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

radial symmetry of positive solutions in nonlinear polyharmonic dirichlet problems we extend the symmetry result of gidas ni nirenberg to semilinear polyharmonic dirichlet problems in the unit ball in the proof we develop a new variant of the method of moving planes relying on fine estimates for the green function of the polyharmonic operator we also consider minimizers for subcritical higher order sobolev embeddings for embeddings into weighted spaces with a radially symmetric weight function we show that the minimizers are at least axially symmetric this result is sharp since we exhibit examples of minimizers which do not have full radial symmetry

rearrangements of functions let g be measurable non negative functions on r and let tf be their equimeasurable symmetric decreasing rearrangements let f r r r be continuous and suppose that the associated rectangle function defined by f r f a c f b d f a d f b c for r x y r2 a x b c y d is non negative then f g d f tf g d where is lebesgue measure the concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces and the inequality holds in the general case if f x y x y where is convex and 0 0 then we obtain tf g d g d in particular if x x p 1 p then we find that the operator s tf is a contraction on lp for 1 p

steiner symmetric extremals in polya szego type inequalities abstract the cases of equality are analyzed in steiner symmetrization inequalities for dirichlet type integrals in particular minimal assumptions are determined under which functions attaining equality are necessarily steiner symmetric

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

sobolev inequalities the poisson semigroup and analysis on the sphere sn abstract hypercontractive estimates are obtained for the poisson semigroup on lp sn such estimates are determined by a sharp logarithmic sobolev inequality that relates the entropy of a function on sn to its smoothness

symmetry of constrained minimizers of the cahn hilliard energy on the torus abstract we establish sufficient conditions for a function on the torus to be equal to its steiner symmetrization and apply the result to so called volume constrained minimizers of the cahn hilliard energy the resulting connectedness of superlevel sets is used in two dimensions together with the bonnesen inequality to quantitatively estimate the sphericity of minimizers we also show how two point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the cahn hilliard model

affine orlicz polya szego principle for log concave functions abstract the affine l p polya szego principle significantly strengthens the classical polya szego principle it is an open problem whether there exists an affine orlicz polya szego principle which includes the affine l p polya szego principle as special case in this paper an affine orlicz polya szego principle for log concave functions is established by using functional steiner symmetrizations

rearrangement inequalities for functionals with monotone integrands the inequalities of hardy littlewood and riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement it is known that these inequalities extend to integrands of the form f u1 um where f is supermodular in particular they hold when f has nonnegative mixed second derivatives i jf for all i j this paper concerns the regularity assumptions on f and the equality cases it is shown here that extended hardy littlewood and riesz inequalities are valid for supermodular integrands that are just borel measurable under some nondegeneracy conditions all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant i e measure preserving maps for the hardy littlewood inequality translations for the riesz inequality the proofs rely on monotone changes of variables in the spirit of sklar s theorem

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

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

symmetrization symmetric stable processes and riesz capacities let x t be a symmetric stable process killed on exiting an open subset d of r n we prove a theorem that describes the behavior of its transition probabilities under polarization we show that this result implies that the probability of hitting a given set b in the complement of d in the first exit moment from d increases when d and b are polarized it can also lead to symmetrization theorems for hitting probabilities green functions and riesz capacities one such theorem is the following among all compact sets k in r n with given volume the balls have the least capacity 0 2

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

sensitivity analysis for szegő polynomials szeg polynomials are orthogonal with respect to an inner product on the unit circle numerical methods for weighted least squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of szeg polynomials this paper discusses the conditioning of several mappings involving szeg polynomials and thereby sheds light on the sensitivity of some approximation problems involving trigonometric polynomials

groundstates for a local nonlinear perturbation of the choquard equations with lower critical exponent abstract we prove the existence of ground state solutions by variational methods to the nonlinear choquard equations with a nonlinear perturbation u u i u n 1 u n 1 u f x u in r n where n 1 i is the riesz potential of order 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions

nodal solutions for the choquard equation abstract we consider the general choquard equations u u i u p u p 2 u where i is a riesz potential we construct minimal action odd solutions for p n n n n 2 and minimal action nodal solutions for p 2 n n 2 we introduce a new minimax principle for least action nodal solutions and we develop new concentration compactness lemmas for sign changing palais smale sequences the nonlinear schrodinger equation which is the nonlocal counterpart of the choquard equation does not have such solutions

