non reciprocity in nonlinear elastodynamics abstract reciprocity is a fundamental property of linear time invariant lti acoustic waveguides governed by self adjoint operators with symmetric green s functions the break of reciprocity in lti elastodynamics is only possible through the break of time reversal symmetry on the micro level and this can be achieved by imposing external biases adding nonlinearities or allowing for time varying system properties we present a volterra series based asymptotic analysis for studying spatial non reciprocity in a class of one dimensional 1d time invariant elastic systems with weak stiffness nonlinearities we show that nonlinearity is neither necessary nor sufficient for breaking reciprocity in this class of systems rather it depends on the boundary conditions the symmetries of the governing linear and nonlinear operators and the choice of the spatial points where the non reciprocity criterion is tested extension of the analysis to higher dimensions and time varying systems is straightforward from a mathematical point of view but not in terms of new non reciprocal physical phenomena whereas the connection of non reciprocity and time irreversibility can be studied as well finally we show that suitably defined non reciprocity measures enable optimization and can provide physical understanding of the nonlinear effects in the dynamics enabling one to establish regimes of maximum nonlinearity we highlight the theoretical developments by means of a numerical example

linear fourier and stochastic analysis to extend the traditional fourier theory of stationary processes some new boundedness notions for processes and for random measures are introduced this leads for these processes and measures to plancherel and hausdorff young type formulae and to a decomposition theory via dilations and multiplications various applications of our methods are also presented

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

schrodinger operators with boundary singularities hardy inequality pohozaev identity and controllability results abstract the aim of this paper is two folded firstly we study the validity of a pohozaev type identity for the schrodinger operator a x 2 r in the situation where the origin is located on the boundary of a smooth domain r n n 1 showing some applications to semi linear elliptic equations the problem we address is very much related to optimal hardy poincare inequalities with boundary singularities which have been investigated in the recent past in various papers in view of that the proper functional framework is described and explained secondly we use the pohozaev identity to derive the method of multipliers and we apply it to study the exact boundary controllability for the wave and schrodinger equations corresponding to the singular operator a in particular this complements and extends well known results by vancostenoble and zuazua 2009 38 who discussed the same issue in the case of interior singularity

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

discrete time subharmonic modelling and analysis traditionally the volterra time and frequency domain analysis tools cannot be applied to severely non linear systems in this paper a new method of building a time domain narx miso model for a class of severely siso non linear systems that exhibit subharmonics is introduced and it is shown how this allows the volterra time and frequency domain analysis to be extended to this class of non linear systems the new approach is based on decomposing the original single input based on a fourier analysis to provide a set of modified input signals which have the same period as the output signal a miso narx model can then be constructed from the decomposed multiple inputs and the single output signal the resulting miso model is shown to meet the basic requirement for the existence of a volterra series representation from which important frequency domain properties can be derived explained and discussed this is done by first introducing the derivation of generalized frequency response functions gfrfs from tim

harmonizability v boundedness 2 p boundedness of stochastic processes some new classes of discrete time non stationary processes related to the harmonizable andv bounded classes are introduced a few characterizations are obtained which in turn unify thev bounded theory our main results depend on a special form of grothendieck s inequality

sharp hardy inequalities in the half space with trace remainder term abstract in this paper we deal with a class of inequalities which interpolate the kato inequality and the hardy inequality in the half space starting from the classical hardy s inequality in the half space r n r n 1 0 we show that if we replace the optimal constant n 2 2 4 with a smaller one 2 2 4 2 n then we can add an extra trace term similar to that one that appears in the kato inequality the constant in the trace remainder term is optimal and it tends to zero when goes to n while it is equal to the optimal constant in kato s inequality when 2 the approach is based on a very classical method of calculus of variation due to weierstrass and developed by hilbert that usually is considered to prove that the solutions of the euler lagrange equation associated to a functional are in fact extremals in this paper we will show how this method is well suited also to functionals that have no extremals

comparison results for pdes with a singular potential several comparison results are obtained for solutions to linear elliptic and parabolic equations with a singular potential solutions to these equations are singular in many cases and our results roughly say that they all have comparable singularities provided that they belong to an appropriate space we formulate the hypothesis on the potential in terms of an inequality which in the case of the well known inverse square potential is a consequence of an improvement of hardy s inequality due to vazquez and zuazua

