 9 and 16 only give proper holomorphic immersions of the disc through a given discrete set in 12 the main problem of course is that in dimension two one cannot remove self intersections o f complex curves by small deformations we do not know whether our theorem holds for non runge pseudoconvex domains in cn our methods do not seem to extend to this case in this direction it was proved in 6 that for every finite subset z in an arbitrary connected pseudoconvex domain f2 c c n n 1 there exist proper holomorphic mappings f d d o f the disc into 12 such that z c f d it is likely that a refinement of the construction in 6 gives proper holomorphic immersions f d f2 whose image contains a given discrete subset of 12 an example

corrigendum to the zariski topology on sets of semistar operations without finite type assumptions j algebra 513 2018 27 49 abstract we correct three issues in the paper the zariski topology on sets of semistar operations without finite type assumptions

a mixed finite element method for approximating incompressible materials a mixed finite element method is proposed for approximating the deformation u of a plane incompressible material satisfying the condition det nabla u 1 for piecewise linear shape functions it is proved that the discrete solutions exist and converge with nearly quasi optimal rates in l infty the corresponding nonlinear system is iteratively solved by a multigrid method

a numerical method for computing singular minimizers a numerical method with truncation methods as a special case for computing singular minimizers in variational problems is described it is proved that the method can avoid lavrentiev phenomenon and detect singular minimizers the convergence of the method is also established numerical results on a 2 d problem are given

the zariski topology on sets of semistar operations without finite type assumptions abstract we study from a topological point of view spaces of semistar operations without the hypothesis that they are of finite type in particular we analyze when the space of all semistar operations the space of spectral operations and the set of stable operations are spectral spaces

a numerical method for detecting singular minimizers a numerical method for computing minimizers in one dimensional problems of the calculus of variations is described such minimizers may have unbounded derivatives even when the integrand is smooth and regular in such cases because of the lavrentiev phenomenon standard finite element methods may fail to converge to a minimizer the scheme proposed is shown to converge to an absolute minimizer and is tested on an example the effect of quadrature is analyzed the implications for higher dimensional problems and in particular for fracture in nonlinear elasticity are discussed

criteria for the density property of complex manifolds in this paper we suggest new effective criteria for the density property this enables us to give a trivial proof of the original anders en lempert result and to establish almost free of charge the algebraic density property for all linear algebraic groups whose connected components are different from tori or c as another application of this approach we tackle the question asked among others by f forstneri v c about the density of algebraic vector fields on euclidean space vanishing on a codimension 2 subvariety

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

polynomially convex hulls of singular real manifolds we obtain local and global results on polynomially convex hulls of lagrangian and totally real submanifolds of c n with self intersections and open whitney umbrella points

a foliation of the ball by complete holomorphic discs we show that the open unit ball mathbb b n of mathbb c n n 1 admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs

the convergence of regularized minimizers for cavitation problems in nonlinear elasticity consider a nonlinearly elastic body which occupies the region omega subset mathbb r m m 2 3 in its reference state and which is held in tension under prescribed boundary displacements on partial omega let bf x 0 in omega be any fixed point in the body it is known from variational arguments that for sufficiently large boundary displacements there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at bf x 0 in the deformed body this is the phenomenon of cavitation for each epsilon 0 define the regularized domains omega epsilon omega backslash overline b epsilon bf x 0 which contain a preexisting hole of radius epsilon 0 centered on bf x 0 now consider the corresponding mixed displacement traction problem on omega epsilon in which the boundary partial omega is subject to the same boundary displacements and the deformed cavity surface i e the image of partial b epsilon is requi

an a priori error estimate for finite element discretizations in nonlinear elasticity for polyconvex materials under small loads we prove optimal a priori error estimates in w1 p for finite element minimizers of polyconvex energy functionals with small applied loads the proof relies on a quantitative version of zhang s stability estimate k zhang arch rat mech anal 114 1991 95 117

on closed invariant sets in local dynamics abstract we investigate the dynamical behaviour of a holomorphic map on an f invariant subset c of u where f u c k we study two cases when u is an open connected and polynomially convex subset of c k and c u closed in u and when u has a p s h barrier at each of its points and c is not relatively compact in u in the second part of the paper we prove a birkhoff s type theorem for holomorphic maps in several complex variables i e given an injective holomorphic map f defined in a neighborhood of u with u star shaped and f u a runge domain we prove the existence of a unique forward invariant maximal compact and connected subset of u which touches u

