a zero thickness limit of multilayer structures a resonant tunnelling δ potential a zero thickness limit for two terminal and three terminal devices from the quantum electronics domain is analysed the study is focused on heterostructures composed of a single barrier with adjacent one or two prewells the point interactions obtained in this limit are shown to be described by a family of resonant diagonal matrices that connect the two sided boundary conditions at the device origin which are a subclass of the whole four parameter family of point interactions transmission through such a device is absent almost everywhere except at a few points whose number and position can be controlled by a gate voltage applied externally to the barrier subsystem it is remarkable that the existence of resonances in the zero thickness limit occurs only if a squeezing sequence is constructed in a like way in this case the limit describes adequately the resonant behaviour of a barrier well heterostructure with realistic parameters simple analytical expressions obtained for resonance sets are supported by direct numerical calculations of the transmission being in agreement with the results of experiments on semiconductor devices the zero thickness like limiting procedure can be used in the design of nanodevices or contacting quantum wires

conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

removal of boundary singularities of pseudo holomorphic curves with lagrangian boundary conditions we give a proof of the theorem of removing isolated singularities of pseudo holomorphic curves with lagrangian boundary conditions and bounded symplectic area the proof is a combination of some lp type estimates standard techniques of geometric p d e and some ideas from symplectic geometry and calibration theory

the plateau problem in alexandrov spaces we study the plateau problem of finding an area minimizing disk bounding a given jordan curve in alexandrov spaces with curvature these are complete metric spaces with a lower curvature bound given in terms of triangle comparison imposing an additional condition that is satisfied by all alexandrov spaces according to a conjecture of perel man we develop a harmonic map theory from two dimensional domains into these spaces in particular we show that the solution to the dirichlet problem from a disk is holder continuous in the interior and continuous up to the boundary using this theory we solve the plateau problem in this setting generalizing classical results in euclidean space due to j douglas and t rado and in riemannian manifolds due to c b morrey

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

casimir effect and global theory of boundary conditions the consistency of quantum field theories defined on domains with external borders imposes very restrictive constraints on the type of boundary conditions that the fields can satisfy we analyse the global geometrical and topological properties of the space of all possible boundary conditions for scalar quantum field theories the variation of the casimir energy under the change of boundary conditions reveals the existence of singularities generically associated with boundary conditions which either involve topology changes of the underlying physical space or edge states with unbounded below classical energy the effect can be understood in terms of a new type of maslov index associated with the non trivial topology of the space of boundary conditions we also analyse the global aspects of the renormalization group flow t duality and the conformal invariance of the corresponding fixed points

the acoustic limit for the boltzmann equation the acoustic equations are the linearization of the compressible euler equations about a spatially homogeneous fluid state we first derive them directly from the boltzmann equation as the formal limit of moment equations for an appropriately scaled family of boltzmann solutions we then establish this limit for the boltzmann equation considered over a periodic spatial domain for bounded collision kernels appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that converge entropically and hence strongly in l 1 to a unique limit governed by a solution of the acoustic equations for all time provided that its initial fluctuations converge entropically to an appropriate limit associated to any given l 2 initial data of the acoustic equations the associated local conservation laws are recovered in the limit

the classification of cmc foliations of bbb r 3 and bbb s 3 with countably many singularities in this paper we generalize our previous local removable singularity theorem for minimal laminations to the case of weak h laminations with h in bbb r constant in a punctured ball of a riemannian three manifold we also obtain a curvature estimate for any weak cmc foliation with possibly varying constant mean curvature from leaf to leaf of a compact riemannian three manifold n with boundary solely in terms of a bound of the absolute sectional curvature of n and of the distance to the boundary of n we then apply these results to classify weak cmc foliations of bbb r 3 and bbb s 3 with a closed countable set of singularities

