bonnesenʼs inequality for john domains in rn abstract we prove sharp quantitative isoperimetric inequalities for john domains in r n we show that the bonnesen style inequalities hold true in r n under the john domain assumption which rules out cusps our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit we use the sharp quantitative inequalities proved in fusco et al 2008 7 and fuglede 1989 4 to reduce our problem to symmetric domains

the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

well conditioned asymptotic waveform evaluation for finite elements the frequency domain finite element method fem results in matrix equations that have polynomial dependence on the frequency of excitation for a wide band fast frequency sweep technique based on a moment matching model order reduction more process researchers generally take one of two approaches the first is to linearize the polynomial dependence which will either limit the bandwidth of accuracy or require the introduction of extra degrees of freedom and then use a well conditioned krylov subspace technique the second approach is to work directly with the polynomial matrix equation and use one of the available but ill conditioned asymptotic waveform evaluation awe methods for large scale fem simulations introducing extra degrees of freedom and therefore increasing the length of the more vectors and the amount of memory required is not desirable therefore the first approach is not alluring on the other hand an ill conditioned awe process is unattractive this paper presents a novel more technique for polynomial matrix equations that circumvents these problematic issues first this novel process does not require any additional unknowns second this process is well conditioned along with the presentation of the novel algorithm which is called well conditioned awe wcawe numerical examples modeled using the fem are given to illustrate its accuracy

regularity of isoperimetric sets in mathbb r 2 with density we consider the isoperimetric problem in mathbb r 2 with density we show that if the density is mathrm c 0 alpha then the boundary of any isoperimetric set is of class mathrm c 1 frac alpha 3 2 alpha this improves the previously known regularity

stability of polya szegő inequality for log concave functions abstract a quantitative version of polya szeg inequality is proven for log concave functions in the case of steiner and schwarz rearrangements

the sharp quantitative sobolev inequality for functions of bounded variation abstract the classical sobolev embedding theorem of the space of functions of bounded variation bv r n into l n r n is proved in a sharp quantitative form

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

sharp dimension free quantitative estimates for the gaussian isoperimetric inequality we provide a full quantitative version of the gaussian isoperimetric inequality the difference between the gaussian perimeter of a given set and a half space with the same mass controls the gap between the norms of the corresponding barycenters in particular it controls the gaussian measure of the symmetric difference between the set and the half space oriented so to have the barycenter in the same direction of the set our estimate is independent of the dimension sharp on the decay rate with respect to the gap and with optimal dependence on the mass

broadband model order reduction of polynomial matrix equations using single point well conditioned asymptotic waveform evaluation derivations and theory summary to eect a model order reduction more process on a system which has a polynomial matrix equation dependence on the more parameter researchers generally take one of two approaches therst is to linearize the system by introducing extra degrees of freedom and then to solve the resulting expanded linear system with a method such as lanczos or arnoldi the second approach is to work directly with the polynomial system and use a technique such as asymptotic waveform evaluation awe of course each approach has advantages and disadvantages in this paper a new technique will be presented which has some desirable characteristics from both approaches and which is able to circumvent simultaneously some of their disadvantages it can be shown that both the arnoldi and the awe methods are special cases of this new technique finally numerical results will show the viability of the new method which will be called the well conditioned asymptotic waveform evaluation wcawe method copyright 2003 john wiley sons ltd

the quantitative faber krahn inequality for the robin laplacian abstract we prove a quantitative form of the faber krahn inequality for the first eigenvalue of the laplace operator with robin boundary conditions the asymmetry term involves the square power of the fraenkel asymmetry multiplied by a constant depending on the robin parameter the dimension of the space and the measure of the set

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

faber krahn inequalities in sharp quantitative form the classical faber krahn inequality asserts that balls uniquely minimize the first eigenvalue of the dirichlet laplacian among sets with given volume in this paper we prove a sharp quantitative enhancement of this result thus confirming a conjecture by nadirashvili and bhattacharya weitsman more generally the result applies to every optimal poincare sobolev constant for the embeddings w 1 2 0 omega hookrightarrow l q omega

on the incompressible fluid limit and the vortex motion law of the nonlinear schrodinger equation the nonlinear schrodinger equation nls has been a fundamental model for understanding vortex motion in superfluids the vortex motion law has been formally derived on various physical grounds and has been around for almost half a century we study the nonlinear schrodinger equation in the incompressible fluid limit on a bounded domain with dirichlet or neumann boundary condition the initial condition contains any finite number of degree 1 vortices we prove that the nls linear momentum weakly converges to a solution of the incompressible euler equation away from the vortices if the initial nls energy is almost minimizing we show that the vortex motion obeys the classical kirchhoff law for fluid point vortices similar results hold for the entire plane and periodic cases and a related complex ginzburg landau equation we treat as well the semi classical wkb limit of nls in the presence of vortices in this limit sound waves propagate through steady vortices

