on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

dimensions of spline spaces over t meshes a t mesh is basically a rectangular grid that allows t junctions in this paper we propose a method based on bezier nets to calculate the dimension of a spline function space over a t mesh when the order of the smoothness is less than half of the degree of the spline functions a dimension formula is derived which involves only the topological quantities of the t mesh the construction of basis functions is briefly discussed furthermore the dimension formulae for t meshes after mesh operations such as edge insertion and mesh merging are also obtained

the quantum relativistic harmonic oscillator spectrum of zeros of its wave functions the distribution of zeros of the wave functions of the quantum relativistic harmonic oscillator is studied an analytic approximate expression for this distribution is derived within the framework of the wkb approximation using potential theory techniques it is proved that this wkb approach gives the right asymptotic distribution which is explicitly calculated

a ground state alternative for singular schrodinger operators let a be a quadratic form associated with a schrodinger operator l a v on a domain r d i fa is nonnegative on c 0 then either there is w 0 such that

existence of solutions for a nonlocal problem arising in plasma physics abstract this paper deals with a nonlocal problem arising in plasma physics this problem involves the terms u u u x and u u x which are neither local nor continuous and monotone by using the galerkin approximate method and the topological degree theory the existence of solutions will be proved

triangulation based isogeometric analysis of the cahn hilliard phase field model abstract this paper presents triangulation based isogeometric analysis of the cahn hilliard phase field model the cahn hilliard phase field model is governed by a time dependent fourth order partial differential equation the corresponding primal variational form involves second order operators making it difficult to be directly analyzed with traditional c 0 finite element analysis in this paper we construct c 1 bernstein bezier simplicial elements through macro element techniques including various triangle split based macro elements in both 2d and 3d space we extend triangulation based isogeometric analysis to solving the primal variational form of the cahn hilliard equation we validate our method by convergence analysis showing the nodal and degree of freedom advantages over c 0 finite element analysis we then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two dimensional squares and three dimensional cubes we incorporate an adaptive time stepping scheme in these numerical experiments our numerical study demonstrates that triangulation based isogeometric analysis offers optimal convergence and time step stability is applicable to complex geometry and allows local refinement

comparison results and improved quantified inequalities for semilinear elliptic equations in this paper we prove some pointwise comparison results between the solutions of some second order semilinear elliptic equations in a domain omega of mathbb r n and the solutions of some radially symmetric equations in the equimeasurable ball omega the coefficients of the symmetrized equations in omega satisfy similar constraints as the original ones in omega we consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient moreover we show some improved quantified comparisons when the original domain is not a ball the method is based on a symmetrization of the second order terms

schrodinger operators with few bound states we show that whole line schrodinger operators with finitely many bound states have no embedded singular spectrum in contradistinction we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast we also prove the following result for one and two dimensional schrodinger operators h with bounded positive ground states given a potential v if both h v are bounded from below by the ground state energy of h then v 0

a new approach to queer differential equations abstract queer differential equations first arose in the work of harold grad on controlled thermonuclear fusion these relate in particular to models for the slow adiabatic evolution and resistive diffusion of a plasma these are queer in that these share aspects of partial ordinary and functional differential equations in this document the authors give a new way of thinking of these equations by relating these to free boundary problems this is the first of a series of papers intending to demonstrate that solutions of such a queer differential equation can be thought of as limits of solutions of free boundary problems with n free boundaries a long term hope is that this work will complement and further refine existing numerical schemes for finding such solutions

on the existence of the free boundary for some problems arising in plasma physics we study some conditions for the existence of a free boundary for two different bidimensional models arising in the magnetic confinement of a plasma we derive estimates on the size and location of the region surrounded by the free boundary known as plasma region for both models we also derive sufficient conditions for the non existence of the free boundary

