the sub and supersolution method for variational hemivariational inequalities abstract the well known method of sub and supersolutions is a powerful tool for proving existence and comparison results for initial and or boundary value problems of nonlinear ordinary differential equations as well as for nonlinear partial differential equations of elliptic and parabolic type the main goal of this paper is to extend the idea of the sub and supersolution method in a natural and systematic way to quasilinear elliptic variational hemivariational inequalities owing to the intrinsic asymmetry of the latter where the problems are stated as inequalities rather than as equalities an appropriate generalization of the notion of sub and supersolutions to variational hemivariational inequalities is by no means straightforward the obtained results of this paper complement the development of the sub and supersolution method for nonsmooth variational problems presented in a recent monograph by s carl vy k le and d motreanu

degree of mapping for nonlinear mappings of monotone type abstract a classical degree function is constructed for pseudomonotone mappings from a reflexive banach space to its dual using galerkin approximations this generalizes the leray schauder degree when the banach space is a hilbert space and yields a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

on the optimality of simple schedules for networks with multiple half duplex relays this paper studies networks that consist of n half duplex relays assisting the communication between a source and a destination in isit 12 brahma et al conjectured that in gaussian half duplex diamond networks i e without a direct link between the source and the destination and with n non interfering relays an approximately optimal relay scheduling policy i e achieving the cut set upper bound to within a constant gap uniformly over all channel gains has at most n 1 active states i e at most n 1 out of the 2 n possible relay listen transmit configurations have a strictly positive probability such relay scheduling policies were referred to as simple in itw 13 we conjectured that simple approximately optimal relay scheduling policies exist for any gaussian half duplex multi relay network irrespectively of the topology this paper formally proves this more general version of the conjecture and shows it holds beyond gaussian noise networks in particular for any class of memoryless half duplex n relay networks with independent noises and for which independent inputs are approximately optimal in the cut set upper bound an approximately optimal simple relay scheduling policy exists the key step of the proof is to write the minimum of the submodular cut set function by means of its lovasz extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution this together with the saddle point property of min max problems and the existence of optimal basic feasible solutions for linear programs proves the conjecture as an example for n relay gaussian networks with independent noises where each node is equipped with multiple antennas and where each antenna can be configured to listen or transmit irrespectively of the others the existence of an approximately optimal simple relay scheduling policy with at most n 1 active states irrespectively of the total number of antennas in the system is proved

affineness of deligne lusztig varieties for minimal length elements abstract we prove that the deligne lusztig varieties associated to elements of the weyl group which are of minimal length in their twisted class are affine our proof differs from the proof of he and orlik rapoport and it is inspired by the case of regular elements which correspond to the varieties involved in broue s conjectures

on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces abstract in this paper we consider a sub supersolution method for variational inequalities with leray lions operators in sobolev spaces with variable exponents existence and qualitative properties of solutions between sub and supersolutions are established

nonresonance conditions for generalised ϕ laplacian problems with jumping nonlinearities abstract we consider the boundary value problem 0 1 x u x u x f x u x u x a e x 0 1 0 2 c 00 u 0 c 01 u 0 c 10 u 1 c 11 u 1 where c j 0 c j 1 0 for each j 0 1 and f 0 1 r 2 r are caratheodory functions with suitable additional properties the differential operator generated by the left hand side of 0 1 together with the boundary conditions 0 2 is a generalisation of the usual p laplacian and also of the so called laplacian which corresponds to x s t t with an odd increasing homeomorphism for the p laplacian problem and more particularly the semilinear case p 2 nonresonance conditions which ensure the solvability of the problem 0 1 0 2 have been obtained in terms of either eigenvalues for non jumping f or the fucik spectrum or half eigenvalues for jumping f of the p laplacian in this paper under suitable growth conditions on and f we extend these conditions to the general problem 0 1 0 2

a browder degree theory from the nagumo degree on the hilbert space of elliptic super regularization abstract let x be a real reflexive separable locally uniformly convex banach space with locally uniformly convex dual space x let q h x be a linear compact injection according to browder and ton such that q h x where h is a real separable hilbert space a degree mapping d on x is constructed from the nagumo degree d n a on h by d t f g 0 lim t 0 d n a i 1 t q t t f q q 1 g 0 where g x is open and bounded t t is the resolvent t 1 t j 1 1 of a strongly quasibounded maximal monotone operator t x d t 2 x with 0 t 0 and f g x is demicontinuous bounded and of type s a range of sums result is also given using the skrypnik degree theory in order to further exhibit the methodology of elliptic super regularization