on a property of the nodal set of least energy sign changing solutions for quasilinear elliptic equations in this note we prove the payne type conjecture about the behaviour of the nodal set of least energy sign changing solutions for the equation delta p u f u in bounded steiner symmetric domains omega subset of r n under the zero dirichlet boundary conditions the nonlinearity f is assumed to be either superlinear or resonant in the latter case least energy sign changing solutions are second eigenfunctions of the zero dirichlet p laplacian in omega we show that the nodal set of any least energy sign changing solution intersects the boundary of omega the proof is based on a moving polarization argument

nonlinear elliptic equations with subhomogeneous potentials abstract we prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation u v y 1 y k u g u in r n where x z y 1 y k r n 0 r n 1 r n k r n with n 3 k 1 n 0 0 and n i 2 for i 0 the nonlinearity g and the potential v are respectively a continuous function not necessarily superlinear at infinity and a positive measurable function not necessarily homogeneous but satisfying a subhomogeneity condition which implies vanishing at infinity and singularity at least at the origin this also yields the existence of nonrotating solitary waves and vortices with a critical frequency for nonlinear schrodinger and klein gordon equations with singular cylindrical potentials

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

gaussian quadrature formulae on the unit circle let be a probability measure on 0 2 in this paper we shall be concerned with the estimation of integrals of the form i f 1 2 02 f ei d for this purpose we will construct quadrature formulae which are exact in a certain linear subspace of laurent polynomials the zeros of szego polynomials are chosen as nodes of the corresponding quadratures we will study this quadrature formula in terms of error expressions and convergence as well as its relation with certain two point pade approximants for the herglotz riesz transform of furthermore a comparison with the so called szego quadrature formulae is presented through some illustrative numerical examples

a remark on comparison results via symmetrization in this paper we study the converse of comparison results for solutions to linear second order elliptic equations namely in the inequalities proved by g talenti and others we study the case of equality and prove that equalities are achieved only in the spherical situation we also present some applications of these results to semilinear elliptic equations

rearrangement and polarization abstract the paper has two main goals the first is to take a new approach to rearrangements on certain classes of measurable real valued functions on r n rearrangements are maps that are monotonic up to sets of measure zero and equimeasurable i e they preserve the measure of super level sets of functions all the principal known symmetrization processes for functions such as steiner and schwarz symmetrization are rearrangements and these have a multitude of applications in diverse areas of the mathematical sciences the second goal is to understand which properties of rearrangements characterize polarization a special rearrangement that has proved particularly useful in a number of contexts in order to achieve this new results are obtained on the structure of measure preserving maps on convex bodies and of rearrangements generally

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

symmetry of solutions to the optimal exit time control problem in this paper we study the solutions to the optimal exit time control problem such a problem tries to find the state feedback control law with a fixed cost that can keep the state of a randomly perturbed system inside a subset of the state space called the safe set for as long as possible on average by formulating the problem as an optimization problem with pde constraints and using symmetrization techniques we show that when the safe set is a ball the optimal feedback control if it exists must be radially symmetric furthermore we show that among all safe sets with a fixed volume the ball is the best in that it yields the most efficient optimal exit time control the proofs make essential use of the general isoperimetric inequality

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

second order derivatives and rearrangements the classical polya szego principle states that the lebesgue norm of the gradient of a weakly differentiable function on r decaying to 0 at infinity decreases under spherically symmetric rearrangement we present extensions of this principle to second order derivatives in two different directions on one hand when dimension n 1 we show that the space of those functions whose second order distributional derivative is a measure with finite total variation is closed under the operation of decreasing rearrangement moreover a polya szego type principle is proved for such a variation on the other hand in any dimension an inequality between the decreasing rearrangement of the second order derivatives of a function and the first order derivative of its decreasing rearrangement is established as a consequence a new unified approach to second order sobolev type inequalities in rearrangement invariant spaces via 1 dimensional hardy type inequalities is derived dipartimento di matematica e applicazioni per l architettura universita di firenze piazza ghiberti 27 50122 firenze italy e mail cianchi cesit1 unifi it