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

characterization of electromagnetic devices via reduced order models abstract efficient procedures are presented for simultaneously characterizing the time and frequency domain behavior of 3d electromagnetic devices the procedures work in the complex frequency domain on either the finite element or boundary element formulation of maxwell s equations various approximation techniques are used to derive reduced order models describing the system transfer functions of the 3d device two different methods for evaluating reduced order models are presented one is called asymptotic waveform evaluation awe and is combined with the finite element method the other is called adaptive lanzcos pade sweep alps and is combined with the boundary element method the resulting reduced order models provide the frequency domain behavior of the device over a broad bandwidth using the inverse laplace transform these reduced order models can also provide the time domain behavior of the device several numerical examples have been run using commercial eda software to demonstrate that this solution procedure is a highly efficient and accurate way to characterize the electromagnetic performance of real life devices

on the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems it is the purpose of this paper to discuss the equality sign in d and in related inequalities in particular we shall give an answer to the following question does the equality in d imply that u u this question has been raised by g polya g szego in their book on isoperimetric inequalities l l p 186 they state that their method of proving d unfortunately leaves little hope for discussing the sign of equality another comment on this question can be found in 8 where e lieu conjectures that probably a strict version of d is true as it turns out the doubts of g polya g szeg were indeed justified because there are functions u for which equality holds in d while u does not coincide with u in section 1 of this paper i give an example of such a function on the other hand the conjecture of e lieb is also reasonable since generically i can give a positive answer to the question above there is a dense subset of nonnegative functions in wol 2 1 1 for which the equality in d implies

vortices in complex scalar fields abstract consider a spatially extended field which evolves in time according to a pde the solutions contain particle like defects whose motions are parts of the full dynamics we inquire into the formulation of an asymptotic particle field defect dynamics which derives from the full pde we carry out this program for two complex scalar field equations in two space dimensions the nonlinear schrodinger equation and the nonlinear heat equation the topological defects are zeros of the complex scalar field with nonzero integer winding numbers called vortices vortices evolving under the nonlinear schrodinger equation behave like point vortices in ideal fluid pairs of vortices evolving under the nonlinear heat equation with like opposite winding numbers undergo a repulsive attractive interaction

improved lp mixed volume inequality for convex bodies abstract a sharp quantitative version of the l p mixed volume inequality is established this is achieved by exploiting an improved jensen inequality this inequality is a generalization of pinsker csiszar kullback inequality for the tsallis entropy finally a sharp quantitative version of the l p brunn minkowski inequality is also proved as a corollary

on the schrodinger poisson slater system behavior of minimizers radial and nonradial cases this paper is motivated by the study of a version of the so called schrodinger poisson slater problem delta u omega u lambda left u 2 star frac 1 x right u u p 2 u where u in h 1 mathbb r 3 we are concerned mostly with p in 2 3 the behavior of radial minimizers motivates the study of the static case 0 among other things we obtain a general lower bound for the coulomb energy which could be useful in other frameworks the radial and nonradial cases turn out to yield essentially different situations

existence and stability for a non local isoperimetric model of charged liquid drops we consider a variational problem related to the shape of charged liquid drops at equilibrium we show that this problem never admits local minimizers with respect to l1 perturbations preserving the volume however we prove that the ball is stable under small c1 1 perturbations when the charge is small enough

on the stability of the polygonal isoperimetric inequality abstract we obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths the variance of its radii and its deviation from being convex our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices

the ionization conjecture in thomas fermi dirac von weizsacker theory we prove that in thomas fermi dirac von weizsacker theory a nucleus of charge z 0 can bind at most z c electrons where c is a universal constant this result is obtained through a comparison with thomas fermi theory which as a by product gives bounds on the screened nuclear potential and the radius of the minimizer a key ingredient of the proof is a novel technique to control the particles in the exterior region which also applies to the liquid drop model with a nuclear background potential c 2017 wiley periodicals inc