on the affineness of distinguished deligne lusztig varieties abstract we consider deligne lusztig varieties in the variety of parabolic subgroups with a fixed type in this paper we give a criterion for the affineness of distinguished deligne lusztig varieties extending and refining the original deligne lusztig criterion in particular we show that distinguished deligne lusztig varieties are affine except possibly for small q and that all distinguished deligne lusztig varieties associated to rank 2 groups are affine

quasilinear elliptic inclusions of hemivariational type extremality and compactness of the solution set abstract we consider the dirichlet boundary value problem for an elliptic inclusion governed by a quasilinear elliptic operator of leray lions type and a multivalued term which is given by the difference of clarke s generalized gradient of some locally lipschitz function and the subdifferential of some convex function problems of this kind arise e g in mechanical models described by nonconvex and nonsmooth energy functionals that result from nonmonotone multivalued constitutive laws our main goal is to characterize the solution set of the problem under consideration in particular we are going to prove that the solution set possesses extremal elements with respect to the underlying natural partial ordering of functions and that the solution set is compact the main tools used in the proofs are abstract results on pseudomonotone operators truncation and special test function techniques zorn s lemma as well as tools from nonsmooth analysis

on the degree theory for general mappings of monotone type degree theory has been developed as a tool for checking the solution existence of nonlinear equations in his classic paper published in 1983 browder developed a degree theory for mappings of monotone type f t w heref is a mapping of class s from a bounded open set in a reflexive banach space x into its dual x a ndt is a maximal monotone mapping from x into x this breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations in this paper we continue to develop the results of browder on the degree theory for mappings of monotone type f t by enlarging the class of maximal monotone mappings and pseudo monotone homotopies we obtain some new results of the degree theory for such mappings 2007 elsevier inc all rights reserved

minimal length elements of coxeter groups let w be a coxeter group and wj a finite parabolic subgroup we present here a geometric new proof of x he s result on minimal length elements of an orbit of wj which acts on w in a certain way

extension of the leray schauder degree for abstract hammerstein type mappings abstract we introduce a new extension of the classical leray schauder topological degree in a real separable reflexive banach space the new class of mappings for which the degree will be constructed is obtained essentially by replacing the compact perturbation by a composition of mappings of monotone type it turns out that the class contains the leray schauder type maps as a proper subclass the new class is not convex thus preventing the free application of affine homotopies however there exists a large class of admissible homotopies including subclass of affine ones so that the degree can be effectively used we shall construct the degree and prove that it is unique we shall generalize the borsuk theorem of the degree for odd mappings and show that the principle of omitted rays remains valid to illuminate the use of the new degree we shall briefly consider the solvability of abstract hammerstein type equations and variational inequalities

network simplification in half duplex building on submodularity this paper explores the network simplification problem in the context of gaussian half duplex diamond networks specifically given an n relay diamond network this problem seeks to derive fundamental guarantees on the capacity of the best k relay subnetwork as a function of the full network capacity simplification guarantees are presented in terms of a particular approximate capacity termed independent gaussian ig approximate capacity that characterizes the network capacity to within an additive gap which is independent of the channel coefficients and operating snr the main focus of this work is when k n 1 relays are selected out of n relays in a diamond network first a simple algorithm is proposed which selects all relays except the one with the minimum ig approximate half duplex capacity it is shown that the selected n 1 relay subnetwork has an ig approximate half duplex capacity that is at least 1 2 of the ig approximate half duplex capacity of the full network and that for the proposed algorithm this guarantee is tight furthermore this work proves the following tight fundamental guarantee there always exists a subnetwork of k n 1 relays that have an ig approximate half duplex capacity that is at least equal to n 1 n of the ig approximate half duplex capacity of the full network finally these results are extended to derive lower bounds on the fraction guarantee when k in 1 n relays are selected the key steps in the proofs lie in the derivation of properties of submodular functions which provide a combinatorial handle on the network simplification problem for gaussian half duplex diamond networks

the leray schauder approach to the degree theory for s perturbations of maximal monotone operators in separable reflexive banach spaces abstract the purpose of this paper is to demonstrate the fact that the topological degree theory of leray and schauder may be used for the development of the topological degree theory for bounded demicontinuous s perturbations f of strongly quasibounded maximal monotone operators t in separable reflexive banach spaces certain basic homotopy properties and the extension of this degree theory to possibly unbounded strongly quasibounded perturbations f are shown to hold this work uses the well known embedding of browder and ton and extends the work of berkovits who developed this theory for the case t 0 besides being an interesting mathematical problem the existence of such a degree theory may conceivably become useful in situations where the use of the leray schauder degree via infinite dimensional compactness is necessary

on the degree theory for mappings of monotone type and applications abstract f e browder has recently extended the concept of the classical topological degree for mappings of monotone type we introduce a new construction based on the leray schauder degree we also consider some existence and surjectivity results which illuminate the use of degree theory

