effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

multiple solutions for a henon like equation on the annulus abstract for the equation u x 2 u p 1 1 x 3 we prove the existence of two solutions for large and of two additional solutions when p is close to the critical sobolev exponent 2 2 n n 2 a symmetry breaking phenomenon appears showing that the least energy solutions cannot be radial functions

uniform concentration compactness for sobolev spaces on variable domains abstract we present a new method for proving existence results in shape optimization problems involving the eigenvalues of the dirichlet laplace operator this method brings together the convergence theory and the concentration compactness principle given a sequence of open sets a n n n in r n not necessarily bounded but of uniformly bounded measure we prove a concentration compactness result in l l 2 r n for the sequence of resolvent operators r a n n n where r a n l 2 r n h 1 0 a n r a n 1

an eigenvalue problem for hemivariational inequalities with combined nonlinearities on an infinite strip in this paper a class ofeigenvalue problems f hemivariational inequalities is studied which is defined on domains ofthe type r is a bounded open subset of r m m 1 and it involves

hylomorphic solitons for the benjamin ono and the fractional kdv equations abstract this paper concerns with the existence of solitons namely stable solitary waves for the benjamin ono and the fractional kdv equations

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

upper bound of weak limitation for the gross pitaevskii equation abstract this paper is concerned with the blow up solutions of gross pitaevskii equation we obtain the upper bound of weak limitation for the blow up solutions by using the method of cazenave 2003 3 as well as the concentration compact principle

on the lowest eigenvalue of general operators for the intersection of two domains e lieb invent math 74 1983 441 448 has proved that ifaandbare two bounded domains in rn then there exists a translation ysuch that 1 a yb 1 a 1 b where 1is the first eigenvalue of the laplacian with dirichlet boundary conditions here we extend this result to elliptic operators in divergence froml div q x c x with mixed dirichlet neumann boundary conditions ona if l a is the corresponding eigenvalue we show that there exists a translation ysuch that l a yb l a 1 b ift q x 2for allx a rn one can further improve the estimate for non isotropic operators where can be large by taking into account rotations ofb in that case a similar inequality holds ift q x 2 whereqis a mean value ofqin different directions

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

uniqueness of positive ground state solutions of the logarithmic schrodinger equation we prove the uniqueness of positive ground state solutions of the problem frac d 2 u dr 2 frac n 1 r frac du dr u ln u 0 u r 0 forall r ge 0 and u r u r to 0 0 as r to infty this equation is derived from the logarithmic schrodinger equation rm i psi t delta psi u ln left u 2 right and also from the classical equation frac partial u partial t delta u u left u p 1 right u for each n ge 1 a positive ground state solution is u 0 r exp left frac r 2 4 frac n 2 right 0 le r infty we combine u 0 r with energy estimates and associated ricatti equation estimates to prove that for each n in left 1 9 right u 0 r is the only positive ground state we also investigate the stability of u 0 r several open problems are stated

phase ordering after a deep quench the stochastic ising and hard core gas models on a tree consider a low temperature stochastic ising model in the phase coexistence regime with markov semigroup p t a fundamental and still largely open problem is the understanding of the long time behavior of p t when the initial configuration is sampled from a highly disordered state e g a product bernoulli measure or a high temperature gibbs measure exploiting recent progresses in the analysis of the mixing time of monte carlo markov chains for discrete spin models on a regular b ary tree open image in new window we study the above problem for the ising and hard core gas independent sets models on open image in new window if is a biased product bernoulli law then under various assumptions on the bias and on the thermodynamic parameters we prove almost sure weak convergence of p t to an extremal gibbs measure pure phase and show that the limit is approached at least as fast as a stretched exponential of the time t in the context of randomized algorithms and if one considers the glauber dynamics on a large finite tree our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution

scaling limits of the schelling model the schelling model of segregation introduced by schelling in 1969 as a model for residential segregation in cities describes how populations of multiple types self organize to form homogeneous clusters of one type in this model vertices in an n dimensional lattice are initially assigned types randomly as time evolves the type at a vertex v has a tendency to be replaced with the most common type within distance w of v we present the first mathematical description of the dynamical scaling limit of this model as w tends to infinity and the lattice is correspondingly rescaled we do this by deriving an integro differential equation for the limiting schelling dynamics and proving almost sure existence and uniqueness of the solutions when the initial conditions are described by white noise the evolving fields are in some sense very rough but we are able to make rigorous sense of the evolution in a key lemma we show that for certain gaussian fields h the supremum of the occupation density of h phi at zero taken over all 1 lipschitz functions phi is almost surely finite thereby extending a result of bass and burdzy in the one dimensional case we also describe the scaling limit of the limiting clusters obtained at time infinity thereby resolving a conjecture of brandt immorlica kamath and kleinberg