unique isoperimetric foliations of asymptotically flat manifolds in all dimensions the question of isoperimetry what is the largest amount of volume that can be enclosed by a given amount of area can be traced back to antiquity 1 the first mathematically rigorous results are as recent as the nineteenth century the question of isoperimetry and its close relative the analysis of minimal surfaces are two of the model problems of the geometric calculus of variations the list of geometries where an explicit answer to the question of isoperimetry is available is short we provide an overview of available results in appendix h in this paper and 22 we extend this list by a class of riemannian manifolds m g for which we describe all large isoperimetric regions completely we refer the reader to sect 2 for the precise definitions of all terms in the statement of our first main theorem

on periodic critical points and local minimizers of the ohta kawasaki functional abstract in this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem arising in the modeling of diblock copolymers in the main result by means of a purely variational procedure we show that it is possible to construct locally minimizing periodic critical points whose shape resemble that of any given strictly stable constant mean curvature periodic hypersurface along the way we establish several auxiliary results of independent interest

on an isoperimetric problem with a competing nonlocal term ii the general case this paper is the continuation of a previous paper h knupfer and c b muratov comm pure appl math 66 2013 1129 1162 we investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance in this work we treat the case of a general space dimension we obtain basic existence results for minimizers with sufficiently small masses for certain ranges of the exponent in the kernel we also obtain nonexistence results for sufficiently large masses as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality the physically important special case of three space dimensions and coulombic repulsion is included in all the results mentioned above in particular our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter in all cases the minimal energy scales linearly with mass for large masses even if the infimum of energy cannot be attained 2014 wiley periodicals inc

isoperimetry and stability properties of balls with respect to nonlocal energies we obtain a sharp quantitative isoperimetric inequality for nonlocal s perimeters uniform with respect to s bounded away from 0 this allows us to address local and global minimality properties of balls with respect to the volume constrained minimization of a free energy consisting of a nonlocal s perimeter plus a non local repulsive interaction term in the particular case s 1 the s perimeter coincides with the classical perimeter and our results improve the ones of knuepfer and muratov comm pure appl math 66 7 1129 1162 2013 comm pure appl math 2014 concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term more precisely their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts

complex ginzburg landau equations and dynamics of vortices filaments and codimension 2 submanifolds here we study the asymptotic behavior of solutions to the complex ginzburg landau equations and their associated heat flows in arbitrary dimensions when the ginzburg landau parameter tends to infinity we prove that the energies of solutions in the flow are concentrated at vortices in two dimensions filaments in three dimensions and codimension 2 submanifolds in higher dimensions moreover we show the dynamical laws for the motion of these vortices filaments and codimension 2 submanifolds away from the energy concentration sets we use some measure theoretic arguments to show the strong convergence of solutions in both static and heat flow cases 1998 john wiley sons inc

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

sharp and rigid isoperimetric inequalities in metric measure spaces with lower ricci curvature bounds we prove that if x mathsf d mathfrak m is a metric measure space with mathfrak m x 1 having in a synthetic sense ricci curvature bounded from below by k 0 and dimension bounded above by n in 1 infty then the classic levy gromov isoperimetric inequality together with the recent sharpening counterparts proved in the smooth setting by milman for any k in mathbb r n ge 1 and upper diameter bounds holds i e the isoperimetric profile function of x mathsf d mathfrak m is bounded from below by the isoperimetric profile of the model space moreover if equality is attained for some volume v in 0 1 and k is strictly positive then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions finally we also establish the almost rigidity if the equality is almost attained for some volume v in 0 1 and k is strictly positive then the space must be mgh close to a spherical suspension to our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying ricci curvature lower bounds examples of spaces fitting our assumptions include measured gromov hausdorff limits of riemannian manifolds satisfying ricci curvature lower bounds alexandrov spaces with curvature bounded from below finsler manifolds endowed with a strongly convex norm and satisfying ricci curvature lower bounds the result seems new even in these celebrated classes of spaces

on the hydrodynamic limit of ginzburg landau wave vortices in this paper we study the hydrodynamic limit of the finite ginzburg landau wave vortices which was established in 16 unlike the classical vortex method for incompressible euler equations we prove here that the densities approximated by the vortex blob method associated with the ginzburg landau wave vortices tend to the solutions of the pressure less compressible euler poisson equations the convergence of such approximation is proved before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates 2002 john wiley sons inc

