beyond the entropy power inequality via rearrangements a lower bound on the renyi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements for the special case of boltzmann shannon entropy this lower bound is better than that given by the entropy power inequality several applications are discussed including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of levy processes

comparison results for nonlinear degenerate dirichlet and neumann problems with general growth in the gradient abstract this paper deals with both dirichlet and neumann problems for a class of nonlinear degenerate elliptic equations with general growth in the gradient first we give an existence result of a spherically symmetric solution to the symmetrized problems with data depending only on the radials second we prove that the solutions of the original problems can be compared under a rearrangement with the solutions of the symmetrized problems

q deformed chern characters for quantum groups suq n in this paper we introduce an n n matrix eab in the quantum groups suq n to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on suq n such as the q deformed structure constant ckij and the q deformed transposition operator from the q gauge covariant condition we study the generalized q deformed killing form and construct the mth q deformed chern class pm for the quantum groups suq n in terms of the q deformed homotopy operator we are able to compute the q deformed chern simons q2m 1 by the condition dq2m 1 pm furthermore the q deformed cocycle hierarchy the q deformed gauge covariant lagrangian and the q deformed yang mills equation are derived

nonlinear aggregation diffusion equations radial symmetry and long time asymptotics we analyze under which conditions equilibration between two competing effects repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction occurs this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses all stationary states with suitable regularity are shown to be radially symmetric by means of continuous steiner symmetrization techniques calculus of variations tools allow us to show the existence of global minimizers among these equilibria finally in the particular case of newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation diffusion equations towards this unique equilibrium profile up to translations as t rightarrow infty

existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data in this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is formula maths where is a bounded open subset of r n n 2 ap is the so called p laplace operator 1 p n is a radon measure with bounded variation on 0 p 1 0 p 1 and c and b belong to the lorentz spaces l p 1 r n p 1 r and l n 1 respectively in particular we prove the existence under the assumptions that p 1 c belongs to the lorentz space ln p 1 r n p 1 r and c n r is small enough

minimization of the ground state for two phase conductors in low contrast regime in this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain in order to minimize the first eigenvalue the ground state of a dirichlet operator it is known when the domain is a ball that the solution is radial and it was conjectured that the optimal distribution of the materials consists of putting the material with the highest conductivity in a ball around the center we show that this conjecture is not true in general for this we consider the particular case where the two conductivities are close to each other low contrast regime and we perform an asymptotic expansion with respect to the difference of conductivities we find that the optimal solution is the union of a ball and an outer ring when the amount of the material with the higher density is large enough

estimates for fully anisotropic elliptic equations with a zero order term abstract integral estimates for weak solutions to a class of dirichlet problems for nonlinear fully anisotropic elliptic equations with a zero order term are obtained by symmetrization techniques the anisotropy of the principal part of the operator is governed by a general n dimensional young function of the gradient which is not necessarily of polynomial type and need not satisfy the 2 condition

new polya szego type inequalities and an alternative approach to comparison results for pde s abstract we prove some polya szego type inequalities which involve couples of functions and their rearrangements our inequalities reduce to the classical polya szego principle when the two functions coincide as an application we give a different proof of a comparison result for solutions to dirichlet boundary value problems for laplacian equations proved in 1

symmetric rearrangements around infinity with applications to levy processes we prove a new rearrangement inequality for multiple integrals which partly generalizes a result of friedberg and luttinger arch ration mech 61 35 44 1976 and can be interpreted as involving symmetric rearrangements of domains around infty as applications we prove two comparison results for general levy processes and their symmetric rearrangements the first application concerns the survival probability of a point particle in a poisson field of moving traps following independent levy motions we show that the survival probability can only increase if the point particle does not move and the traps and the levy motions are symmetrically rearranged this essentially generalizes an isoperimetric inequality of peres and sousi geom funct anal 22 4 1000 1014 2012 for the wiener sausage in the second application we show that the q capacity of a borel measurable set for a levy process can only decrease if the set and the levy process are symmetrically rearranged this result generalizes an inequality obtained by watanabe z wahrsch verw gebiete 63 487 499 1983 for symmetric levy processes

symmetrization results for classes of nonlinear elliptic equations with q growth in the gradient abstract in this paper we study the dirichlet problem for a class of nonlinear elliptic equations in the form a u h x u du g x u where the principal term is a leray lions operator defined on w 0 1 p comparison results are obtained between the rearrangement of a solution u of dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric

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

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

a polya szego inequality in a fermi metric abstract let n k n with n 2 and 1 k n given a positive function c r n k we form the riemannian metric g on r n associated to the differential expression d s 2 d x 2 x 2 d y 2 where we write r n x x y with x r n k and y r k let be a log convex measure on r k with smooth density and the product measure l n k on r n where c r n k is a positive function we obtain a polya szego inequality of the form r n f u j g u d r n f u s j g u s d for sobolev functions u where the operation s refers to the k n steiner symmetrisation with respect to the gradient operator g is associated to the metric g and the mapping j may be seen as interpolating between the tangent space at x and r n the nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses as an application we derive a polya szego inequality in the hyperbolic plane that takes the above form