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

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

decay of heat semigroups in l and applications to nonlinear parabolic problems in unbounded domains we characterize all domains of rn such that the heat semigroup decays in l l or l l1 as t namely we prove that this property is equivalent to the poincare inequality and that it is also equivalent to the solvability of u f in l for all f l in particular under mild regularity assumptions on these properties are equivalent to the geometric condition that has finite inradius next we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains first we consider the quenching problem for singular parabolic equations we prove that the solution in quenches in finite time no matter how small the nonlinearity is if and only if does not fulfill the poincare inequality second for the semilinear heat equation with a power nonlinearity we prove roughly speaking that the trivial solution is stable in l or in l1 if and only if has finite inradius

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

unperturbed schelling segregation in two or three dimensions schelling s models of segregation first described in 1969 am econ rev 59 488 493 1969 are among the best known models of self organising behaviour their original purpose was to identify mechanisms of urban racial segregation but his models form part of a family which arises in statistical mechanics neural networks social science and beyond where populations of agents interact on networks despite extensive study unperturbed schelling models have largely resisted rigorous analysis prior results generally focusing on variants in which noise is introduced into the dynamics the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory young in individual strategy and social structure an evolutionary theory of institutions princeton university press princeton 1998 a series of recent papers brandt et al in proceedings of the 44th annual acm symposium on theory of computing stoc 2012 2012 barmpalias et al in 55th annual ieee symposium on foundations of computer science philadelphia 2014 j stat phys 158 806 852 2015 has seen the first rigorous analyses of 1 dimensional unperturbed schelling models in an asymptotic framework largely unknown in statistical mechanics here we provide the first such analysis of 2 and 3 dimensional unperturbed models establishing most of the phase diagram and answering a challenge from brandt et al in proceedings of the 44th annual acm symposium on theory of computing stoc 2012 2012

tipping points in 1 dimensional schelling models with switching agents schelling s spacial proximity model was an early agent based model illustrating how ethnic segregation can emerge unwanted from the actions of citizens acting according to individual local preferences here a 1 dimensional unperturbed variant is studied under switching agent dynamics interpretable as being open in that agents may enter and exit the model following the authors work barmpalias et al focs 2014 and that of brandt et al proceedings of the 44th acm symposium on theory of computing stoc 2012 2012 rigorous asymptotic results are established the dynamic allows either type to take over almost everywhere tipping points are identified between the regions of takeover and staticity in a generalization of the models considered in 1 and 3 the model s parameters comprise the initial proportions of the two types along with independent values of the tolerance for each type this model comprises a 1 dimensional spin 1 model with spin dependent external field as well as providing an example of cascading behaviour within a network

a variational inequality on the half line abstract multiple solutions are obtained for a variational inequality defined on the half line 0 our approach is based on a key embedding result as well as on the non smooth critical point theory for szulkin type functionals

the focusing singularity of the davey stewartson equations for gravity capillary surface waves abstract we analyze the self focusing of gravity capillary surface waves modeled by the davey stewartson equations this system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two dimensional schrodinger equation with a non local cubic nonlinearity with accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical as in the usual two dimensional cubic schrodinger equation which is the deep water limit of the davey stewartson equations we also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation

semilinear equations on fractal blowups abstract the paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal where imbedding of the energy space into l p is not compact in absence of invariant transformations involved in conventional concentration compactness argument the paper develops convergence reasoning based on the fractal s self similarity

existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli abstract we consider the nonlinear elliptic problem u u p in r u 0 in r u 0 in r where p 1 and r x r n r x r 1 with n 3 it is known that as r the number of nonequivalent solutions of the above problem goes to when p 1 n 2 n 2 n 3 here we prove the same phenomenon for any p 1 by finding o n 1 symmetric clustering bump solutions which concentrate near the set x 1 x n r x n 0 for large r 0

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

on the cauchy problem for the nonlinear schrodinger equations with time dependent linear loss gain abstract this paper is devoted to the cauchy problem for the nonlinear schrodinger equation with time dependent loss gain which reads i u t u u u i a t u 0 this equation appears in the recent studies of bose einstein condensates and optical systems we obtain some global existence and blow up results which depend on the size of the loss gain coefficient in particular we prove the global existence for the energy critical nonlinearity by scaling and compactness arguments we also discuss asymptotic profiles and concentration properties of blow up solutions

asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli we study the asymptotic behavior of positive solutions of the semilinear elliptic equation u u 0 in a u 0 on a where a x of rn a x a 1 are expanding annuli as a and is positive and superlinear at both 0 and we first show that there are a priori bounds for some positive solutions ua x as a then if we fix any direction after a suitable translation of ua the limiting solutions are non negative solutions on the infinite strip we can obtain more detailed descriptions of these limits if ua is radially symmetric least energy or least energy with a particular symmetry