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

some remarks on rearrangement for nonlocal functionals we prove that a nonlocal functional approximating the standard dirichlet p norm fails to decrease under two point rearrangement furthermore we get other properties related to this functional such as decay and compactness

a weak strong convergence property and symmetry of minimizers of constrained variational problems in mathbb r n we prove a weak strong convergence result for functionals of the form rnj x u du dx on w1 p along equiintegrable sequences we will then use it to study cases of equality in the extended polya szego inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints

on starshaped rearrangement and applications a radial symmetrization technique is investigated and new properties are proven the method transforms functions u into new functions u with starshaped level sets and is therefore called starshaped rearrangement this rearrangement is in general not equimeasurable a circumstance with some surprising consequences the method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems in particular the lipschitz continuity of free boundaries can now be easily obtained in a new fashion introduction in 26 g szego introduced the concept of radial symmetrization this is a nonequimeasurable rearrangement which transforms functions u q r q c r2 with starshaped level sets qc x e flu x c into functions u with symmetric starshaped level sets later 2 3 15 16 22 his approach was generalized to transform functions with level sets containing the origin into starshaped and not necessarily symmetric level sets and to higher dimensions we call a set d c rn starshaped with respect to xo if x e d implies tx 1 t xo e d for every t e 0 1 we call a set d starshaped if it is starshaped with respect to the origin therefore a more appropriate name for this rearrangement might be starshaped rearrangement in particular since the notion of radial symmetrization can be easily confused with circular or spherical symmetrization or spherically symmetric rearrangement one of the standard results in the theory of rearrangements is that the capacity of a condenser decreases under rearrangement i e j vu x lpdx 1 vu x ipdx for p 1 where u 1 on ql u 0 on 9q and u o c q c q e 0 0 j ivu p x ipdx for p 1 provided q c rn is a bounded starshaped domain containing ue 0 and u 0 1 u e c0 1 q u 1 in u 0 and u 0 on iq another convenient feature of other rearrangements is their equimeasurability a rearrangement u u is called equimeasurable if the n dimensional lebesgue received by the editors march 26 1984 and in revised form september 24 1985 1980 mathematics subject classification primary 35j20 35r35 35b05 26d20 lthis research was supported by the deutsche forschungsgemeinschaft dfg and by e reid 1986 american mathematical society 0002 9947 86 1 00 25 per page

the longest shortest fence and sharp poincare sobolev inequalities we prove a long standing conjecture concerning the fencing problem in the plane among planar convex sets of given area the disc and only the disc maximizes the length of the shortest area bisecting curve although it may look intuitive the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so called auerbach triangle

on population resilience to external perturbations we study a spatially explicit harvesting model in periodic or bounded environments the model is governed by a parabolic equation with a spatially dependent nonlinearity of kolmogorov petrovsky piskunov type and a negative external forcing term delta using sub and supersolution methods and the characterization of the first eigenvalue of some linear elliptic operators we obtain existence and nonexistence results as well as results on the number of stationary solutions we also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude in particular we define two critical values delta and delta 2 such that if delta is smaller than delta the population density converges to a significant state which is everywhere above a certain small threshold whereas if delta is larger than delta 2 the population density converges to a remnant state everywhere below this small threshold our results are shown to be useful for studying

on the construction of szego polynomials abstract the chebyshev and stieltjes procedures are algorithms for computing recursion coefficients for polynomials that are orthogonal with respect to an inner product defined on part of the real axis the chebyshev procedure is an implementation of a map from moments to recursion coefficients of orthogonal polynomials the modified chebyshev procedure is an implementation of a map from modified moments to recursion coefficients the latter map is generally much better conditioned than the former one the conditioning of these maps has been studied by gautschi this paper is concerned with analogues of the chebyshev and stieltjes procedures when the inner product is defined on the unit circle polynomials orthogonal with respect to such an inner product are known as szeg polynomials and the analogue of the chebyshev procedure is known as schur s algorithm this algorithm implements a map from moments to recursion coefficients for szeg polynomials our analysis shows that this map generally is much better conditioned than the map implemented by the chebyshev procedure and suggests that schur s algorithm is as insensitive to errors in the data as the modified chebyshev procedure thus roughly moments associated with inner products defined on the unit circle correspond to modified moments associated with inner products defined on the real axis