on the picard principle for δ μ given a local kato measure on mathbb r d setminus 0 d ge 2 let mathcal h 0 delta mu u be the convex cone of all continuous real solutions u 0 to the equation u u 0 on the punctured unit ball u satisfying lim x to 1 u x 0 it is shown that mathcal h 0 delta mu u ne 0 if and only if the operator f mapsto int u g cdot y f y d mu y where g denotes the green function on u is bounded on mathcal l 2 u mu and has a norm which is at most one moreover extremal rays in mathcal h 0 delta mu u are characterized and it is proven that satisfies the picard principle on u that is that mathcal h 0 delta mu u consists of one ray provided there exists a suitable sequence of shells in u such that on these shells is either small or not too far from being radial further it is shown that the verification of the picard principle can be localized several results on l2 sub eigenfunctions and 3g inequalities which are used in the paper but may be of independent interest are proved at the end of the paper

sharp decay estimates of lq norms for nonnegative schrodinger heat semigroups abstract let h v be a nonnegative schrodinger operator on l 2 r n where n 3 and v is a radially symmetric nonpositive function in r n decaying quadratically at the space infinity for any 1 p q we denote by e t h q p the operator norm of the schrodinger heat semigroup e t h from l p r n to l q r n in this paper under suitable conditions on v we give the exact and optimal decay rates of e t h q p as t for all 1 p q the decay rates of e t h q p depend on whether the operator h is subcritical or critical and on the behavior of the positive harmonic function for the operator h

smooth macro elements on powell sabin 12 splits macro elements of smoothness c r are constructed on powell sabin 12 splits of a triangle for all r 0 these new elements complement those recently constructed on powell sabin 6 splits and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary value problems and for interpolation of hermite data

polar factorization and pseudo rearrangements applications to polya szego type inequalities abstract we are interested in the polar factorization of a function f defined in an open bounded set r n it is well known that there exists a measure preserving map such that f f where f is the decreasing rearrangement of f we prove that under suitable assumptions besides the classical polar factorization of f we have f f u where f u is a pseudo rearrangement of f with respect to the measurable function u and is the measure preserving map such that u u as an application we characterize those functions that realize equality in the polya szego inequality

on criticality and ground states of second order elliptic equations ii abstract in this article we examine a necessary and sufficient condition for the existence of a positive solution of the equation p tw u 0 in r n where p is a second order alliptic operator which is critical in w c 0 and t 0 is small enough

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

lp norms of nonnegative schrodinger heat semigroup and the large time behavior of hot spots this paper is concerned with the cauchy problem for the heat equation with a potential p tu u v x uin rn 0 u x 0 x in rn where t t n 3 l2 rn and v v x is a smooth nonpositive and radially symmetric function having quadratic decay at the space infinity in this paper we assume that the schrodinger operator h v is nonnegative on l2 rn and give the exact power decay rates of lq norm q 2 of the solution e th of p as t furthermore we study the large time behavior of the solution of p and its hot spots

on green functions of second order elliptic operators on riemannian manifolds the critical case abstract let p be a second order linear elliptic operator with real coefficients which is defined on a noncompact and connected riemannian manifold m it is well known that the equation p u 0 in m admits a positive supersolution which is not a solution if and only if p admits a unique positive minimal green function on m and in this case p is said to be subcritical in m if p does not admit a positive green function but admits a global positive super solution then such a solution is called a ground state of p in m and p is said to be critical in m we prove for a critical operator p in m the existence of a green function which is dominated above by the ground state of p away from the singularity moreover in a certain class of green functions such a green function is unique up to an addition of a product of the ground states of p and p under some further assumptions we describe the behavior at infinity of such a green function this result extends and sharpens the celebrated result of p li and l f tam concerning the existence of a symmetric green function for the laplace beltrami operator on a smooth and complete riemannian manifold m