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

lattice boltzmann method for the compressible euler equations the lattice boltzmann model for the compressible euler equations is proposed together with its rigorous theoretical background the proposed model has completely overcome the defects of the previous model that the specific heat ratio cannot be chosen freely the macroscopic variables obtained from the solution are shown to satisfy in the limit of the small knudsen number the compressible euler equations if the variation of the solution is moderate this is the case where no shock waves or contact discontinuities appear in contrast when the solution makes steep variation at several localized regions due to the appearance of shock waves and contact discontinuities the corresponding macroscopic variables satisfy the weak form of the euler equations their derivation is carried out rigorously by taking into account the scale of variation of the solution correctly this is the first study that has laid the theoretical foundation of the lattice boltzmann model for the simulation of flows with shock waves and contact discontinuities numerical examples and the error estimates are also given which are consistent with the above theoretical arguments

the dirichlet problem for h systems with small boundary data blowup phenomena and nonexistence results given h 3 of class c1 and bounded we consider a sequence un of solutions of the h system open image in new window in the unit open disc open image in new window satisfying the boundary condition un n on open image in new window in the first part of this paper assuming that un is bounded in h1 open image in new window 3 we study the behavior of un when the boundary data n shrink to zero we show that either un 0 strongly in h1 open image in new window 3 or un blows up at least one h bubble namely a nonconstant conformal solution of the h system on 2 under additional assumptions on h we can obtain more precise information on the blow up in the second part of this paper we investigate the multiplicity of solutions for the dirichlet problem on the disc with small boundary datum we detect a family of nonconstant functions h even close to a nonzero constant in any reasonable topology for which the dirichlet problem cannot admit a large solution at a mountain pass level when the boundary datum is small

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

a delta well with a mass jump in this paper it is shown that a one dimensional hamiltonian with an attractive delta potential at the origin plus a mass jump at the same point cannot have a bound state as is the case with the ordinary attractive delta potential with constant mass unless another term is added into the potential in the form of a derivative of the dirac delta at the origin the consistency of this singular potential with two terms is guaranteed by choosing suitable matching conditions at the singular point for the wavefunctions furthermore it is proved that the self adjointness of the hamiltonian with both singular interactions determines the coefficient of the derivative of the delta in a unique manner under these conditions the bound state and its energy are obtained and it is checked that the correct results in the limit of equal masses are obtained

conical casimir pistons with hybrid boundary conditions in this paper we compute the casimir energy and force for massless scalar fields endowed with hybrid boundary conditions in the setting of the bounded generalized cone by using spectral zeta function regularization methods we obtain explicit expressions for the casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston our general formulas are subsequently specialized to the case in which the piston is modeled by a d dimensional sphere in this particular situation explicit results are given for d 2 3 4 5

an onsager like relation for the lattice boltzmann method an onsager like relation is proposed as a new criterion for constructing and analyzing the lattice boltzmann lb method for lb models obeying the relation we analyze their linearized stability establish their diffusive limit and find new constraints for those with free parameters the new relation seems of fundamental importance for the lb method

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

structure theorems for singular minimal laminations in this paper we apply our local removable singularity theorem and local structure theorems for embedded minimal surfaces and minimal laminations in r 3 proven in 16 15 to obtain global structure theorems for certain possibly singular minimal laminations of r 3 we will use theorems 1 3 and theorem 1 6 below in 14 to prove that a complete embedded minimal surface in r 3 with finite genus and a countable number of ends is proper theorem 1 6 will also be applied in 13 to obtain bounds on the index and the topology of complete embedded minimal surfaces of fixed genus and finite topology in r 3

a stability notion for lattice boltzmann equations lattice boltzmann equations are usually constructed to satisfy some physical requirements such as galilean invariance and isotropy to possess a velocity independent pressure and no compressible effects and so on in this paper a stability requirement is introduced as a new criterion for the constructions with this requirement we derive some relations of parameters for several lattice boltzmann models interestingly these relations are satisfied by many choices of parameters used in the literature

weighted mathbb l 2 stability of the lattice boltzmann method this article is concerned with the linearized stability of the lattice boltzmann method both on periodic domains and on bounded domains with the bounce back rule used at the boundaries under a structural hypothesis we prove that a weighted mathbb l 2 norm of the solutions to the linearized lattice boltzmann method is decreasing with time moreover we show that the structural hypothesis holds true for many lattice boltzmann models