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

infinitely many non radial solutions to a critical equation on annulus abstract in this paper we build infinitely many non radial sign changing solutions to the critical problem p u u 4 n 2 u in u 0 on on the annulus x r n a x b n 3 in particular for any integer k large enough we build a non radial solution which look like the unique positive solution u 0 to p crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 n 2 2 u 0 r 0 max a r b r n 2 2 u 0 r

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

the sharp threshold and limiting profile of blow up solutions for a davey stewartson system abstract the blow up solutions of the cauchy problem for the davey stewartson system which is a model equation in the theory of shallow water waves are investigated firstly the existence of the ground state for the system derives the best constant of a gagliardo nirenberg type inequality and the variational character of the ground state secondly the blow up threshold of the davey stewartson system is developed in r 3 thirdly the mass concentration is established for all the blow up solutions of the system in r 2 finally the existence of the minimal blow up solutions in r 2 is constructed by using the pseudo conformal invariance the profile of the minimal blow up solutions as t t blow up time is in detail investigated in terms of the ground state

travelling wave solutions for the discrete sine gordon equation with nonlinear pair interaction the focus of study is the nonlinear discrete sine gordon equation where the nonlinearity refers to a nonlinear interaction of neighbouring atoms the existence of travelling heteroclinic homoclinic and periodic waves is shown the asymptotic states are chosen such that the action functional is finite the proofs employ variational methods in particular a suitable concentration compactness lemma combined with direct minimisation and mountain pass arguments

self organized segregation on the grid we consider an agent based model with exponentially distributed waiting times in which two types of agents interact locally over a graph and based on this interaction and on the value of a common intolerance threshold tau decide whether to change their types this is equivalent to a zero temperature ising model with glauber dynamics an asynchronous cellular automaton with extended moore neighborhoods or a schelling model of self organized segregation in an open system and has applications in the analysis of social and biological networks and spin glasses systems some rigorous results were recently obtained in the theoretical computer science literature and this work provides several extensions we enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size epsilon 0 to size approx 0 134 namely we show that for 0 433 tau 1 2 and by symmetry 1 2 tau 0 567 the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood we further extend the interval leading to expected large segregated regions to size approx 0 312 considering almost segregated regions namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows in this case we show that for 0 344 tau le 0 433 and by symmetry for 0 567 le tau 0 656 the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood this behavior is reminiscent of supercritical percolation where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster the exponential bounds that we provide also imply that complete segregation where agents of a single type cover the whole grid does not occur with high probability for p 1 2 and the range of intolerance considered

large deviations for the chemical distance in supercritical bernoulli percolation the chemical distance d x y is the length of the shortest open path between two points x and y in an infinite bernoulli percolation cluster in this work we study the asymptotic behavior of this random metric and we prove that for an appropriate norm depending on the dimension and the percolation parameter the probability of the event 0 x d 0 x x 1 e 1 e exponentially decreases when x 1 tends to infinity from this bound we also derive a large deviation inequality for the corresponding asymptotic shape result

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

a semidefinite programming approach to optimal unambiguous discrimination of quantum states we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states we show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem based on this formulation we develop a set of necessary and sufficient conditions for an optimal quantum measurement we show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well known algorithms for solving semidefinite programs which are guaranteed to converge to the global optimum using the general conditions for optimality we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal we refer to this measurement as the equal probability measurement epm we then show that for any state set the prior probabilities of the states can be chosen such that the epm is optimal finally we consider state sets with strong symmetry properties and equal prior probabilities for which the epm is optimal we first consider geometrically uniform gu state sets that are defined over a group of unitary matrices and are generated by a single generating vector we then consider compound gu state sets which are generated by a group of unitary matrices using multiple generating vectors where the generating vectors satisfy a certain weighted norm constraint

existence of nontrivial homoclinic orbits for fourth order difference equations we obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equations by using mountain pass theorem a weak convergence argument and a discrete version of lieb s lemma

on the existence of positive radial solutions for nonlinear elliptic equations in annular domains abstract we study the existence of positive radially symmetric solutions of u g x f u 0 in annulus with dirichlet dirichlet neumann boundary conditions we show that the equation has a positive radial solution on any annulus if f and f are positive and f is superlinear at 0 and