some new results related to lorentz gγ spaces and interpolation abstract we compute the k functional related to some couple of spaces as small or classical lebesgue space or lorentz marcinkiewicz spaces completing the results of 11 this computation allows to determine the interpolation space in the sense of peetre for such couple it happens that the result is always a g space since this last space covers many spaces the motivations of such study are various among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain with data in the space of the integrable function with respect to the distance function to the boundary of

spectral properties of solutions of hypergeometric type differential equations abstract the second order differential equation x y x y y 0 is usually called equation of hypergeometric type provided that are polynomials of degree not higher than two and one respectively and is a constant their solutions are commonly known as hypergeometric type functions htfs in this work a study of the spectrum of zeros of those htfs for which 1 2 1 v r and are independent of is done within the so called semiclassical or wkb approximation specifically the semiclassical or wkb density of zeros of the htfs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy applications to the gaussian and confluent hypergeometric functions as well as to hermite functions are shown

wkb approach to zero distribution of solutions of linear second order differential equations given a second order linear differential equation y z s z y z 0 the distribution of zeros of its solutions is defined by v y z 0 z where z stands for the dirac delta at the point z some techniques of approximation of the restriction of v to r directly from s z are considered in particular for the wkb method error bounds are provided and some related results established in the second part formulas for the appropriate scaling in the holonomic case are given as an illustration we obtain the asymptotic distribution of the real zeros of some families of polynomials

new hardy inequalities and behaviour of linear elliptic equations abstract in this study we want to emphasize the role of some hardy inequalities in the blow up phenomena of the very weak solution of a linear equation in the sense of brezis thus we present here some new hardy inequalities related to some extended sobolev spaces such that sobolev hardy spaces sobolev zygmund spaces or other non standard weighted spaces firstly we apply those results then provide two applications of these inequalities secondly we improve recent results by showing that the blow up phenomena of the gradient can also occur in hardy spaces the hardy inequalities for sobolev zygmund spaces are obtained via an integral formula estimating the oscillation in a ball of radius r of a general function u in the usual sobolev space this formula involves the notion of relative rearrangement we shall give a pointwise estimate for the solution u of linear equation u div f for a bounded function f using the distance function

norm estimate of green operator perturbation of green function and integrability of superharmonic functions an operator norm estimate of the green operator for the laplace equation on an open set in the euclidean space is given in terms of capacity this applies to the perturbation of the green function for the schrodinger equation a sufficient condition for the green function for the schrodinger equation to have decay near the boundary comparable to that for the laplace equation is given in terms of an integrable condition for the potential of the schrodinger equation this has an application to the martin boundary of a smooth domain with respect to the schrodinger equation also it is related to the cranston mcconnell inequality about the life time estimate and the integrability of nonnegative superharmonic functions 1991 mathematics subject classification 31a35 31b35 31c35

existence of solutions for degenerate quasilinear elliptic equations abstract this paper deals with a class of degenerate quasilinear elliptic equations of the form div a x u u g div f where a x u u is allowed to be degenerate with the unknown u we prove existence of bounded solutions under some hypothesis on f and g moreover we prove that there exists a renormalized solution in the case where g l 1 and f l p n

upper and lower bounds on the dimension of multivariate spline spaces we give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangu lations in mathbb r k the bounds are optimal in a certain sense

multivariate weak spline function space in this paper b net method for studying multivariate weak spline is discussed and the dimension of w21 i1 1 is presented several b splines on type 1 triangulation are also obtained

sharp two sided heat kernel estimates of twisted tubes and applications we prove on diagonal bounds for the heat kernel of the dirichlet laplacian delta d omega in locally twisted three dimensional tubes in particular we show that for any fixed x the heat kernel decays for large times as rm e e 1t t 3 2 where e1 is the fundamental eigenvalue of the dirichlet laplacian on the cross section of the tube this shows that any suitably regular local twisting speeds up the decay of the heat kernel with respect to the case of straight untwisted tubes moreover the above large time decay is valid for a wide class of subcritical operators defined on a straight tube we also discuss some applications of this result such as sobolev inequalities and spectral estimates for schrodinger operators delta d omega v

