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

on a class of degenerate parabolic equations with dynamic boundary conditions we consider a quasi linear parabolic possibly degenerate equation with nonlinear dynamic boundary conditions the corresponding class of initial and boundary value problems has already been studied previously proving well posedness of weak solutions and the existence of the global attractor assuming that the nonlinearities are subcritical to a given exponent the goal of this article is to show that the previous analysis can be redone for supercritical nonlinearities by proving an additional l estimate on the solutions in particular we derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved even when both the nonlinear sources contribute in opposite directions then we show how to construct a trajectory attractor for the weak solutions of the associated parabolic system and prove that any solution belonging to the attractor is bounded which implies uniqueness finally we also prove for the semilinear reaction diffusion equation with nonlinear dynamic boundary conditions that the fractal dimension of the global attractor is of the order n 1 as diffusion 0 in any space dimension n 2 improving some recent results in gal 2012 23

the existence of multiple equilibrium points in a global attractor for some p laplacian equation abstract in this paper we are mainly concerned with some properties of the global attractor for some p laplacian equation with a lyapunov function in a banach space under some suitable assumptions we prove the existence of multiple equilibrium points in a global attractor for some p laplacian equation

attractors for the semilinear reaction diffusion equation with distribution derivatives in unbounded domains abstract in this paper we prove the existence of global attractors for a nonlinear reaction diffusion equation with a nonlinearity having a polynomial growth of arbitrary order p 1 p 2 and with distribution derivatives in the inhomogeneous term the global attractors are obtained in l 2 r n and l p r n respectively a new a priori estimate method has been used since the solutions of the equation have no higher regularity and the semigroup associated with the solutions is not continuous in l p r n the results are new and appear to be optimal

random attractors for singular stochastic evolution equations abstract the existence of random attractors for singular stochastic evolution equations see perturbed by general additive noise is proven the drift is only assumed to satisfy the standard assumptions of the variational approach to see with compact embeddings in the gelfand triple and singular coercivity for ergodic monotone contractive random dynamical systems it is proven that the random attractor consists of a single random point in case of real linear multiplicative noise finite time extinction is obtained applications include stochastic generalized fast diffusion equations and stochastic generalized singular p laplace equations perturbed by levy noise with jump measure having finite first and second moments

global attractors for p laplacian equation abstract the existence of a l 2 w 0 1 p l q global attractor is proved for the p laplacian equation u t div u p 2 u f u g on a bounded domain r n n 3 with dirichlet boundary condition where p 2 the nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c 1 u q k f u u c 2 u q k and f u l where q 2 is arbitrary there is no other restriction on p and q the asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method called asymptotic a priori estimate

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

asymptotic behaviour of non linear parabolic equations with monotone principal part abstract we consider non linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non lipschitz perturbations the case of globally lipschitz perturbations of a maximal monotone operator has been addressed in boll un mat ital b 8 2 2000 693 706 in the case of perturbations which are not globally lipschitz the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors we overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed the results developed here are applied to the case when the subdifferential operator is the p laplacian to obtain existence of attractors and the existence of periodic solutions

the frobenius norm and the commutator in an earlier paper we conjectured an inequality for the frobenius norm of the commutator of two matrices this conjecture was recently proved by seak weng vong and xiao qing jin and independently also by zhiqin lu we here give a completely different proof of this inequality prove some related results and embark on the corresponding question for unitarily invariant norms

on the z2 index of the global attractor for a class of p laplacian equations by means of the z 2 index we study the geometrical property of the global attractor for a class of symmetric p laplacian equations u t div vertical bar del u vertical bar p 2 del u vertical bar u vertical bar r 2 u lambda vertical bar u vertical bar s 2 u f x u 0 where lambda 0 2 s p r we first provide under suitable assumptions the existence result for the symmetric global attractor a then when 2 s p r we prove that the z 2 index of the global attractor is infinite as a corollary we obtain that the fractal dimension of the global attractor is infinite for when 2 s p r we provide the lower bound of the z 2 index of the global attractor c 2010 elsevier ltd all rights reserved