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

the index of a critical point for densely defined operators of type _ in banach spaces the purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type s l acting from a real reflexive and separable banach space x into x this index is defined via a degree theory for such operators which has been recently developed by the authors the calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point this linearization is a new tool even for continuous everywhere defined operators which are not necessarily frechet differentiable various cases of operators are considered unbounded nonlinear operators with unbounded linearization bounded nonlinear operators with bounded linearization and operators in hilbert spaces examples and counterexamples are given in l p p 2 illustrating the main results the associated bifurcation problem for a pair of operators is also considered the main results of the paper are substantial extensions and improvements of the classical results of leray and schauder for continuous operators of leray schauder type as well as the results of skrypnik for bounded demicontinuous mappings of type s applications to nonlinear dirichlet problems have appeared elsewhere

deligne lusztig varieties and period domains over finite fields abstract we prove that the drinfeld space is essentially the only deligne lusztig variety which is at the same time a period domain over a finite field we also discuss an affineness criterion for deligne lusztig varieties and its relation to cohomology vanishing theorems

multiple solution results for elliptic neumann problems involving set valued nonlinearities abstract the main goal of this paper is to present multiple solution results for elliptic inclusions of clarke s gradient type under nonlinear neumann boundary conditions involving the p laplacian and set valued nonlinearities to be more precise we study the inclusion p u f x u u p 2 u in with the boundary condition u p 2 u a u p 1 b u p 1 g x u on we prove the existence of two constant sign solutions and one sign changing solution depending on the parameters a and b our approach is based on truncation techniques and comparison principles for elliptic inclusions along with variational tools like the nonsmooth mountain pass theorem the second deformation lemma for locally lipschitz functionals as well as comparison results of local c 1 minimizers and local w 1 p minimizers of nonsmooth functionals

topological degree for s mappings with maximal monotone perturbations and its applications to variational inequalities abstract this paper is concerned with the topological degree for mappings of class s with maximal monotone perturbations several results and remarks concerning the evaluation of this degree are given in particular it is shown that the local degree for the generalized gradient of nonsmooth functional at the local minimizer is equal to one as applications two examples of elliptic variational inequalities are given where the multiple existence of solutions is discussed

topological degree for a class of operators and applications abstract we define the topological degree for a class of operators and use it to solve a strongly singular nonlinear elliptic equation

on the affineness of deligne lusztig varieties abstract we prove that the deligne lusztig variety associated to minimal length elements in any conjugacy class of the weyl group is affine which was conjectured by orlik and rapoport in s orlik m rapoport deligne lusztig varieties and period domains over finite fields arxiv 0705 1646

a new topological degree theory for perturbations of the sum of two maximal monotone operators abstract let x be an infinite dimensional real reflexive banach space with dual space x and g x open and bounded assume that x and x are locally uniformly convex let t x d t 2 x be maximal monotone and strongly quasibounded s x d s x maximal monotone and c x d c x strongly quasibounded w r t s and such that it satisfies a generalized s condition w r t s assume that d s l d t d c where l is a dense subspace of x and 0 t 0 s 0 0 a new topological degree theory is introduced for the sum t s c with degree mapping d t s c g 0 the reason for this development is the creation of a useful tool for the study of a class of time dependent problems involving three operators this degree theory is based on a degree theory that was recently developed by kartsatos and skrypnik just for the single valued sum s c as above

on the homotopy property of topological degree for maximal monotone mappings abstract let e be a real reflexive banach space e the dual space of e and e an open bounded subset and let t i d t i 2 e i 1 2 be two maximal monotone mappings such that d t 1 d t 2 and 0 t 0 1 tt 1 1 t t 2 d t 1 d t 2 under some additional assumptions we prove that deg t 1 d t 1 0 deg t 2 d t 2 0