criticality theory for schrodinger operators with singular potential abstract this paper extends a weighted spectral gap classification for schrodinger operators v to a large class of locally integrable potentials v called balanced potentials here harnack s inequality will not hold in general for such potentials in addition to the standard potentials in the space l l o c p for some p n 2 this class contains potentials for which v is locally bounded with no restriction on v or more generally those for which the positive and negative singularities are separated as well as those for which the interaction of the positive and negative singularities is not too strong we also establish for such potentials under the additional assumption that v l l o c p for some p n 2 versions of the agmon allegretto piepenbrink principle characterising the subcritical critical operators in terms of the cone of positive distributional super solutions

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

estimates for green function and poisson kernels of higher order dirichlet boundary value problems pointwise estimates are derived for the kernels associated to the polyharmonic dirichlet problem on bounded smooth domains as a consequence one obtains optimal weighted l p l q regularity estimates for weights involving the distance function

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

dimensions of biquadratic spline spaces over t meshes this paper discusses the dimensions of spline spaces over t meshes of low degree two new concepts are proposed an extension of t meshes and spline spaces with homogeneous boundary conditions in the dimensional analysis the key strategy is linear space embedding with the operator of the mixed partial derivative a lower bound on the dimension of the biquadratic spline spaces over general t meshes is provided furthermore by making full use of the level structure of hierarchical t meshes a dimension formula of biquadratic spline space over hierarchical t meshes is proved additionally a topological explanation of the dimension formula is provided

on positivity criticality and the spectral radius of the shuttle operator for elliptic operators in this paper we study the shuttle operator for a second order linear elliptic operator p on a noncompact manifold x zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the laplacian in ir n 3 zhao was motivated by works of chung and varadhan which consider one dimensional schrodinger operators the main purpose of the paper is to extend the above studies we prove that in the general case the spectral radius of the shuttle operator is strictly less equal or strictly greater than 1 if and only if the operator p is respectively subcritical critical or supercritical in x we demonstrate the usefulness of the characterization for proving theorems our approach is purely analytic this research was partially supported by the israel science foundation administered by the israel academy of sciences and humanities and by the fund for the promotion of research at the technion

criticality and the λ property for elliptic equations abstract let l denote a nonnegative second order elliptic expression defined on c 0 and let l be an associated self adjoint operator l is said to have the property in iff l for some fixed positive function and any c 0 l is termed subcritical iff a generalized green s function may be defined for l in various applications of these and closely related concepts may be found in the literature it is the main purpose of this article to show that l is subcritical iff l has the property several corollaries and an example are given as consequences of this equivalence

jacobi sobolev type orthogonal polynomials second order differential equation and zeros abstract we obtain an explicit expression for the sobolev type orthogonal polynomials qn associated with the inner product p q 1 1 p x q x p x dx a 1 p 1 q 1 b 1 p 1 q 1 a 2 p 1 q 1 b 2 p 1 q 1 where p x 1 x 1 x is the jacobi weight function 1 a1 b1 a2 b2 0 and p q p the linear space of polynomials with real coefficients the hypergeometric representation 6f5 and the second order linear differential equation that such polynomials satisfy are also obtained the asymptotic behaviour of such polynomials in 1 1 is studied furthermore we obtain some estimates for the largest zero of qn x such a zero is located outside the interval 1 1 we deduce his dependence of the masses finally the wkb analysis for the distribution of zeros is presented

rearrangement relatif dans les equations elliptiques quasi lineaires avec un second membre distribution application a un theoreme d existence et de regularite abstract in this paper we are concerned with quasilinear elliptic equations that is p au f u u t in r u w01 p l where a is an operator of leray lions type which is defined on w 0 1 p l 1 f is a non linear map having a suitable growth and t is a distribution of w 1 r r n p 1 and r p p 1 using the techniques of the relative rearrangement ann scuola norm sup pisa cl sci 4 in press we give a precise a priori estimate of the solution u of p in l norm these estimates allow us to prove an existence theorem for p and to get the holder continuity of the solution u