stability in shape optimization with second variation we are interested in the question of stability in the eld of shape optimization precisely we prove that under structural assumptions on the hessian of the considered shape functions the necessary second order minimality conditions imply that the shape hessian is coercive for a given norm moreover under an additional continuity condition for the second order derivatives we derive precise optimality results in the class of regular perturbations of a domain these conditions are quite general and are satised for the volume the perimeter the torsional rigidity and the rst dirichlet eigenvalue of the laplace operator as an application we provide non trivial examples of inequalities obtained in this way

multifrequency pade approximation via jordan continued fraction expansion the multifrequency pade approximation of transfer functions is performed via the jordan type continued fraction expansion an efficient algorithm that requires no complex algebra is derived for expanding a transfer function into a jordan continued fraction about arbitrary points s or j omega sub j on the imaginary axis of the s plane also derived is a forward inversion algorithm for inverting a multifrequency jordan continued fraction expansion into a rational form the algorithms presented are amenable for obtaining a family of frequency response matched models of different orders for a high order transfer function via a single set of computations

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

lanczos versus subspace iteration for solution of eigenvalue problems in this paper we consider solution of the eigen problem in structural analysis using a recent version of the lanczos method and the subspace method the two methods are applied to examples and we conclude that the lanczos method has advantages which are too good to overlook

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

a quantitative isoperimetric inequality for fractional perimeters abstract recently frank and seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from sobolev seminorms of fractional order this isoperimetric inequality is improved here in a quantitative form

on the dynamical law of the ginzburg landau vortices on the plane we study the ginzburg landau equation on the plane with initial data being the product of n well separated 1 vortices and spatially decaying perturbations if the separation distances are o e 1 e 1 we prove that the n vortices do not move on the time scale o e 2 e e o log 1 e instead they move on the time scale o e 2 log 1 according to the law x j x jw w l6 j logjxl x jj x j j j 2 r 2 the location of the j th vortex the main ingredients of our proof consist of estimating the large space behavior of solutions a monotonicity inequality for the energy density of solutions and energy comparisons combining these we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law c 1999 john wiley sons inc

on an isoperimetric problem with power law potentials and external attraction abstract in this paper an isoperimetric problem with power law potentials and external attraction is considered using concentration compactness lemma for sets of finite perimeter which was introduced by frank and lieb we prove the existence of minimizers for the isoperimetric problem in the special case of quadratic attraction we also obtain that the ball is the unique minimizer of the isoperimetric problem

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

droplet minimizers of an isoperimetric problem with long range interactions we give a detailed description of the geometry of single droplet patterns in a non local isoperimetric problem in particular we focus on the sharp interface limit of the ohta kawasaki free energy for diblock copolymers regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long range interactions exploiting fine properties of the regularity theory for minimal surfaces we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns where u is the order parameter of a two phases system confined in r n and and m are two nonnegative numerical parameters the two terms in the energy mimic attractive short range and repulsive long range energies between the phases more precisely the first term is local it favors minimal interface area and drives the system toward a partition into few pure phases while the second term involving a coulomb like kernel g is non local and favor a fine mixing of the phases a detailed description of the energy is given in 1 the competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns whose geometry strongly depends on the choice of the parameters and m e g spherical spots cylinders gyroids and lamellae 0 1 the ohta kawasaki functional for diblock copolymers the model we consider arises as a simplification of a ginzburg landau functional proposed by ohta and kawasaki in their pioneering paper 41 as a possible description of a diblock copolymers dbc system even though it is questionable whether such an energy actually describes dbcs see choksi and ren 15 muratov 39 and niethammer and oshita 40 nevertheless it is a first and mathematically non trivial attempt to capture some of the main features of these systems for such a reason it deserved over the last twenty years great attention from both the mathematical and the physical

nonexistence of a minimizer for thomas fermi dirac von weizsacker model 1 x y dx dy a nonexistence result similar to theorem 1 also holds theorem 2 there exists constant v0 0 such that the variational problem 4 inf v e does not have a minimizer if v v0 if the coulomb term is not present in 1 and 3 a standard modica mortela type result establishes a link between the two functional through gamma convergence when the volume constraint m is large this is no longer valid for the energy functionals 1 and 3 due to the coulomb term the connection between the two functionals is just on the formal level

