asymptotic behavior of the growth fragmentation equation with bounded fragmentation rate we are interested in the large time behavior of the solutions to the growth fragmentation equation we work in the space of integrable functions weighted with the principal dual eigenfunction of the growth fragmentation operator this space is the largest one in which we can expect convergence to the steady size distribution although this convergence is known to occur under fairly general conditions on the coefficients of the equation we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded first we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the dyson phillips series then we extend it to general kernels by using the notion of quasi compactness and the fact that it is a topological invariant

on the net reproduction rate of continuous structured populations with distributed states at birth we consider a nonlinear structured population model with a distributed recruitment term the question of the existence of non trivial steady states can be treated at least in three different ways one approach is to study spectral properties of a parametrised family of unbounded operators the alternative approach which we develop here is based on the reformulation of the partial differential equation as an integral equation in this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity finally we discuss a third approach which is based on a finite rank approximation of the recruitment operator

bulk and interfacial energy densities for structured deformations of continua to provide a basis for the application of techniques in the calculus of variations to non classical deformations of continua these structured deformations of continua are suitable for describing deformations of materials whose kine matics warrants analysis at both the macroscopic and microscopic levels the motivation for this work lies in the study of equilibrium configura tions of crystals with defects in a defective crystal the macroscopic defor mation together with the referential bravais lattice configuration do not suace to describe fully the configuration of a deformed body phenomena such as slips vacancies and dislocations may be present in the deformed bravais lattice basis thus preventing the use of the cauchy born hypoth esis as described below in a perfect crystal it is postulated that the crystal lattice consists of identical atoms located at all positions vectors

on a nonlocal reaction diffusion advection system modelling the growth of phytoplankton with cell quota structure abstract phytoplankton species in a water column compete for mineral nutrients and light and the existing models usually neglect differences in the nutrient content and the amount of light absorbed of individuals in this current paper we examine a size structured and nonlocal reaction diffusion advection system which describes the dynamics of a single phytoplankton species in a water column where the species depends simply on light for its growth our model is under the assumption that the amount of light absorbed by individuals is proportional to cell size which varies for populations that reproduce by simple division into two equally sized daughters we first establish the existence of a critical death rate and our analysis indicates that the phytoplankton survives if and only if its death rate is less than the critical death rate the critical death rate depends on a general reproductive rate the characteristics of the water column e g turbulent diffusion rate sinking depth cell growth cell division and cell size

existence of 2d skyrmions in this paper we consider the 2d skyrme model e u frac 1 2 int limits r 2 du 2dx frac lambda 4 int limits r 2 du wedge du 2dx frac mu 16 int limits r 2 u bf n 4dx where and 0 are positive coupling constants and n 0 0 1 is the north pole of s2 we derive a lower bound of 2d skyrme model using this estimate we prove the existence of 2d skyrmion for any positive coupling constants

dynamics of structured populations with inherited properties abstract a mathematical model of an age structured proliferating cell population is analyzed the model allows the inheritance of cycle length from mother cells to daughter cells it is proved that the solutions have the property of asynchronous exponential growth provided the transition kernel of inheritance satisfies smoothness and positivity requirements the method of proof uses the theory of semigroups of operators and the theory of positive operators

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

exponential decay for the fragmentation or cell division equation abstract we consider a classical integro differential equation that arises in various applications as a model for cell division or fragmentation in biology it describes the evolution of the density of cells that grow and divide we prove the existence of a stable steady distribution first positive eigenvector under general assumptions in the variable coefficients case we also prove the exponential convergence for large times of solutions toward such a steady state

energy splitting substantial inequality and minimization for the faddeev and skyrme models in this paper we prove that the faddeev energy e 1 at the unit hopf charge is attainable the proof is based on utilizing an important inequality called the substantial inequality in our previous paper which describes how the faddeev energy splits into its sublevels in terms of energy and topology when compactness fails with the help of an optimal sobolev estimate of the faddeev energy lower bound and an upper bound of e 1 we show that e 1 is attainable for the two dimensional skyrme model we prove that the substantial inequality is also valid which allows us to greatly improve the range of the coupling parameters for the existence of unit charge solitons previously guaranteed in a smaller range of the coupling parameters by the validity of the concentration compactness method

existence of two dimensional skyrmions via the concentration compactness method in this paper we prove the existence of a unit charge energy minimizer in the two dimensional skyrme model by the method of concentration compactness 2004 wiley periodicals inc

an eigenvalue problem related to cell growth abstract in this paper the eigenvalues of the operator corresponding to the partial differential equation which describes the evolution of a population reproducing by simple fission are investigated this is done by transforming the eigenvalue problem to an integral equation the theory concerning positive operators on a banach space appears to be very useful

exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition the coagulation fragmentation equation describes the concentration f i t of particles of size i in nn 0 at time t geq 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break up we show that when the rate of fragmentation is sufficiently stronger than that of coalescence f i t i in nn 0 tends to an unique equilibrium as t tends to infinity although we suppose that the initial datum is sufficiently small we do not assume a detailed balance or reversibility condition the rate of convergence we obtain is furthermore exponential