the fractional schrodinger equation with general nonnegative potentials the weighted space approach abstract we study the dirichlet problem for the stationary schrodinger fractional laplacian equation s u v u f posed in bounded domain r n with zero outside conditions we consider general nonnegative potentials v l l o c 1 and prove well posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f important properties of the solutions such as its boundary behaviour are derived the case of super singular potentials that blow up near the boundary is given special consideration since it leads to so called flat solutions we comment on related literature

spectral properties of the biconfluent heun differential equation abstract the spectrum of zeros of the polynomial solutions of the biconfluent heun differential equation is investigated by two different methods first the spectral newton sums i e the sums of the r th powers of the zeros are given in a rigorous and recurrent way second the density of zeros i e the number of zeros per unit interval is calculated in an explicit way within the so called semiclassical or bkw approximation this is done by using a general theorem which applies to a linear second order differential equation under certain conditions applications to the radial schrodinger equations associated to some quantum mechanical systems rotating harmonic oscillator confinement potentials as well as to the bessel differential equation are also shown

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

bivariate spline spaces and minimal determining sets abstract the paper describes bivariate spline spaces and minimal determining sets shows applications of these concepts presents two new triangular finite elements describes public domain software for the analysis of spline spaces and concludes with a concise history of these subjects the software described in this paper is available on http www math utah edu alfeld mds that page also contains links to colored versions of the figures in this paper

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

new properties of small lebesgue spaces and their applications we prove some new properties of the small lebesgue spaces introduced by fiorenza in f combining these properties with the poincare sobolev inequalities for the relative rearrangement see r1 we derive some new and precises estimates either for small lebesgue sobolev spaces or for quasilinear equations with data in the small lebesgue spaces

criticality and ground states for second order elliptic equations abstract let p be a second order elliptic operator in a domain r n suppose that p admits a positive minimal green function g p in it is the main purpose of this paper to prove that if w c is a small perturbation of p in then max w x 0 min w x 0 t u u g p outside a compact set in

generalized transition fronts for one dimensional almost periodic fisher kpp equations this paper investigates the existence of generalized transition fronts for fisher kpp equations in one dimensional almost periodic media assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction we show that such fronts exist if and only if their average speed is above an explicit threshold this hypothesis is satisfied in particular when the reaction term does not depend on x or in some cases is small enough moreover except for the threshold case the fronts we construct and their speeds are almost periodic in a sense when our hypothesis is no longer satisfied such generalized transition fronts still exist for an interval of average speeds with explicit bounds our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues

on some degenerate and nondegenerate quasilinear elliptic systems with nonhomogeneous dirichlet boundary condition on considere des systemes elliptiques sous forme diagonale du type d u f x u u g ou d est une matrice diagonale on montre que l on peut obtenir la norme l pourvu qu il y ait une bonne structure du systeme

multivalued fixed point index and nonlocal problems involving relative rearrangement abstract we apply the fixed point index theory for multivalued problems to nonlocal problems involving relative rearrangement mappings these applications need an extension of the almgren lieb theorem and an improvement of the strong continuity of the relative rearrangement mapping on arbitrary open sets bounded or not the proofs of these results are based on some pointwise equations relating the relative rearrangement of the gradient and the weighted de giorgi perimeter

spline element methods allowing multiple level hanging nodes abstract in this paper a systematic algorithm of spline element method allowing multiple level hanging nodes on triangular mesh is proposed the classical smoothness condition of the piecewise polynomials on triangular mesh is generalized to the case of meshes with hanging nodes and is treated as the linear constraints of the system equations we then derive a concise formulation for such linear constraints according to the special hierarchical configurations of the hanging nodes moreover an efficient linear equation solver is proposed when applied to solve the elliptic partial differential equations numerical examples are illustrated to show the accuracy of the proposed methods and the comparisons are made between different levels of hanging nodes