pointwise estimates for the ground state of singular dirichlet fractional laplacian we establish sharp pointwise estimates for the ground states of some singular fractional laplace operators on relatively compact euclidean subsets the considered operators are of the type where is the fractional laplacian on an open subset of with zero exterior condition and the intrinsic ultracontractivity property for such an operators is discussed as well and a sharp large time asymptotic for their heat kernels is derived

optimizing improved hardy inequalities let be a bounded domain in rn n 3 containing the origin motivated by a question of brezis and vazquez we consider an improved hardy inequality with best constant b that we formally write as n 22 21 x 2 bv x we first give necessary conditions on the potential v under which the previous inequality can or cannot be further improved we show that the best constant b is never achieved in h01 and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in h01 our analysis reveals that the original inequality can be repeatedly improved by adding on the right hand side specific potentials this leads to an infinite series expansion of hardy s inequality the series obtained is in some sense optimal in establishing these results we derive various sharp improved hardy sobolev inequalities

nonexistence of distributional supersolutions of a semilinear elliptic equation with hardy potential abstract in this paper we study nonexistence of non negative distributional supersolutions for a class of semilinear elliptic equations involving inverse square potentials

prediction of stable processes spectral and moving average representations for stable processes which are fourier transforms of processes with independent increments we obtain a wold decomposition we characterize their regularity and singularity and in the discrete parameter case we derive their linear predictors in sharp contrast with the gaussian case regular stable processes which are fourier transforms of processes with independent increments are not moving averages of stable motion

application of higher order spectral analysis to cubically nonlinear system identification in this study digital higher order spectral analysis and frequency domain volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time invariant systems up to third order the primary focus is on consideration of random excitation of nonlinear systems and thus the approach makes extensive use of higher order spectral analysis to determine the frequency domain volterra kernels which correspond to linear quadratic and cubic transfer functions although the volterra model is nonlinear in terms of its input it is linear in terms of its unknown transfer functions thus a least squares approach is used to determine the optimal in a least squares sense set of linear quadratic and cubic transfer functions of particular practical note is the fact that the approach of this paper is valid for non gaussian as well as gaussian random excitation it may also be utilized for multitone inputs the complexity of the problem addressed in this paper arises from two principal causes 1 the necessity to work in a 3d frequency space to describe cubically nonlinear systems and 2 the necessity to characterize the non gaussian random excitation by computing higher order spectral moments up to sixth order a detailed description of the approach used to determine the nonlinear transfer functions including considerations necessary for digital implementation is presented

on the best possible remaining term in the hardy inequality we give a necessary and sufficient condition on a radially symmetric potential v on a bounded domain omega of n that makes it an admissible candidate for an improved hardy inequality of the following type for every element in h 1 0 omega integral omega vector differential u 2 dx n 2 2 2 integral omega u 2 x 2 dx or c integral omega v x u 2 dx a characterization of the best possible constant c v is also given this result yields easily the improved hardy s inequalities of brezis vazquez brezis h vazquez jl 1997 blow up solutions of some nonlinear elliptic problems revista mat univ complutense madrid 10 443 469 adimurthi et al adimurthi chaudhuri n ramaswamy n 2002 an improved hardy sobolev inequality and its applications proc am math soc 130 489 505 and filippas tertikas filippas s tertikas a 2002 optimizing improved hardy inequalities j funct anal 192 186 233 as well as the corresponding best constants our approach clarifies the issue behind the lack of an optimal improvement while yielding the following sharpening of known integrability criteria if a positive radial function v satisfies lim inf r o ln r integral r o sv s ds infinity then there exists rho rho omega 0 such that the above inequality holds for the scaled potential v rho x v x rho on the other hand if lim r 0 ln r integral r o sv s ds infinity then there is no rho 0 for which the inequality holds for v rho