the numerical computation of singular minimizers in two dimensional elasticity in this paper we use a spectral collocation method for the computation of singular minimizers in two dimensional elasticity in particular we are interested in computing cavitating solutions those that open a hole at the center and determining whether or not they are globally stable in the energy sense we describe the spectral collocation method which is used in conjunction with a richardson extrapolation iteration and discuss various aspects related to the convergence of the method we obtain a variety of results for certain ranges of parameters that show how the size of the cavitation depends on these parameters

a fatou bieberbach domain in mathbb c 2 which is not runge since a paper by rosay and rudin trans am math soc 310 47 86 1988 there has been an open question whether all fatou bieberbach domains are runge we give an example of a fatou bieberbach domain in mathbb c 2 which is not runge the domain provides yet a negative answer to a problem of bremermann

density property for hypersurfaces uv p bar x we study hypersurfaces of mathbb c n 2 bar x u v given by equations of form uv p bar x where the zero locus of a polynomial p is smooth reduced the main result says that the lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the lie algebra of all algebraic vector fields consequences of this result for some conjectures of affine algebraic geometry and for the oka grauert gromov principle are discussed

error analysis of a dual parametric bi quadratic fem in cavitation computation in elasticity the orientation preservation conditions and approximation errors of a dual parametric bi quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed the analytical results allow us to establish based on an error equidistribution principle an optimal meshing strategy for the method in cavitation computation numerical results are in good agreement with the analytical results

an interpolation theorem for proper holomorphic embeddings given a stein manifold x of dimension n 1 a discrete sequence a j subset x and a discrete sequence b j subset mathbb c m where m ge n left frac 3n 2 right 1 there exists a proper holomorphic embedding f colon x hookrightarrow mathbb c m satisfying f a j b j for every j 1 2

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

oka properties of ball complements let n 1 be an integer we prove that holomorphic maps from stein manifolds x of dimension n to the complement mathbb c n setminus l of a compact convex set l subset mathbb c n satisfy the basic oka property with approximation and interpolation if l is polynomially convex then the same holds when 2 dim x le n we also construct proper holomorphic maps immersions and embeddings x rightarrow mathbb c n with additional control of the range thereby extending classical results of remmert bishop and narasimhan

carleman approximation by holomorphic automorphisms of ℂ n we approximate smooth maps defined on non compact totally real manifolds by holomorphic automorphisms of c

some topological considerations on semistar operations abstract we consider properties and applications of a new topology called the zariski topology on the space sstar a of all the semistar operations on an integral domain a we prove that the set of all overrings of a endowed with the classical zariski topology is homeomorphic to a subspace of sstar a the topology on sstar a provides a general theory through which we see several algebraic properties of semistar operation as very particular cases of our construction moreover we show that the subspace sstar f a of all the semistar operations of finite type on a is a spectral space

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

ten remarks on nonconvex minimisation for phase transition simulations nonconvex minimisation problems are encountered in many applications such as phase transitions in solids or liquids but also in optimal design tasks or micromagnetism in contrast to rubber type elastic materials and many other variational problems in continuum mechanics the minimal energy may be not attained in the sense of sobolev functions the nonrank 1 convex minimisation problem m is ill posed the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures their limit is a measure called gradient young measure gym and describes the effective energy density w qc the quasiconvex envelope of the original energy density w this gives rise to a relaxed minimisation problem r which is well posed in the sense that the minimum is attained the paper compares computational aspects of the two problems m and r neither adaptive finite element methods nor effective solvers may work for discrete versions of m but theoretical and numerical evidence supports that they work for r the drawback of r is that the relaxed energy density w qc is not always given by a known closed form formula instead a numerical relaxation has to be involved which is of the form m and hence yields a computational challenge other semiconvexity notions are requested and give rise to a new category of algorithms the demand for quasiconvexification algorithms is highly motivated by recent models in finite plasticity as the time discretisation in the latter models typically leads to nonrank 1 convex minimisation problems 2004 published by elsevier b v

algebraic density property of danilov gizatullin surfaces a danilov gizatullin surface is an affine surface v which is the complement of an ample section s for the ruling of a hirzebruch surface the remarkable theorem of danilov and gizatullin states that the isomorphism class of v depends only on the self intersection number s s in this paper we apply the theorem of danilov gizatullin to prove that the lie algebra generated by the complete algebraic vector fields on v coincides with the set of all algebraic vector fields of v