symmetry and symmetry breaking for ground state solutions of some strongly coupled elliptic systems abstract we consider the ground state solutions of the lane emden system with henon type weights u x v q 1 v v x u p 1 u in the unit ball b of r n with dirichlet boundary conditions where n 1 0 p q 0 and 1 p 1 1 q 1 n 2 n we show that such ground state solutions u v always have definite sign in b and exhibit a foliated schwarz symmetry with respect to a unit vector of r n we also give precise conditions on the parameters p and q under which the ground state solutions are not radially symmetric

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

a connection between quadrature formulas on the unit circle and the interval 1 1 abstract we establish a relation between gauss quadrature formulas on the interval 1 1 that approximate integrals of the form i f 1 1 f x x d x and szeg quadrature formulas on the unit circle of the complex plane that approximate integrals of the form i f f e i d the weight x is positive on 1 1 while the weight is positive on it is shown that if cos sin then there is an intimate relation between the gauss and szeg quadrature formulas moreover as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to x and orthogonal szeg polynomials with respect to inclusion of gauss lobatto and gauss radau formulas is natural

positive interpolatory quadrature formulas and para orthogonal polynomials we establish a relation between quadrature formulas on the interval 1 1 that approximate integrals of the form j f 11 f x x dx and szego quadrature formulas on the unit circle that approximate integrals of the form i f f ei d the functions x and are assumed to be weight functions on 1 1 and respectively and are related by cos sin it is well known that the nodes of szego formulas are the zeros of the so called para orthogonal polynomials bn z n z n z 1 n z and n z being the orthogonal and reciprocal polynomials respectively with respect to the weight function furthermore for 1 we have recently obtained gauss type quadrature formulas on 1 1 see bultheel et al j comput appl math 132 1 2000 1 in this paper making use of the para orthogonal polynomials with 1 a one parameter family of interpolatory quadrature formulas with positive coefficients for j f is obtained along with error expressions for analytic integrands finally some illustrative numerical examples are also included

a generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators abstract an extended iteration method for calculating the periodic solutions of nonlinear oscillator equations is given the procedure is illustrated by applying it to two truly nonlinear oscillator differential equations

sieved para orthogonal polynomials on the unit circle abstract sieved orthogonal polynomials on the unit circle were introduced independently by ismail and li 1992 15 and marcellan and sansigre 1991 19 we look at the para orthogonal polynomials chain sequences and quadrature formulas that follow from the kernel polynomials of sieved orthogonal polynomials on the unit circle

rearrangement inequalities in the discrete setting and some applications abstract we prove extended hardy littlewood riesz and polya szeg inequalities in the discrete case we also establish cases of equality in these inqualities and give some applications of our result

optimization problems on general classes of rearrangements this paper is concerned with maximization and minimization problems of the energy integral associated to p laplace equations depending on functions that belong to a class of rearrangements we prove existence and uniqueness results and present some features of optimal solutions the radial case is discussed in detail we also prove a result of uniqueness for a class of p laplace equations under non standard assumptions

on a comparison theorem via symmetrisation talenti s theorem allows us to estimate the norms of the solution of an elliptic boundary value problem posed on a bounded open set in terms of those of the symmetrised problem posed on the ball centred at 0 and such that we are interested in the limiting case when equality is achieved between the norms of the two solutions we show then that and that the solution is radially symmetric and decreasing the novelty of this exposition is that the method used is very elementary compared to the usual arguments

functions of bounded variation and rearrangements the action of the symmetric rearrangement of functions of bounded variation is investigated in particular pointwise regularizing effects are established polya szeg type inequalities with a characterization of equality cases are also proved