on the convergence of the volterra series representation of the duffing s oscillators subjected to harmonic excitations abstract this paper is concerned with the investigation of the convergence issue of the volterra series representations of the duffing s oscillator subjected to harmonic inputs a simple criterion is proposed to determine the upper limit of the magnitude of the harmonic inputs a comparison between the new proposed criterion and a criterion suggested by tomlinson et al g r tomlinson g manson g m lee a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series journal of sound and vibration 190 1996 751 762 was carried out the results show that the new criterion can provide a more accurate prediction about the convergence of the volterra series representation of the duffing s oscillator

continuous time non linear system identification in the frequency domain a new algorithm for identifying non linear differential equation models is introduced the algorithm avoids the computation of derivatives by using the generalized frequency response functions to reconstruct the model it is shown that the model can be constructed sequentially by building in first the linear terms then the quadratic terms and so on and provides unbiased estimates in the presence of noise a simulated example and the identification of a model relating the action of waveforces on an offshore structure are included to demonstrate the procedure

on schrodinger operators with multipolar inverse square potentials abstract positivity essential self adjointness and spectral properties of a class of schrodinger operators with multipolar inverse square potentials are discussed in particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

optimal hardy rellich inequalities maximum principle and related eigenvalue problem in this paper we deal with three types of problems concerning the hardy rellich s embedding for a bilaplacian operator first we obtain the hardy rellich inequalities in the critical dimension n 4 then we derive a maximum principle for fourth order operators with singular terms then we study the existence non existence simplicity and asymptotic behavior of the first eigenvalue of the hardy rellich operator 2 n 2 n 4 2 16 q x x 4 under various assumptions on the perturbation q

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

analysis of nonlinear oscillators using volterra series in the frequency domain the volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems both in the time and the frequency domain the volterra series is associated with the so called weakly nonlinear systems but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid volterra series representation for a given nonlinear differential equation barrett 1965 1 proposed a time domain criterion to prove that the volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity in this paper this time domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation a common and severe nonlinear phenomenon called jump a behavior associated with the duffing oscillator and the multi valued properties of the response solution is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators to predict the onset point of the jump and the volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

bessel pairs and optimal hardy and hardy rellich inequalities we give necessary and sufficient conditions on a pair of positive radial functions v and w on a ball b of radius r in rn n 1 so that the following inequalities hold for all u in c 0 infty b label one int limits b v x nabla u 2 dx geq int limits b w x u 2dx label two int limits b v x delta u 2 dx geq int limits b w x nabla u 2 dx n 1 int limits b left frac v x x 2 frac v r x x right nabla u 2dx this characterization makes a very useful connection between hardy type inequalities and the oscillatory behaviour of certain ordinary differential equations and helps in the identification of a large number of such couples v w that we call bessel pairs as well as the best constants in the corresponding inequalities this allows us to improve extend and unify many results old and new about hardy and hardy rellich type inequalities such as those obtained by caffarelli et al compos math 53 259 275 1984 brezis and vazquez revista mat univ complutense madrid 10 443 469 1997 wang and willem j funct anal 203 550 568 2003 adimurthi et al proc am math soc 130 489 505 2002 and many others

the heat equation with a singular potential of concern is the singular problem du dt au c x 2 iz x tz x 0 u0 x and its generalizations here c 0 x g rn t 0 and and t 0 are nonnegative and not both identically zero there is a dimension dependent constant c n such that the problem has no solution for c ct n for c s ct a necessary and sufficient conditions are found for and u0 so that a nonnegative solution exists

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

stable processes moving averages versus fourier transforms we study the relationship between fourier transforms and moving averages of stable processes the two classes have a large intersection however as soon as the noise is required to be bounded they become disjoint

resonance phenomenon for a gelfand type problem we consider positive radially symmetric solutions of u eu 1 inb u 0on b where b is the unit ball in rn n 3 and 0 is a parameter we establish infinite multiplicity of regular solutions for 3 n 9 and some and we obtain a bound for the morse index and the number of solutions when n 10