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

quantile and probability curves without crossing this paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions also known as the quantile crossing problem bassett and koenker 1982 the method consists in sorting or monotone rearranging the original estimated non monotone curve into a monotone rearranged curve we show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples establish a functional delta method for rearrangement related operators and derive functional limit theory for the entire rearranged curve and its functionals we also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals our limit results are generic in that they apply to every estimator of a monotone function provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions consequently our results apply to estimation of other econometric functions with monotonicity restrictions such as demand production distribution and structural distribution functions we illustrate the results with an application to estimation of structural distribution and quantile functions using data on vietnam veteran status and earnings

on a nonlocal stationary free boundary problem arising in the confinement of a plasma in a stellarator geometry we prove the existence and some qualitative properties of the solution to a two dimensional free boundary problem modeling the magnetic confinement of a plasma in a stellarator configuration the nonlinear elliptic partial differential equation on the plasma region was obtained from the three dimensional magnetohydrodynamic system by hender carreras in 1984 by using averaging arguments and a suitable system of coordinates boozer s vacuum coordinates the free boundary represents the separation between the plasma and vacuum regions and the model is described by an inverse type problem some nonlinear terms of the equation are unknown using the zero net current condition for the stellarator configurations we reformulate the problem with the help of the notion of relative rearrangement leading to a new problem involving nonlocal terms in the equation we use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution

existence of solutions for nonlinear elliptic degenerate equations abstract in this paper we study the problem div a x u u div u g x u f in in the setting of the weighted sobolev space w 0 1 p the main novelty of our work is l estimates on the solutions and the existence of a weak and renormalized solution

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

electrostatic models for zeros of polynomials old new and some open problems we give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well known families of polynomials and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity the leading role is played by the differential equation satisfied by these polynomials some new developments applications and open problems are presented

an elliptic parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator we prove the existence and the regularity of weak solutions of a nonlocal elliptic parabolic free boundary problem involving the notions of relative rearrangement and monotone rearrangement the problem arises in the study of the dynamics of a magnetically confined fusion plasma in a stellarator device when the dimensional analysis on the characteristic times suggests to neglect the inertial acceleration in presence of a time dependent magnetic field

some properties of the relative rearrangement abstract we develop some new properties of the relative rearrangement some of these properties generalize well known results known as the hardy littlewood inequality

algebraic and spectral properties of some quasiorthogonal polynomials encountered in quantum radiation the nodal structure of the wavefunctions of a large class of quantum mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials it is known that these polynomials i may be described by an arbitrary linear combination of two orthogonal polynomials pn x and ii have real and simple zeros here the three term recurrence relation the second order differential equation and the distribution of zeros of quasiorthogonal polynomials of the classical class i e when pn x is a jacobi laguerre or hermite polynomial are derived and analyzed specifically the exact values of the newton sum rules and the wkb density of zeros of these polynomials are found

on optimization problems with prescribed rearrangements on considere certains resultats concernant des fonctions avec des distributions imposees et des prroblemes d optimisation qui y sont relies

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

stability of martin boundary under non local feynman kac perturbations recently the authors showed that the martin boundary and the minimal martin boundary for a censored or resurrected stable process y in a bounded c 1 1 open set d with 1 2 can all be identified with the euclidean boundary d of d under the gaugeability assumption we show that the martin boundary and the minimal martin boundary for the schrodinger operator obtained from y through a non local feynman kac transform can all be identified with d in other words the martin boundary and the minimal martin boundary are stable under non local feynman kac perturbations moreover an integral representation of nonnegative excessive functions for the schrodinger operator is explicitly given these results in fact hold for a large class of strong markov processes as are illustrated in the last section of this paper as an application the martin boundary for censored relativistic stable processes in bounded c 1 1 smooth open sets is studied in detail