harnack s inequality for a nonlinear eigenvalue problem on metric spaces abstract we prove harnack s inequality for first eigenfunctions of the p laplacian in metric measure spaces the proof is based on the famous moser iteration method which has the advantage that it only requires a weak 1 p poincare inequality as a by product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains

density theorems for sampling and interpolation in the bargmann fock space we give a complete description of sampling and interpolation in the bargmann fock space based on a density concept of beurling roughly speaking a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von neumann lattice and similarly a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von neumann lattice

a minimization method and applications to the study of solitons abstract roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time a soliton is a solitary wave which exhibits some strong form of stability so that it has a particle like behavior in this paper we prove a general abstract theorem theorem 26 which allows to prove the existence of a class of solitons such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons then we apply the abstract theory to problems related to the nonlinear schrodinger equation nse and to the nonlinear klein gordon equation nkg

on the sign of the mountain pass solution we consider a class of superlinear elliptic problems on a bounded domain in r n i t is known that besides the zero solution such superlinear problems possess at least one positive and one negative solution of mountain pass type as well as a sign changing solution moreover the least energy solutions ground states are among those one sign solutions we study the eect of the nonlinearity on the sign of such ground states and the corresponding role played by the interior geometry of and the geometry of 2001 elsevier science ltd all rights reserved

three dimensional vortices in abelian gauge theories abstract in this paper we consider an abelian gauge theory in r 4 equipped with the minkowski metric this theory leads to a system of equations the klein gordon maxwell equations which provide models for the interaction between the electromagnetic field and matter a three dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid under suitable assumptions we prove the existence of vortex solutions

asymptotically radial solutions in expanding annular domains in this paper we consider the problem left begin array ll delta u u p quad rm in omega r u 0 quad quad quad rm on partial omega r quad quad quad 0 1 end array right where p 1 and r is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as r longrightarrow infty the main goal of the paper is to prove for large r the existence of a positive solution to 0 1 which is close to the positive radial solution in the corresponding diffeomorphic annulus the proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions

cauchy problem for the l2 critical nonlinear schrodinger equation below h1 abstract in this paper we consider solutions of the l 2 critical focusing nonlinear schrodinger equation on r 2 for the initial data in h s s 1 mainly about the l 2 concentration phenomena for blow up solutions and global well posedness for small initial data

distinguishability of countable quantum states and von neumann lattice the condition for distinguishability of a countably infinite number of pure states by a single measurement is given distinguishability is to be understood as the possibility of an unambiguous measurement for a finite number of states it is known that the necessary and sufficient condition of distinguishability is that the states are linearly independent for an infinite number of states several natural classes of distinguishability can be defined we give a necessary and sufficient condition for a system of pure states to be distinguishable it turns out that each level of distinguishability naturally corresponds to one of the generalizations of linear independence to families of infinite vectors as an important example we apply the general theory to von neumann s lattice a subsystem of coherent states which corresponds to a lattice in the classical phase space we prove that the condition for distinguishability is that the area of the fundamental region of the lattice is greater than the planck constant and also find subtle behavior on the threshold these facts reveal the measurement theoretical meaning of the planck constant and give a justification for the interpretation that it is the smallest unit of area in the phase space the cases of uncountably many states and of mixed states are also discussed

least energy solutions of the emden fowler equation in hollow thin symmetric domains abstract in this paper the emden fowler equation is studied in a hollow thin domain which is invariant under the action of a closed subgroup of the orthogonal group then it is proved that if the domain is thin enough a least energy solution is not group invariant

compactness via symmetrization consider two types of translation invariant functionals i and j on rm and a sequence of functions fn whose corresponding symmetric rearrangements f n are convergent we show that fn themselves converge up to translations if either limn i fn limn i fn or limn j fn limn j fn these compactness results lead to applications in variational problems and stability problems in stellar dynamics

faber krahn type inequalities and uniqueness of positive solutions on metric measure spaces abstract we consider a general class of metric measure spaces equipped with a strongly local regular dirichlet form and provide a lower bound on the hitting time probabilities of the associated hunt process using these estimates we establish i a generalization of the classical lieb s inequality and ii uniqueness of nonnegative super solutions to semi linear elliptic equations on metric measure spaces finally using heat kernel estimates we generalize the local faber krahn inequality recently obtained in 28 to local and non local dirichlet spaces