the discrete spectrum of the spinless one dimensional salpeter hamiltonian perturbed by δ interactions we rigorously define the self adjoint one dimensional salpeter hamiltonian perturbed by an attractive of strength centred at the origin by explicitly providing its resolvent our approach is based on a coupling constant renormalization a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the in two and three dimensions we show that the spectrum of the self adjoint operator consists of the absolutely continuous spectrum of the free salpeter hamiltonian and an eigenvalue given by a smooth function of the parameter the method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self adjoint hamiltonian consists of two eigenvalues namely the ground state energy and that of the excited antisymmetric state we investigate in detail the dependence of these two eigenvalues on the two parameters of the model that is to say both the aforementioned strength and the separation distance with regard to the latter a remarkable phenomenon is observed differently from the well behaved schrodinger case the 1d salpeter hamiltonian with two identical dirac distributions symmetrically situated with respect to the origin does not converge as the separation distance shrinks to zero to the one with a single centred at the origin having twice the strength however the expected behaviour in the limit in the norm resolvent sense can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself

on the stability structure for lattice boltzmann schemes the stability structure for lattice boltzmann schemes has been introduced in banda et al 2006 16 junk and yong 2007 14 to analyze the stability of numerical algorithms the first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis its second goal is to illustrate and apply the results to different classes of lattice boltzmann collision operators in particular we formulate an equivalence condition just recently also reported in yong 2008 18 that guarantees the existence of a pre stability structure it is then illustrated by several examples how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators finally we point out some shortcomings of the stability structure approach arising in certain cases

two point one dimensional delta delta prime interactions non abelian addition law and decoupling limit in this contribution to the study of one dimensional point potentials we prove that if we take the limit on a potential of the type we obtain a new point potential of the type when u0 and u1 are related to v0 v1 w0 and w1 by a law with the structure of a group this is the borel subgroup of we also obtain the non abelian addition law from the scattering data the spectra of the hamiltonian in the decoupling cases emerging in the study are also described in full it is shown that for the values of the couplings the singular kurasov matrices become equivalent to dirichlet at one side of the point interaction and robin boundary conditions at the other side

comparison of analysis techniques for the lattice boltzmann method we show that the chapman enskog expansion can be viewed as a special instance of a general expansion procedure which also encompasses other methods like the regular error expansion and multi scale techniques and that any two expansions which properly describe the lattice boltzmann solution necessarily coincide up to higher order terms for a model problem both the regular error expansion and the chapman enskog expansion are carried out it turns out that the classical chapman enskog method leads to an unstable equation at super burnett order in a parameter regime for which the underlying lattice boltzmann algorithm is stable however our approach naturally allows us to consider variants of the super burnett equation which do not suffer from instabilities the article concludes with a detailed comparison of the chapman enskog and the regular error expansion

multiple solutions to the plateau problem for nonconstant mean curvature we prove the existence of at least two solutions to the plateau problem for any curvature function h close enough in l norm to a nonzero constant h 0 such that h 0 1 where denotes the l norm of the boundary condition

gromov s compactness theorem for pseudo holomorphic curves we give a complete proof for gromov s compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary

a remarkable spectral feature of the schrodinger hamiltonian of the harmonic oscillator perturbed by an attractive δ interaction centred at the origin double degeneracy and level crossing we rigorously define the self adjoint hamiltonian of the harmonic oscillator perturbed by an attractive interaction of strength centred at 0 the bottom of the confining parabolic potential by explicitly providing its resolvent our approach is based on a coupling constant renormalization related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the interaction in two and three dimensions the way the interaction enters in our hamiltonian corresponds to the one originally discussed for the free hamiltonian instead of the harmonic oscillator one by p s ba it should not be confused with the potential perturbation of the harmonic oscillator discussed e g in a recent paper by gadella glasser and nieto also introduced by p s ba as a perturbation of the one dimensional free laplacian and recently investigated in that context by golovaty hryniv and zolotaryuk we investigate in detail the spectrum of our perturbed harmonic oscillator the spectral structure differs from that of the one dimensional harmonic oscillator perturbed by an attractive interaction centred at the origin the even eigenvalues are not modified at all by the interaction moreover all the odd eigenvalues regarded as functions of exhibit the rather remarkable phenomenon called level crossing after first producing the double degeneracy of all the even eigenvalues for the value b being the beta function