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

quantitative isoperimetry a la levy gromov on a riemannian manifold with a positive lower bound on the ricci tensor the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius the deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise these results are actually obtained in the more general context of possibly non smooth metric measure spaces with curvature dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method

proof of the ionization conjecture in a reduced hartree fock model the ionization conjecture for atomic models states that the ionization energy and maximal excess charge are bounded by constants independent of the nuclear charge we prove this for the hartree fock model without the exchange term

energy estimates and cavity interaction for a critical exponent cavitation model we consider the minimization of in a perforated domain of among maps that are incompressible det and invertible and satisfy a dirichlet boundary condition u g on if the volume enclosed by g is greater than any such deformation u is forced to map the small holes be ai onto macroscopically visible cavities which do not disappear as e 0 we restrict our attention to the critical exponent p n where the energy required for cavitation is of the order of and the model is suited therefore for an asymptotic analysis v1 vm denote the volumes of the cavities in the spirit of the analysis of vortices in ginzburg landau theory we obtain estimates for the renormalized energy showing its dependence on the size and the shape of the cavities on the initial distance between the cavitation points a1 am and on the distance from these points to the outer boundary based on those estimates we conclude for the case of two cavities that either the cavities prefer to be spherical in shape and well separated or to be very close to each other and appear as a single equivalent round cavity this is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence 2012 wiley periodicals inc

stability for a gns inequality and the log hls inequality with application to the critical mass keller segel equation starting from the quantitative stability result of bianchi and egnell for the 2 sobolev inequality we deduce several dierent stability results for a gagliardo nirenberg sobolev inequality in the plane then exploiting the connection between this inequality and a fast diusion equation we get stability for the log hls inequality finally using all these estimates we prove a quantitative convergence result for the critical mass keller segel system

the ionization conjecture in hartree fock theory we prove the ionization conjecture within the hartree fock theory of atoms more precisely we prove that if the nuclear charge is allowed to tend to infinity the maximal negative ionization charge and the ionization energy of atoms nevertheless remain bounded moreover we show that in hartree fock theory the radius of an atom properly defined is bounded independently of its nuclear charge

on the isoperimetric problem with respect to a mixed euclidean gaussian density abstract the isoperimetric problem with respect to the product type density e x 2 2 d x d y on the euclidean space r h r k is studied in particular existence symmetry and regularity of minimizers is proved in the special case k 1 also the shape of all the minimizers is derived finally a conjecture about the minimality of large cylinders in the case k 1 is formulated

traveling wave solutions of the schrodinger map equation we first construct traveling wave solutions for the schrodinger map in r 2 m t m m m 3 e 3 in 2 of the form m x 1 x 2 t where m has exactly two vortices at approximately 1 2 0 r 2 of degree 1 we use a perturbative approach that gives a complete characterization of the asymptotic behavior of the solutions with a few modifications a similar construction yields traveling wave solutions of schrodinger map equations in higher dimensions 2010 wiley periodicals inc

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ for convex bodies d in rn the deviation d from spherical shape is estimated from above in terms of the dimensionless isoperimetric deficiency a of d as follows d f a for a sufficiently small here f is an explicit elementary function vanishing continuously at 0 the estimate is sharp as regards the order of magnitude of f the dimensions n 2 and 3 present anomalies as to the form of f in the planar case n 2 the result is contained in an inequality due to t bonnesen a qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies d in rn in the sense that as d varies d p 0 for a a 0 the proof of the estimate d f a is based on a related estimate in the case of domains not necessarily convex that are supposed a priori to be nearly spherical in a certain sense

sharp stability theorems for the anisotropic sobolev and log sobolev inequalities on functions of bounded variation abstract combining rearrangement techniques with gromov s proof via optimal mass transportation of the 1 sobolev inequality we prove a sharp quantitative version of the anisotropic sobolev inequality on b v r n we also deduce as a corollary of this result a sharp stability estimate for the anisotropic 1 log sobolev inequality

from the ginzburg landau model to vortex lattice problems we introduce a coulombian renormalized energy w which is a logarithmic type of interaction between points in the plane computed by a renormalization we prove various of its properties such as the existence of minimizers and show in particular using results from number theory that among lattice configurations the triangular lattice is the unique minimizer its minimization in general remains open

study of island formation in epitaxially strained films on unbounded domains we consider a variational model related to the formation of islands in heteroepitaxial growth on unbounded domains we first derive the scaling regimes of the minimal energy in terms of the volume of the film and the amplitude of the crystallographic misfit for small volumes non existence of minimizers is then proven this corresponds to the experimentally observed wetting effect on the other hand we show the existence of minimizers for large volumes we finally study the asymptotic behavior of the optimal shapes