on the variational iteration method and other iterative techniques for nonlinear differential equations abstract a variety of iterative methods for the solution of initial and or boundary value problems in ordinary and partial differential equations is presented these iterative procedures provide the solution or an approximation to it as a sequence of iterates for initial value problems it is shown that these iterative procedures can be written in either an integral or differential form the integral form is governed by a volterra integral equation whereas the differential one can be obtained from the volterra representation by simply differentiation it is also shown that integration by parts variation of parameters adjoint operators green s functions and the method of weighted residuals provide the same volterra integral equation and that this equation in turn can be written as that of the variational iteration method it is therefore shown that the variational iteration method is nothing else by the picard lindelof theory for initial value problems in ordinary differential equations and banach s fixed point theory for initial value problems in partial differential equations and the convergence of these iterative procedures is ensured provided that the resulting mapping is lipschitz continuous and contractive it is also shown that some of the iterative methods for initial value problems presented here are special cases of the bellman kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives but such a condition is not required by the techniques presented in this paper for boundary value problems it is shown that one may use the iterative procedures developed for initial value problems but the resulting iterates may not satisfy the boundary conditions and two new iterative methods governed by fredholm integral equations are proposed it is shown that the resulting iterates satisfy the boundary conditions if the first one does so the iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of differential two and three level iterative techniques harmonic balance procedures and standard and modified linstedt poincare techniques the method is also applied to two boundary value problems

convergence acceleration of series through a variational approach abstract by means of a variational approach we find new series representations both for well known mathematical constants such as and the catalan constant and for mathematical functions such as the riemann zeta function the series that we have found are all exponentially convergent and provide quite useful analytical approximations with limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical functions

szegő lobatto quadrature rules gauss type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as gauss radau and gauss lobatto quadrature rules respectively efficient algorithms are available for their computation szego quadrature rules are analogs of gauss quadrature rules for the integration of periodic functions they integrate exactly trigonometric polynomials of as high degree as possible szego quadrature rules have a free parameter which can be used to prescribe one node this paper discusses an analog of gauss lobatto rules i e szego quadrature rules with two prescribed nodes we refer to these rules as szego lobatto rules their properties as well as numerical methods for their computation are discussed

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

a quantitative polya szego principle andrea cianchidipartimento di matematica e applicazioni per l architetturapiazza ghiberti 27 50122 firenze italye mail cianchi uni itluca espositodipartimento di ingegneria dell informazione e matematica applicatavia ponte don melillo 84084 fisciano sa italye mail esposito diima unisa itnicola fuscodipartimento di matematica e applicazionivia cintia 80126 napoli italye mail n fusco unina itcristina trombettidipartimento di matematica e applicazionivia cintia 80126 napoli italye mail cristina unina itjune 27 2006

desingularization of vortices for the euler equation we study the existence of stationary classical solutions of the incompressible euler equation in the planes that approximate singular stationary solutions of this equation the construction is performed by studying the asymptotics of equation varepsilon 2 delta u varepsilon u varepsilon q frac kappa 2 pi log frac 1 varepsilon p with dirichlet boundary conditions and q a given function we also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices

szegő type quadrature formulas abstract we consider the approximation of integrals with respect to measures supported on the unit circle by means of positive quadrature formulas with maximal domain of exactness and up to three preassigned nodes the quadrature formulas are exact in a new nested sequence of subspaces of laurent polynomials that allows us to extend analogous properties to the usual quadrature formulas on the real line estimations for the error of these quadrature formulas depending on the regularity of the integrand are obtained

positive solutions to a fourth order elliptic problem by the lusternik schnirelmann category abstract in this paper we consider the fourth order problem 2 u u s 1 u u 2 2 u in u u 0 in u u 0 on where is a smooth bounded domain in r n n 5 and 2 2 n n 4 we assume 2 s 1 2 in case n 8 and 2 2 s 1 2 for the critical dimensions n 5 6 7 then we prove that if has a rich topology described by its lusternik schnirelmann category then the problem has multiple solutions at least as many as cat in case the parameter 0 is sufficiently small