derivation and analysis of lattice boltzmann schemes for the linearized euler equations we derive lattice boltzmann lbm schemes to solve the linearized euler equations in 1d 2d and 3d with the future goal of coupling them to an lbm scheme for navier stokes equations and a finite volume scheme for linearized euler equations the derivation uses the analytical maxwellian in a bgk model in this way we are able to obtain second order schemes in addition we perform an l 2 stability analysis numerical results validate the approach

existence of energy minimal diffeomorphisms between doubly connected domains the paper establishes the existence of homeomorphisms between two planar domains that minimize the dirichlet energy

lattice boltzmann model for high order nonlinear partial differential equations in this paper a general lattice boltzmann lb model is proposed for the high order nonlinear partial differential equation with the form ensuremath partial t ensuremath phi ensuremath sum k 1 m ensuremath alpha k ensuremath partial x k mathrm ensuremath pi k ensuremath phi 0 1 ensuremath le k ensuremath le m ensuremath le 6 ensuremath alpha k are constant coefficients mathrm ensuremath pi k ensuremath phi are some known differential functions of ensuremath phi as some special cases of the high order nonlinear partial differential equation the classical m kdv equation kdv burgers equation mathrm k n n burgers equation kuramoto sivashinsky equation and kawahara equation can be solved by the present lb model compared to the available lb models the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved in addition we also conducted a detailed chapman enskog analysis and found that the high order nonlinear partial differential equation can be correctly recovered from the proposed lb model finally a large number of simulations are performed and it is found that the numerical results agree with the analytical solutions and usually the present model is also more accurate than the existing lb models h lai and c ma sci china ser g 52 1053 2009 h lai and c ma phys a amsterdam 388 1405 2009 for high order nonlinear partial differential equations

a structured approach to the construction of stable linear lattice boltzmann collision operator abstract we introduce a structured approach to the construction of linear bgk type collision operators ensuring that the resulting lattice boltzmann methods are stable with respect to a weighted l 2 norm the results hold for particular boundary conditions including periodic bounce back and bounce back with flipping of sign boundary conditions this construction uses the equivalent moment space definition of bgk type collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency the presented structured approach is then applied to the 3d isothermal linearized euler equations with non vanishing background velocity finally convergence results in the strong discrete l norm highlight the suitability of the structured approach introduced in this manuscript

a monotonicity formula for stationary biharmonic maps we give a rigorous proof of the monotonicity formula of s y a chang l wang and p yang 3 for extrinsically stationary biharmonic maps of class w2 2

stationary partitioning of convex bodies on considere une version particuliere du probleme de partage soit 7b b un corps convexe dans r 3 de volume m 3 7b b une surface de type disque s est a l interieur de 7b b divisant cet interieur en deux cellules simplement connexes 7b b et 7b b de volumes m 3 7b b m 3 7b b m 3 7b b 1 m 3 7b b 0 1 on considere comme fixee et on etudie les cas ou s a l aire la plus petite possible

quartic parameters for acoustic applications of lattice boltzmann scheme using the taylor expansion method we show that it is possible to improve the lattice boltzmann method for acoustic applications we derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy the corresponding discrete equations are solved with the help of symbolic manipulation the solutions are obtained in the case of d3q27 lattice boltzmann scheme various numerical tests support the coherence of this approach

local removable singularity theorems for minimal laminations in this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a riemannian three manifold this removable singularity theorem is the key result used in our proof that a complete embedded minimal surface in r 3 with quadratic decay of curvature has nite total curvature