global attractors for one dimensional p laplacian equation abstract we study the existence of the l 2 r w 1 p r global attractor for an evolution p laplacian equation under additional restrictions on force and source terms we improve the result of a kh khanmamedov existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain j math anal appl 316 2006 601 615 and correct the result of m yang c sun c zhong existence of a global attractor for a p laplacian equation in r n nonlinear anal 66 2007 1 13 in the one dimensional case

existence of a global attractor for a p laplacian equation in rn abstract in this paper we prove the existence of the l 2 r n w 1 p r n l q r n global attractor for the p laplacian equation u t div u p 2 u u p 2 u f u g with a more general nonlinear term f f 1 a x f 2 in r n where a l 1 r n l r n and f 1 f 2 satisfy the arbitrary q order polynomial growth condition without any restriction on q p q 2 p 2

global attractors for nonlinear parabolic equations with nonstandard growth and irregular data abstract we investigate the large time behavior of solutions to the following nonlinear parabolic equations u t div u p x 2 u f x u g in r u 0 on r u x 0 u 0 x in where u 0 g l 1 we first provide the existence and uniqueness of an entropy solution for the problem then through some delicate analysis we establish some regularity results on the solution by which we prove the existence of a global attractor for the solution semigroup

comparison results for nonlinear parabolic equations with monotone principal part in this work we prove an abstract comparison principle for abstract quasi linear parabolic problems with monotone principal part this result is applied to parabolic problems having the p laplacian as principal part and with non lipschitz perturbations

h1 random attractors for stochastic reaction diffusion equations with additive noise abstract in this paper a new theorem obtained by li and guo y li b guo random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations j differential equations 245 7 2008 1775 1800 is employed to some reaction diffusion equations with additive noise we prove the unique existence of d random attractors in h 0 1 for the corresponding random dynamical system we also obtain the existence of a unique random equilibrium for the generated random dynamical system under some restrictive assumption on the coefficients

uniform attractors for non autonomous p laplacian equations abstract in this paper by the galerkin method we give the existence of solutions for the non autonomous p laplacian equation u t div u p 2 u f u g t after that we explore the asymptotic behavior of the equation the existence and the structures of the l 2 l 2 uniform attractor and the l 2 l q w 0 1 p uniform attractor are proved under the conditions below the nonlinear term f is supposed to satisfy the polynomial condition of arbitrary order c 1 u q k f u u c 2 u q k and f u l where q 2 is arbitrary and the external force g t in l loc 2 r l s s 2 is translation bounded and uniformly bounded in l s with respect to t r

variance bounds with an application to norm bounds for commutators abstract murthy and sethi m n murthy v k sethi sankhya ser b 27 1965 201 210 gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable we generalise this bound to the complex case and more importantly to the matrix case in doing so we make contact with several geometrical and matrix analytical concepts such as the numerical range and introduce the new concept of radius of a matrix we also give a new and simplified proof for a sharp upper bound on the frobenius norm of commutators recently proven by bottcher and wenzel a bottcher d wenzel the frobenius norm and the commutator linear algebra appl 429 2008 1864 1885 and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced as an immediate application of our variance bounds we obtain stronger versions of bottcher and wenzel s upper bound

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

impressions of convexity an illustration for commutator bounds abstract we determine the sharpest constant c p q r such that for all complex matrices x and y and for schatten p q and r norms the inequality xy yx p c p q r x q y r is valid the main theoretical tool in our investigations is complex interpolation theory

long time random dynamics of stochastic parabolic p laplacian equations on rn abstract in this article we study the long time random dynamics of non autonomous stochastic parabolic p laplacian equations driven by a wiener type multiplicative noise by a new asymptotic a priori estimate method not involving the plus or minus sign of the nonlinearity the omega limit compact of solutions in l p r n l q r n is proved for any p q 2 and n 1 where q is the polynomial growth exponent of the nonlinearity if 2 p n by a boot strap technique we obtain the higher order integrability of difference of solutions near the initial time as an application we show that the obtained l 2 random attractor is attracting in the topology of l 2 r n for any 0