isogeometric analysis with bezier tetrahedra abstract this paper presents an approach for isogeometric analysis of 3d objects using rational bezier tetrahedral elements in this approach both the geometry and the physical field are represented by trivariate splines in bernstein bezier form over the tetrahedrangulation of a 3d geometry given a nurbs represented geometry either untrimmed or trimmed we first convert it to a watertight geometry represented by rational triangular bezier splines rtbs for trimmed geometries a compatible subdivision scheme is developed to guarantee the watertightness the rtbs geometry preserves exactly the original nurbs surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs from the watertight rtbs geometry a bezier tetrahedral partition is generated automatically by imposing continuity constraints on bezier ordinates of the elements we obtain a set of global c r smooth basis functions and use it as the basis for analysis numerical examples demonstrate that our method achieves optimal convergence in c r spaces and can handle complicated geometries

continuity and convergence in rational triangular bezier spline based isogeometric analysis abstract this paper presents a method for isogeometric analysis using rational triangular bezier splines rtbs where optimal convergence rates are achieved in this method both the geometry and the physical field are represented by bivariate splines in bernstein bezier form over the triangulation of a domain from a given physical domain bounded by nurbs curves a parametric domain and its triangulation are constructed by imposing continuity constraints on bezier ordinates we obtain a set of global c r smooth basis functions convergence analysis shows that isogeometric analysis with such c r rtbs basis can deliver the optimal rate of convergence provided that the c r geometric map remains unchanged during the refinement process this condition can be satisfied by constructing a pre refinement geometric map that is sufficiently smooth numerical experiments verify that optimal rates of convergence are achieved for poisson and linear elasticity problems

on the differentiability of very weak solutions with right hand side data integrable with respect to the distance to the boundary we study the differentiability of very weak solutions v is an element of l 1 omega of v l phi 0 f phi 0 for all phi is an element of c 2 omega over bar vanishing at the boundary whenever f is in l 1 omega delta with delta dist x partial derivative omega and l is a linear second order elliptic operator with variable coefficients we show that our results are optimal we use symmetrization techniques to derive the regularity in lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function f over tilde of f

on uniqueness and nonuniqueness of the positive cauchy problem for parabolic equations with unbounded coefficients where i xi a nonnegative solution u x t of the equation lu 0 in xt x 0 t with a nonnegative initial data u x 0 u0 x is called a solution of the positive cauchy problem such a solution can be thought of as the temperature field at the time t in an n dimensional body which occupies the domain x the coefficients aij x are the components of the matrix of the thermal conductivities while the function 1 x is the heat capacity of a unit hypervolume hence nonnegative solutions of the cauchy problem are natural objects to study in particular it is of interest to know whether the

instability in the dimension of spaces of bivariate piecewise polynomials of degree 2 r and smoothness order r the spaces of bivariate splines on a particular triangulation are considered the dimension of the space of splines with degree 2 and with smoothness order 1 on this triangulation has long been known to depend on the geometry of the triangulation this result is extended to the space of splines of degree 2r and smoothness order r the dimension of the spline space is shown to have a similar dependence on geometry for all values of r geqq 1

on critical and subcritical sturm liouville operators we study critical and subcritical sturm liouville operators k 1 ddx p ddx v in l2 a b k dx a b r defined as friedrichs extensions under most general conditions on k p and v in particular we prove the equivalence of the notions of criticality as introduced by simon and murata for an important class of potentials v

the generic dimension of the space of c 1 splines of degree d 8 on tetrahedral decompositions this paper considers the linear space of globally differentiable piecewise polynomial functions defined on a three dimensional polyhedral domain which has been partitioned into tetrahedra combining bernstein bezier methods and combinatorial and geometric techniques from rigidity theory this paper gives an explicit expression for the generic dimension of this space for sufficiently large polynomial degrees d geq 8 this is the first general dimension statement of its kind