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

nontrivial solutions for perturbations of a hardy sobolev operator on unbounded domains in this paper we study the existence of nontrivial solution of the problem pu d x p u p 2u f u in and u 0 on where rn k is a bounded domain with smooth boundary in rk 1 k n existence is established using mountain pass lemma and concentration of compactness principle

null controllability for a heat equation with a singular inverse square potential involving the distance to the boundary function this article is devoted to the analysis of control properties for a heat equation with a singular potential 2 defined on a bounded c2 domain rn where is the distance to the boundary function more precisely we show that for any 1 4 the system is exactly null controllable using a distributed control located in any open subset of while for 1 4 there is no way of preventing the solutions of the equation from blowing up the result is obtained applying a new carleman estimate

ginzburg landau minimizers with prescribed degrees capacity of the domain and emergence of vortices let omega subset mathbb r 2 be a simply connected domain let omega be a simply connected subdomain of omega and set a omega setminus overline omega let j be the class of complex valued maps on the annular domain a with degree 1 both on partial omega and on partial omega we consider the variational problem of minimizing the ginzburg landau energy e lambda among all maps in j because only the degrees of the map are prescribed on the boundary the set j is not closed under weak h 1 convergence we show that the attainability of the minimum of e lambda over j is determined by the value of cap a where cap a is the h 1 capacity of the domain a in contrast it is known that the existence of minimizers of e lambda among the maps with a prescribed dirichlet boundary data does not depend on this geometric characteristic when cap a ge pi such an a is denoted as either subcritical if cap a pi or critical if cap a pi we show that the global minimizers of e lambda exist for each lambda 0 and that they are vortexless when lambda is large assuming that lambda to infty we demonstrate that the minimizers of e lambda converge in h 1 a and even better to an mathbb s 1 valued harmonic map which we explicitly identify when cap a lambda 0 or ii the global minimizers exist for each lambda 0 we conjecture that the second case never occurs further for large lambda we establish that the minimizing sequences minimizers in supercritical domains develop exactly two vortices a vortex of degree 1 near partial omega and a vortex of degree 1 near partial omega

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

diagonals of green s functions and applications abstract we describe the diagonal of the green s function for some sturm liouville operators as it was introduced and studied in a number of papers already and we discuss a number of applications that depend on its properties semilinear elliptic pde s with rotationally invariant and possibly singular coefficients with power type nonlinearities provide a particularly well suited class of examples where the results of this article apply

improved poincare inequalities although the hardy inequality corresponding to one quadratic singularity with optimal constant does not admit any extremal function it is well known that such a potential can be improved in the sense that a positive term can be added to the quadratic singularity without violating the inequality and even a whole asymptotic expansion can be build with optimal constants for each term this phenomenon has not been much studied for other inequalities our purpose is to prove that it also holds for the gaussian poincare inequality the method is based on a recursion formula which allows to identify the optimal constants in the asymptotic expansion order by order we also apply the same strategy to a family of hardy poincare inequalities which interpolate between hardy and gaussian poincare inequalities

a geometrical version of hardy s inequality we prove a version of hardy s type inequality in a domain omega subset of r n which involves the distance to the boundary and the volume of omega in particular we obtain a result which gives a positive answer to a question asked by h brezis and m marcus

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions we present some comparison results for solutions to certain non local elliptic and parabolic problems that involve the fractional laplacian operator and mixed boundary conditions given by a zero dirichlet datum on part of the complementary of the domain and zero neumann data on the rest these results represent a non local generalization of a hopf s lemma for elliptic and parabolic problems with mixed conditions in particular we prove the non local version of the results obtained by davila and davila and dupaigne for the classical case s 1 in 23 24 respectively

a note on hardy s inequalities with boundary singularities abstract let be a smooth bounded domain in r n with n 1 in this paper we study the hardy poincare inequalities with weight function singular at the boundary of in particular we give sufficient conditions so that the best constant is achieved

a variational problem with lack of compactness for h1 2 s1 s1 maps of prescribed degree abstract we consider for maps in h 1 2 s 1 s 1 a family of semi norms equivalent to the standard one we ask whether for such a norm there is some map in h 1 2 s 1 s 1 of prescribed topological degree equal to 1 and minimal norm in general the answer is no due to concentration phenomena the existence of a minimal map is sensitive to small perturbations of the norm we derive a sufficient condition for the existence of minimal maps in particular we prove that for every given norm there are arbitrarily small perturbations of it for which the minimum is attained in case there is no minimizer we determine the asymptotic behavior of minimizing sequences we prove that for such minimizing sequences the energy concentrates near a point of s 1 we describe this concentration in terms of bubbling off of circles