regularity and rothe method error estimates for parabolic hemivariational inequality this paper deals with regularity of solutions to the abstract operator version of parabolic partial differential inclusion of the form u t au t j u t f t with the multivalued term given in the form of clarke subdifferential of a locally lipschitz functional j using the rothe method it is shown that under appropriate assumptions on the data the solution has the increased regularity three regularity theorems are given the first one concerns the regularity of solution in the besov space b2 1 2 0 t h the second theorem provides assumption under which the solution lies in the space h1 0 t h l 0 t v c 0 t vweak the last one shows that if the operator a is strongly monotone and the multivalued term satisfies the relaxed monotonicity assumption then the unique solution also belongs to h1 0 t v w1 0 t h in the last case the error estimates on the rothe method are also proved

new surjectivity results for perturbed weakly coercive operators of monotone type in reflexive banach spaces abstract let x be a real reflexive locally uniformly convex banach space with locally uniformly convex dual space x let t x d t 2 x be maximal monotone and s x 2 x be bounded pseudomonotone let p be a nonnegative integer assume further that there exist nonnegative constants a i i 1 2 p such that v w x i 1 p a i x i x x p 2 for all x d t with sufficiently large x v t x and w s x where 0 0 such that t 0 as t new surjectivity results are given for the operator t s along with weakly coercive type hypothesis on t s the results are new and improve the corresponding theory for coercive operators of monotone type the theory developed herein can be suitably applied in the study of partial differential equations variational and hemi variational inequality problems in appropriate sobolev spaces to demonstrate the applicability of the theory an example of time periodic parabolic partial differential equation which models nonmonotone semipermeability problem is provided

the approximate capacity of half duplex line networks this paper investigates the problem of characterizing the capacity of half duplex hd line networks where a source node communicates to a destination node through a multi hop path of n relays if the relays operate in full duplex fd it is well known that the capacity of the line network equals the minimum among the point to point link capacities in the path in contrast this paper considers a different case where the relays operate in hd in the first part of the paper it is shown that the approximate capacity optimal up to a constant additive gap that only depends on the number of nodes in the network of an hd n relay line network equals half the minimum of the harmonic means of the point to point link capacities of each two consecutive links in the path it is then proved that the n 1 listen transmit states out of the 2 n possible ones sufficient to characterize the approximate capacity can be found in linear time in the second part of the paper it is shown that the problem of finding the path that has the largest hd approximate capacity in a network that can be represented as a graph is np hard however if the number of cycles in the network is polynomial in the number of nodes then a polynomial time algorithm can indeed be designed

minimal length elements in some double cosets of coxeter groups abstract we study the minimal length elements in some double cosets of coxeter groups and use them to study lusztig s g stable pieces and the generalization of g stable pieces introduced by lu and yakimov we also use them to study the minimal length elements in a conjugacy class of a finite coxeter group and prove a conjecture in m geck s kim g pfeiffer minimal length elements in twisted conjugacy classes of finite coxeter groups j algebra 229 2 2000 570 600

generalizations of browder s degree theory the starting point of this paper is the recent important work of f e browder who extended degree theory to operators of monotone type the degree function of browder is generalized to maps of the form t f g where t is maximal monotone f is of class s bounded and g is an u s c compact multifunction it is also generalized to maps of the form f ng with f of class s and ng the nemitsky operator of a multifunction g x r satisfying various types of sign conditions some examples are also included to illustrate the abstract results

degree of mapping for nonlinear mappings of monotone type densely defined mapping abstract the classical degree function constructed earlier for pseudomonotone mappings has been used to develop a broader degree theory of classical type for the sum of a maximal monotone map from a reflexive banach space to its dual together with a bounded pseudomonotone map the proof uses the generalized yosida approximation of the maximal monotone mapping

existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems abstract we consider noncoercive quasilinear elliptic inclusions under multivalued flux boundary conditions involving multifunctions of clarke s generalized gradient our main goal is to provide existence and comparison results which are based on an appropriate generalization of the notions of subsolutions and supersolutions furthermore compactness and extremality results of the solution set enclosed by an ordered pair of sub supersolutions will be proved

on the normalized eigenvalue problems for nonlinear elliptic operators abstract this paper solves the following form of normalized eigenvalue problem a u c u 0 0 and u d where d is a bounded open set in a real infinite dimensional banach space x and both x and its dual x are locally uniformly convex a is an unbounded maximal monotone operator on x the operators c is defined and continuous only on r d such that zero is not in the weak closure of a subset of c u c u this research reveals the fact that such eigenvalue problems do not depend on any properties of c located in r d this remarkable discovery extends theorem 4 in a g kartsatos i v skrypnik normalized eigenvalues for nonlinear abstract and elliptic operators j differential equations 155 1999 443 475 and is applied to the nonlinear elliptic operators under degenerate and singular conditions