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

critical periods of perturbations of reversible rigidly isochronous centers abstract in this paper we discuss bifurcation of critical periods in an m th degree time reversible system which is a perturbation of an n th degree homogeneous vector field with a rigidly isochronous center at the origin we present period bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function this procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods applying our procedure to the case n m 2 we determine the maximum number of critical periods and their location to the case n m 3 we investigate the bifurcation of critical periods up to the first order in e and obtain the expression of the second period bifurcation function when the first one vanishes

multiplicity of positive solutions for a class of elliptic equations in divergence form we prove results concerning the existence and multiplicity of positive solutions for the quasilinear equation div a ex u p 2 u u p 2 u f u in r n u w 1 p r n where 2 p n a is a positive potential and f is a superlinear function we relate the number of solutions with the topology of the set where a attains its minimum the results are proved by using minimax theorems and ljusternik schnirelmann theory

on the continuity of weakly monotone functions abstract a criterion for the continuity of weakly monotone functions is proposed in terms of the decreasing rearrangement of their gradient its use in the proof of the continuity of weakly monotone functions with gradient in rearrangement invariant spaces is also demonstrated in particular weakly monotone functions whose gradient belongs to an orlicz space or to a lorentz space are discussed and results available in the literature in this regard are extended and refined

symmetrization of levy processes and applications abstract it is shown that many of the classical generalized isoperimetric inequalities for the laplacian when viewed in terms of brownian motion extend to a wide class of levy processes the results are derived from the multiple integral inequalities of brascamp lieb and luttinger but the probabilistic structure of the processes plays a crucial role in the proofs

the sharp sobolev type inequalities in the lorentz sobolev spaces in the hyperbolic spaces abstract let w 1 l p q h n 1 q p denote the lorentz sobolev spaces of order one in the hyperbolic spaces h n our aim in this paper is three fold first of all we establish a sharp poincare inequality in w 1 l p q h n with 1 q p which generalizes the result in 41 to the setting of lorentz sobolev spaces second we prove several sharp poincare sobolev type inequalities in w 1 l p q h n with 1 q p n which generalize the results in 45 to the setting of lorentz sobolev spaces finally we provide the improved moser trudinger type inequalities in w 1 l n q h n in the critical case p n with 1 q n which generalize the results in 43 and improve the results in 57 in the proof of the main results we shall prove a polya szego type principle in w 1 l p q h n with 1 q p which maybe is of independent interest

optimal lower bounds for eigenvalues of linear and nonlinear neumann problems in this paper we prove a sharp lower bound for the first non trivial neumann eigenvalue 1 for the p laplace operator p 1 in a lipschitz bounded domain in n our estimate does not require any convexity assumption on and it involves the best isoperimetric constant relative to in a suitable class of convex planar domains our bound turns out to be better than the one provided by the payne weinberger inequality

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

smoothness properties for the optimal mixture of two isotropic materials the compliance and eigenvalue problems the present paper is devoted to obtaining some smoothness results for the solution of two classical control problems relative to the optimal mixture of two isotropic materials in the first one the goal is to maximize the energy in the second one we want to minimize the first eigenvalue of the corresponding elliptic operator at least for the first problem it is well known that it does not have a solution in general thus we deal with a relaxed formulation one of the applications of our results is in fact the nonexistence of a solution for the unrelaxed problem in this sense we improve a classical nonexistence result by murat and tartar for the maximization of the energy which was obtained assuming the solution smooth we also get a counterexample to the existence of a solution for the eigenvalue problem which to our knowledge was an open problem

sharp sobolev inequalities in lorentz spaces for a mean oscillation abstract we exhibit the optimal constant for sobolev inequalities in lorentz spaces for a mean oscillation and its relation with a boundedness of the hardy littlewood maximal operator in sobolev spaces some applications to a scale invariant form of hardy s inequality in a limiting case are also considered

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 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

an optimal design problem for a two phase isolating material in the wall of a cavity abstract we are interested in the optimal distribution of two isolating materials in the wall of a cavity in order to get the best isolation when the amount of the best isolating material is limited we assume that the wall is of width e 0 small the problem can be formulated as the minimization of the first eigenvalue of a certain diffusion operator where the diffusion constant is of order one inside the cavity and of order e in the wall as it is usual in optimal design the problem has not solution in general and therefore it is necessary to work with a relaxed formulation passing to the limit when e tends to zero we get an asymptotic model where the variable control is now in a robin boundary condition instead of the diffusion operator we also present some numerical simulations showing the behavior of the solutions

on low eigenvalues of the laplacian with mixed boundary conditions abstract by means of the so called symmetrization we study the eigenvalue problem for the laplace operator with mixed boundary conditions we obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of bessel functions

equivalent and attained version of hardy s inequality in rn abstract we investigate connections between hardy s inequality in the whole space r n and embedding inequalities for sobolev lorentz spaces in particular we complete previous results due to 1 alvino and 28 talenti by establishing optimal embedding inequalities for the sobolev lorentz quasinorm p q also in the range p q which remained essentially open since 1 attainability of the best embedding constants is also studied as well as the limiting case when q here we surprisingly discover that the hardy inequality is equivalent to the corresponding sobolev marcinkiewicz embedding inequality moreover the latter turns out to be attained by the so called ghost extremal functions of 6 brezis vazquez in striking contrast with the hardy inequality which is never attained in this sense our functional approach seems to be more natural than the classical sobolev setting answering a question raised in 6

extremal energies of laplacian operator different configurations for steady vortices abstract in this paper we study a maximization and a minimization problem associated with a poisson boundary value problem optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions the main contribution of this paper is to determine the optimal solutions at first we determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime secondly for the high contrast regime two optimization algorithms are developed for the minimization problem we prove that our algorithm converges to the global minimizer regardless of the initializer the maximization algorithm is capable of deriving all local maximizers including the global one numerical experiments lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function