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

optimal design in elasticity and plasticity a direct weight minimization subject to compliance constraints or plastic yielding constraints leads to a non convex variational problem both the theoretical and the numerical analysis are unsatisfactory the minimum weight is not achieved by any design and the approximate designs oscillate as the element mesh is refined we look for equivalent relaxed problems with the same minima they come from allowing composite materials constructed in an optimal way from the original materials the constructions are different for elasticity and plasticity but surprisingly the final relaxed problems are in some cases the same

α and β curves sister sister and mother daughter correlations in cell population dynamics abstract the and curves and sister sister and mother daughter cycle length correlations are analyzed for a general model of age and size structured cell population dynamics some basic facts concerning these concepts are established a criterion is given for log a and log a to have asymptotically equal slopes the significance of size dependence upon sister sister and mother daughter correlations is demonstrated some numerical examples are given to illustrate the concepts

on the distributional jacobian of maps from sn into sn in fractional sobolev and holder spaces h brezis and l nirenberg proved that if gk c 0 s n s n and g2 c 0 s n s n n 1 are such that gk g in bmo s n then deggk degg on the other hand if g 2 c 1 s n s n then kronecker s formula asserts that degg 1 jsnj r sn det rg d consequently r sn det rgk d converges to r sn det rg d provided gk g in bmo s n in the same spirit we consider the quantity j g r sn det rg d for all 2 c 1 s n r and study the convergence of j gk in particular we prove that j gk converges to j g for any 2 c 1 s n r if gk converges to g in c 0 s n for some n 1 n surprisingly this result is optimal when n 1 in the case n 1 we prove that if gk g almost everywhere and lim supk 1jgk gjbmo is suciently small then j gk j g for any 2 c 1 s 1 r we also establish bounds for j g which are motivated by the works of j bourgain h brezis and h m nguyen and h m nguyen we pay special attention to the case n 1

existence criteria for baby skyrmions for a wide range of potentials abstract topologically stable field configurations are an important subject of study in various areas such as magnetism condensed matter physics nuclear physics liquid crystals in the baby skyrme models the two dimensional versions of the skyrme model of nuclear physics such structures are called baby skyrmions and they have been widely studied by means of numerical methods the aim of this paper is to give a simple criterion that guarantees the existence of skyrmions with unit topological charge for most baby skyrme models considered in the literature the proof of this criterion is purely analytic and it is based on the concentration of the energy of minimizing sequences

optimal design of fractured media with prescribed macroscopic strain abstract in this work we consider an optimal design problem for two component fractured media for which a macroscopic strain is prescribed within the framework of structured deformations we derive an integral representation for the relaxed energy functional we start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material the relaxed energy densities obtained via a blow up method are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks this model has the far reaching perspective to incorporate elements of plasticity in optimal design of composite media

on the stability of the cell size distribution ii time periodic developmental rates abstract a deterministic model for the growth of a size structured proliferating cell population is analyzed the developmental rates are allowed to vary with time for periodically varying rates stability of the cell size distribution is shown under similar conditions for the growth rate of individual cells as found before in the time homogeneous case strongly positive quasicompact linear operators on banach lattices serve as powerful abstract tools finally the autonomous case is revisited and the conditions for stability found in 1 are relaxed

skyrme models with self dual limits d 2 3 the most general skyrme sigma models in two and three euclidean dimensions described by o 3 and o 4 fields respectively are studied first by numerical methods and analytic proofs of existence are subsequently given particular emphasis is given to the special cases of these models where the topological inequalities can be saturated by self duality equations the o d 1 models in d dimensions exhibit qualitatively similar features

relaxation of quasiconvex functional in bv ω ℝp for integrands f x u u in this paper it is shown that if p x u is a quasiconvex function with linear growth then the relaxed functional in bv p of u to int limits omega f x u x nabla u x dx with respect to the l1 topology has an integral representation of the form begin gathered mathfrak f u int limits omega f x u x nabla u x dx int limits sigma u k x u x u x v x dh n 1 x hfill int limits omega f infty x u x dc u hfill end gathered where du u dx u u v dhn 1l u c u the proof relies on a blow up argument introduced by fonseca muller in the case where u w1 1 and on a recent result by alberti showing that the cantor part c u is rank one valued

on a feynman kac approach to growth fragmentation semigroups and their asymptotic behaviors abstract this work develops further a probabilist approach to the asymptotic behavior of growth fragmentation semigroups via the feynman kac formula which was introduced in a joint article with a r watson 6 here it is first shown that the sufficient condition for a malthusian behavior which was established in 6 is also necessary we then provide a simple criterion to ensure exponential speed of convergence which enables us to treat cases that were not covered previously in the literature