existence and nonexistence of positive radial solutions for some semilinear elliptic problems in annular domains au f x u 0 in s u 0 on xj when n is some annulus and the above conditions in 4 29 are not necessarily satisfied in fact in our main result theorem 2 4 f is not necessarily c lim f t u u may not exist or may be u rn finite and f t u could be zero for u in some finite interval 0 p we also allow lim f t u u 0 corollary 2 6 and lim f t u u may not exist in example 2 3 below u o u 0 neither lim f u u nor lim f u u exists on the other hand we show that u 0 u m

existence and multiplicity results for hemivariational inequalities with two parameters a multiplicity result of ricceri based on a minimax inequality is applied to locally lipschitz functionals and the existence of one or two non trivial solutions for a class of two parameter hemivariational inequalities is proved an application is given to elliptic problems with discontinuous nonlinearities on unbounded domains

a variational approach to the isoperimetric inequality for the robin eigenvalue problem the isoperimetric inequality for the first eigenvalue of the laplace operator with robin boundary conditions was recently proved by daners in the context of lipschitz sets this paper introduces a new approach to the isoperimetric inequality based on the theory of special functions of bounded variation sbv we extend the notion of the first eigenvalue 1 for general domains with finite volume possibly unbounded and with irregular boundary and we prove that the balls are the unique minimizers of 1 among domains with prescribed volume

zero temperature glauber dynamics on mathbb z d we study zero temperature glauber dynamics on mathbb z d which is a dynamic version of the ising model of ferromagnetism spins are initially chosen according to a bernoulli distribution with density p and then the states are continuously and randomly updated according to the majority rule this corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field to one at zero temperature with no external field define p c mathbb z d to be the infimum over p such that the system fixates at with probability 1 it is a folklore conjecture that p c mathbb z d 1 2 for every 2 le d in mathbb n we prove that p c mathbb z d to 1 2 as d

the existence and uniqueness of bound state solutions of a semi linear equation we investigate the existence and uniqueness of bounded radially symmetric solutions in r3 of uf u 0 where f u is continuous piecewise linear and has three distinct real zeros we give a detail

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

traveling waves for nonlinear schrodinger equations with nonzero conditions at infinity for a large class of nonlinear schrodinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity we prove the existence of nontrivial finite energy traveling waves moving with speed c in any space dimension n 3 our results are valid as well for the gross pitaevskii equation and for nls with cubic quintic nonlinearity

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains abstract this paper is concerned with the problem of elliptic equations in annular domains involving derivative terms using nonlinear analysis methods some results regarding existence of solutions are established more precisely at least one radial solution is obtained basing on schauder s fixed point theorem and contraction mapping theorem respectively some iterative technique is also applied which allows us to overcome difficulties that the nonlinearity is related to derivative terms

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

minimization problems in l1 r3 abstract in this paper minimization problems in l1 ir3 are considered these problems arise in astrophysics for the determination of equilibrium configurations of axially symmetric rotating fluids rotating stars under nearly optimal assumptions a minimizer is proved to exist by a direct variational method which uses heavily the symmetry of the problem in order to get some compactness finally by looking directly at the euler equation we give some existence results of solutions of the euler equation even if the infimum is not finite author

concentration properties of blow up solutions and instability results for zakharov equation in dimension two part ii we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right

geometric methods for nonlinear many body quantum systems abstract geometric techniques have played an important role in the seventies for the study of the spectrum of many body schrodinger operators in this paper we provide a formalism which also allows to study nonlinear systems we start by defining a weak topology on many body states which appropriately describes the physical behavior of the system in the case of lack of compactness that is when some particles are lost at infinity we provide several important properties of this topology and use them to write a simple proof of the famous hvz theorem in the repulsive case in the second step we recall the method of geometric localization in fock space as proposed by derezinski and gerard and we relate this tool to our weak topology we then provide several applications we start by studying the so called finite rank approximation which consists in imposing that the many body wavefunction can be expanded using finitely many one body functions we thereby emphasize geometric properties of hartree fock states and prove nonlinear versions of the hvz theorem in the spirit of works of friesecke in the last section we study translation invariant many body systems comprising a nonlinear term which effectively describes the interactions with a second system as an example we prove the existence of the multi polaron in the pekar tomasevich approximation for certain values of the coupling constant

a hemivariational inequality involving p laplacian on an infinite strip abstract this paper deals with an eigenvalue problem for hemivariational inequalities on domains of the type r is a bounded open subset of r n 1 n 2 and it involves concave convex nonlinearities under suitable conditions on the nonlinearities two nontrivial solutions are obtained which belong to a special closed convex cone of w 0 1 p r whenever the eigenvalues are of certain range our approach is variational based on the theories of non smooth analysis our results are a generalization of the case of laplacian from a kristaly et al to the case of p laplacian