convergence analysis of volterra series response of nonlinear systems subjected to harmonic excitation volterra series provides a strong platform for non linear analysis and higher order frequency response functions however limited convergence is an inherent difficulty associated with the series and needs to be addressed rigorously prior to its application to a physical system the power series representation of the response of non linear systems subjected to harmonic excitation is investigated in this study the problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non linear response though the procedure is applicable to general polynomial form non linearity it is illustrated for a duffing oscillator subjected to harmonic excitation a general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non dimensional non linear parameter critical values of this parameter representing the upper limit of excitation level for the convergence are defined for a wide range of excitation frequencies zones of convergence and divergence of the response series are presented graphically for a range of the non dimensional non linear parameter and the number of terms included in the approximation of a response harmonic an algorithm based on ratio test is presented to compute the critical value of the non dimensional non linear parameter results obtained from the suggested algorithm are found to be in close agreement with the exact values the method gives better results compared to previous methods and has wider application in terms of excitation frequency the procedure is also investigated for a two degree of freedom system

semilinear elliptic equations for the fractional laplacian with hardy potential abstract in this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the hardy potential

improving l2 estimates to harnack inequalities we consider operators of the form l l v where l is an elliptic operator and v is a singular potential defined on a smooth bounded domainir n with dirichlet boundary conditions we allow the boundary of to be made of various pieces of different codimension we assume that l has a generalized first eigenfunction of which we know two sided estimates under these assumptions we prove optimal sobolev inequalities for the operator l we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic harnack inequality up to the boundary as well as sharp heat kernel estimates

a hardy moser trudinger inequality abstract in this paper we obtain an inequality on the unit disk b in r 2 which improves the classical moser trudinger inequality and the classical hardy inequality at the same time namely there exists a constant c 0 0 such that b e 4 u 2 h u d x c 0 u c 0 b 0 where h u b u 2 d x b u 2 1 x 2 2 d x this inequality is a two dimensional analog of the hardy sobolev maz ya inequality in higher dimensions which has been intensively studied recently we also prove that the supremum is achieved in a suitable function space which is an analog of the celebrated result of carleson chang for the moser trudinger inequality

multiple critical points for nondifferentiable functionals involving hardy potentials abstract in this paper we study general functionals of the calculus of variations with the presence of a hardy potential we will improve several results obtained in the semilinear framework we will first prove a general weak lower semicontinuity result which will imply the existence of a minimum point whenever the functional is coercive then we will demonstrate existence and multiplicity results of critical points even if our functional is not differentiable we will apply a nonsmooth critical point theory developed in corvellec et al nonlinear anal 1 1993 151 and degiovanni and marzocchi ann mat pura appl 167 1994 73

some remarks on elliptic problems with critical growth in the gradient abstract in this work we analyze existence nonexistence multiplicity and regularity of solution to problem 1 u u u 2 f x in u 0 on where is a continuous nondecreasing positive function and f belongs to some suitable lebesgue spaces

the spectral representation of stable processes harmonizability and regularity we show that symmetric stable moving average processes are not harmonizable however we show that a concept of generalized spectrum holds for alll p bounded processes o p 2 in capep 2 generalized spectrum is a measure and the classical representation follows for strongly harmonizable symmetric stable processes we derive necessary and sufficient conditions for the regularity and the singularity for 0 2 using known results on the invariant subspaces we also get cramer wold decomposition for the case 0 2

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

existence and asymptotic behavior of ground states for quasilinear singular equations involving hardy sobolev exponents abstract we study the existence and decaying rate of solutions for the quasilinear problem p u x f u x g u in r n u 0 in r n u x 0 where p stands for the p laplacian operator 1 p n r n 0 is continuous and not identically zero 0 is a parameter x is the euclidean norm of x 0 p f g 0 0 are continuous and nondecreasing f has sublinear growth and the hardy sobolev exponent p p n n p bounds the growth of g we deal with variational methods and the lower and upper solutions technique