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

asymptotic behaviour of nonlocal p laplacian reaction diffusion problems abstract in this paper we focus on studying the existence of attractors in the phase spaces l 2 and l p among others for time dependent p laplacian equations with nonlocal diffusion and nonlinearities of reaction diffusion type firstly we prove the existence of weak solutions making use of a change of variable which allows us to get rid of the nonlocal operator in the diffusion term thereupon the regularising effect of the equation is shown applying an argument of a posteriori regularity since under the assumptions made we cannot guarantee the uniqueness of weak solutions in addition this argument allows to ensure the existence of an absorbing family in w 0 1 p this leads to the existence of the minimal pullback attractors in l 2 l p and some other spaces as l p relationships between these families are also established

long time behavior for a nonlinear parabolic problem with variable exponents abstract this paper addresses the question of the asymptotic behavior of solutions to the p x laplacian problem u t div u p x 2 u f x u g with general assumptions on f x u and the exponent p x we prove the existence of global attractors in proper spaces then we consider the fractal dimension of global attractors for the problem under suitable conditions we show that the problem admits an infinite dimensional global attractor

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

on schatten p norms of commutators abstract for any odd integer d 3 we determine the sharpest constant c p q r such that x y y x p c p q r x q y r for all x y m d where m d denotes the set of all d d complex matrices p 1 p denotes the schatten p norm on m d and 1 p q r satisfy 1 p 1 q 1 r this is a continuation of the study of the problem considered in wenzel and audenaert 2010 8

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

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

global attractors for the p laplacian equations with nonregular data abstract let be a smooth bounded domain in r n n 3 we consider the long time behavior of solutions to the p laplacian equation u t p u f u g where 2 p n p div u p 2 u is the p laplace operator assume that g and the initial condition u 0 lie in l 1 f r r is of class c 1 and satisfies proper growth conditions firstly we prove the uniqueness of the entropy solution and establish some regularity results then we show the existence of a global attractor a in l r 1 w 0 1 s with s max n p 1 n 1 p r 1 r to obtain the results a decomposition method and a bootstrap technique are used

random dynamics of stochastic p laplacian equations on rn with an unbounded additive noise abstract this article is concerned with the random dynamics of solutions of the non autonomous parabolic p laplacian equations on r n driven by an unbounded additive noise where the nonlinearity has a p q growth exponents including the general polynomial type functions firstly by using an inductive method we establish the higher order integrability of difference of solutions near the initial time for arbitrary space dimension secondly based on this very estimate the asymptotical compactness of solutions in l p r n l q r n for any p q 2 is proved by a new technique instead of the usual truncation estimate method and then we obtain the existence of pullback attractor in l p r n l q r n for p laplacian equations with an unbounded additive noise thirdly we prove that the obtained l 2 pullback attractor is attracting and translation bounded in l r n topology for any 2 and arbitrary space dimension n n 1

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 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

asymptotic regularity for p laplacian equation this paper is devoted to proving some asymptotic regularity of the solutions of the p laplacian equation ut div u p 2 u f u g x p 2 n considered on a bounded domain rn n 3 the nonlinear term f satisfies the polynomial growth condition of arbitrary order c1 u q k f u u c2 u q k where q 2 is arbitrary as an application of the asymptotic regularity results we not only can obtain the existence of a l2 w01 p lq global attractor a immediately but also can show further that a attracts every bounded subsets of l2 under the w01 p lq norm for any 0 furthermore the fractal dimension of a is finite in lq for any 0

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

attractors for parabolic equations with dynamic boundary conditions abstract in this paper we prove existence of global solutions and l 2 l 2 h 1 l p l p global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary where there is no other restriction on p 2

long time behavior for quasilinear parabolic equations involving weighted p laplacian operators abstract in this paper we study the first initial boundary value problem for a class of quasilinear degenerate parabolic equations involving weighted p laplacian operators the long time behavior of solutions to that problem is considered via the concept of global attractors for multi valued semiflows

asymptotic regularity for some dissipative equations abstract this paper is devoted to proving some asymptotic regularity for both reaction diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity which excel the sharp regularity allowed by the corresponding stationary equations equilibrium points based on this regularity the existence of the finite dimensional global and exponential attractors can be obtained easily