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

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

trigonometric orthogonal systems and quadrature formulae quadrature rules with maximal even trigonometric degree of exactness are considered we give a brief historical survey on such quadrature rules special attention is given on an approach given by turetzkii a h turetzkii on quadrature formulae that are exact for trigonometric polynomials east j approx 11 3 2005 337 359 translation in english from uchenye zapiski vypusk 1 149 seria math theory of functions collection of papers izdatel stvo belgosuniversiteta imeni v i lenina minsk 1959 pp 31 54 the main part of the topic is orthogonal trigonometric systems on 0 email protected or on email protected p with respect to some weight functions w x we prove that the so called orthogonal trigonometric polynomials of semi integer degree satisfy a five term recurrence relation in particular we study some cases with symmetric weight functions also we present a numerical method for constructing the corresponding quadratures of gaussian type finally we give some numerical examples also we compare our method with other available methods

generalized averaged szegź quadrature rules szeg quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane however often it is difficult to determine the quadrature error recently spalevic introduced generalized averaged gauss quadrature rules for estimating the quadrature error obtained when applying gauss quadrature over an interval on the real axis we describe analogous quadrature rules for the unit circle that often yield higher accuracy than szeg rules using the same moment information and may be used to estimate the error in szeg quadrature rules

systematic perturbation calculation of integrals with applications to physics in this paper we generalize and improve a method for calculating the period of a classical oscillator and other integrals of physical interest which was recently developed by some of the authors we derive analytical expressions that prove to be more accurate than those commonly found in the literature and test the convergence of the series produced by the approach

a connection between szegő lobatto and quasi gauss type quadrature formulas in this paper we obtain new results on positive quadrature formulas with prescribed nodes for the approximation of integrals with respect to a positive measure supported on the unit circle we revise szeg lobatto rules and we present a characterization of their existence in particular when the measure on the unit circle is symmetric this characterization can be used to recover in a more elementary way a recent characterization result for the existence of positive quasi gauss quasi radau and quasi lobatto rules quasi gauss type due to b beckermann et al some illustrative numerical examples are finally carried out in order to show the powerfulness of our results

the shape of extremal functions for poincare sobolev type inequalities in a ball we study extremal functions for a family of poincare sobolev type inequalities these functions minimize for subcritical or critical p 2 p 2 the quotient u 2 u p u 2 u p among all u h 1 b 0 u h 1 b 0 with b u 0 b u 0 here b is the unit ball in rn r n we show that the minimizers are axially symmetric with respect to a line passing through the origin we also show that they are strictly monotone in the direction of this line in particular they take their maximum and minimum precisely at two antipodal points on the boundary of b we also prove that for p close to 2 minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis and that once the symmetry axis is fixed they are unique up to multiplication by a constant in space dimension two we prove that minimizers are not antisymmetric for large p

stieltjes type polynomials on the unit circle stieltjes type polynomials corresponding to measures supported on the unit circle t are introduced and their asymptotic properties away from t are studied for general classes of measures as an application we prove the convergence of an associated sequence of interpolating rational functions to the corresponding caratheodory function in turn this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the gauss kronrod rule provided that the integrand is analytic in a neighborhood of t

rational szego quadratures associated with chebyshev weight functions in this paper we characterize rational szeg o quadrature formulas associated with chebyshev weight functions by giving explicit expressions for the corresponding para orthogonal rational functions and weights in the quadratures as an application we give characterizations for szeg o quadrature formulas associated with rational modifications of chebyshev weight functions some numerical experiments are finally presented

development of accurate solutions for a classical oscillator abstract we present a method to obtain arbitrarily accurate solutions for conservative classical oscillators the method that we propose here works both for small and large nonlinearities and provides simple analytical approximations a comparison with the standard lindstedt poincare method is presented from which the advantages of our method are clear

on the sharp effect of attaching a thin handle on the spectral rate of convergence abstract consider two domains connected by a thin tube it can be shown that the resolvent of the dirichlet laplacian is continuous with respect to the channel section parameter this in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation under an explicit nondegeneracy condition we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain as an application we prove that again under an explicit nondegeneracy condition the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions

quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity in this paper we investigate the szego radau and szego lobatto quadrature formulas on the unit circle these are n m point formulas for which m nodes are fixed in advance with m 1 and m 2 respectively and which have a maximal domain of validity in the space of laurent polynomials this means that the free parameters free nodes and positive weights are chosen such that the quadrature formula is exact for all powers z j email protected email protected p with p p n m as large as possible