a new topological degree theory for pseudomonotone perturbations of the sum of two maximal monotone operators and applications abstract let x be a real reflexive locally uniformly convex banach space with locally uniformly convex dual space x and g be a nonempty bounded and open subset of x let t x d t 2 x and a x d a 2 x be maximal monotone operators assume further that for each y x there exists a real number y and there exists a strictly increasing function 0 0 with 0 0 t as t satisfying w x y x x y for all x d a w a x and s x 2 x is bounded of type s or bounded pseudomonotone such that 0 t a s d t d a g or 0 t a s d t d a g respectively new degree theory is developed for operators of the type t a s with degree mapping d t a s g 0 the degree is shown to be unique invariant under suitable homotopies the theory developed herein generalizes the asfaw and kartsatos degree theory for operators of the type t s new results on surjectivity and solvability of variational inequality problems are obtained the mapping theorems extend the corresponding results for operators of type t s the degree theory developed herein is used to show existence of weak solution of nonlinear parabolic problem in appropriate sobolev spaces

general comparison principle for quasilinear elliptic inclusions abstract the main goal of this paper is to prove existence and comparison results for elliptic differential inclusions governed by a quasilinear elliptic operator and a multivalued function given by clarke s generalized gradient of some locally lipschitz function these kinds of problems have been treated in the past by various authors including the authors of this paper however in all the works we are aware of additional assumptions on the structure of the elliptic operator and or the generalized clarke s gradient are needed to get comparison results in terms of sub supersolutions comparison principles were obtained recently e g in the case where the elliptic operator is of potential type or clarke s gradient is required to satisfy some one sided growth condition or the sub supersolutions are supposed to satisfy additional properties the novelty of this paper is that we are able to obtain a comparison principle without assuming any of the above restrictions to the best of our knowledge this is the first mathematical treatment of the considered elliptic inclusion in its full generality the obtained results of this paper complement the development of the sub supersolution method for nonsmooth problems presented in a recent monograph by s carl vy k le and d motreanu

gaussian half duplex relay networks improved constant gap and connections with the assignment problem this paper considers a gaussian relay network where a source transmits a message to a destination with the help of n half duplex relays the information theoretic cut set upper bound to the capacity is shown to be achieved to within 1 96 n 2 bits by noisy network coding thereby reducing the previously known gap this gap is obtained as a special case of a more general constant gap result for gaussian half duplex multicast networks it is then shown that the generalized degrees of freedom of this network is the solution of a linear program where the coefficients of the linear inequality constraints are proved to be the solution of several linear programs referred as the assignment problem in graph theory for which efficient numerical algorithms exist the optimal schedule that is the optimal value of the 2 n possible transmit receive configuration states for the relays is investigated and known results for diamond networks are extended to general relay networks it is shown for the case of n 2 relays that only n 1 3 out of the 2 n 4 possible states have a strictly positive probability and suffice to characterize the capacity to within a constant gap extensive experimental results show that for a general n relay network with n leq 8 the optimal schedule has at most n 1 states with a strictly positive probability as an extension of a conjecture presented for diamond networks it is conjectured that this result holds for any half duplex relay network and any number of relays finally a network with n 2 relays is studied in detail to illustrate the channel conditions under which selecting the best relay is not optimal and to highlight the nature of the rate gain due to multiple relays

approximately achieving gaussian relay network capacity with lattice based qmf codes recently a new relaying strategy quantize map and forward qmf scheme has been demonstrated to approximately achieve within an additive constant number of bits the gaussian relay network capacity universally i e for arbitrary topologies channel gains and snrs this was established using gaussian codebooks for transmission and random mappings at the relays in this paper we develop structured lattice codes that implement the qmf strategy the main result of this paper is that such structured lattice codes can approximately achieve the gaussian relay network capacity universally again within an additive constant in addition we establish a similar result for half duplex networks where we demonstrate that one can approximately achieve the capacity using fixed transmit receive tx rx schedules for the relays with no transmit power optimization across the different tx rx states of the network

degree of mapping for nonlinear mappings of monotone type strongly nonlinear mapping abstract the classical degree function developed earlier for pseudomonotone operators from a reflexive banach space to its dual is extended for the special case of sobolev spaces to the sum of a pseudomonotone mapping representing a nonlinear elliptic operator and a strongly nonlinear lower order term