conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

on the monotonicity of optimal transportation plans abstract we obtain analytical properties of the maps that give optimal transportation plans for the l 2 wasserstein distance we take advantage of the monotonicity of such optimal transportation plans to discuss their measurability and continuity as a main result of independent interest we obtain the a e continuity of monotone in the sense of zarantonello operators

independent bernstein sets and algebraic constructions abstract we present the method of constructing algebras and linear spaces of 2 c generators using independent bernstein sets as an application we obtain large algebras of special functions i strongly everywhere surjective functions which are not perfectly everywhere surjective ii nowhere continuous darboux functions iii nowhere continuous compact to compact functions iv functions which are continuous precisely on a fixed closed proper subset of r most conclusions obtained in this paper are improvements of some already known results

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

an elementary proof of the polar factorization of vector valued functions we present an elementary proof of an important result of y brenier br1 br2 namely that vector fields in d satisfying a nondegeneracy condition admit the polar factorization u x s x where is a convex function and s is a measure preserving mapping brenier solves a minimization problem using monge kantorovich theory whereas we turn our attention to a dual problem whose euler lagrange equation turns out to be

a separable space with no remote points in the model obtained by adding w02 side by side sacks reals to a model of ch there is a separable nonpseudocompact space with no remote points to prove this it is also shown that in this model the countable box product of cantor sets contains a subspace of size w02 such that every uncountable subset has density co furthermore assuming the existence of a measurable cardinal k with 2k k a space x is produced with no isolated points but with remote points in vx x it is also shown that a pseudocompact space does not have remote points

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

some applications of mass transport to gaussian type inequalities as discovered by brenier mapping through a convex gradient gives the optimal transport in n in the present article this map is used in the setting of gaussian like measures to derive an inequality linking entropy with mass displacement by a straightforward argument as a consequence logarithmic sobolev and transport inequalities are recovered finally a result of caffarelli on the brenier map is used to obtain gaussian correlation inequalities

existence of infinitely many weak solutions for the p laplacian with nonlinear boundary conditions abstract in this paper we deal with the existence of weak solutions for quasilinear elliptic problem involving a p laplacian of the form p u x u p 2 u f x u in u p 2 u u p 2 u on we consider the above problem under several conditions on f for f superlinear and subcritical with respect to u we prove the existence of infinitely many solutions of the above problem by using the fountain theorem and the dual fountain theorem respectively for the case where f is critical with a subcritical perturbation namely f x u u p 2 u u r 2 u we show that there exists at least a nontrivial solution when p r p and there exist infinitely many solutions when 1 r p by using the mountain pass theorem and the concentration compactness principle respectively

on a reformulation of cournot nash equilibria abstract we present variations on a theme of mas colell and report results on the existence of cournot nash equilibrium distributions in which individual actions and the payoffs are represented by relations that are not necessarily complete or transitive

a proof of a loomis whitney type inequality via optimal transport abstract the paper is devoted to exhibiting a proof of an analytical extension of the well known loomis whitney inequality such a proof is completely independent of the original one and it is based on the technique of optimal transport which leads also to fully characterize the equality case

solution of the monge ampere equation on wiener space for general log concave measures abstract in this work we prove that the unique 1 convex solution of the monge kantorovitch measure transportation problem between the wiener measure and a target measure which has an h log concave density in the sense of feyel and ustunel j funct anal 176 2000 400 428 w r t the wiener measure is also the strong solution of the monge ampere equation in the frame of infinite dimensional frechet spaces we further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of monge kantorovitch and clarify the relation between this concept and the ito solutions of the monge ampere equation

free discontinuities in optimal transport we prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions this theorem is explicit and global it gives a formula representing this zero set as a difference of convex functions which holds throughout the entire domain of the original functions as applications we prove results on the stability of singularities of envelopes of semi convex functions and solutions to optimal transport problems under appropriate perturbations along with global structure theorems on certain discontinuities arising in optimal transport maps for the bilinear cost c x bar x langle x bar x rangle for x bar x in bf r n for targets whose components satisfy additional convexity separation multiplicity and affine independence assumptions we show that these discontinuities occur on submanifolds of the appropriate codimension which are parameterized locally as differences of convex functions dc hence c 2 rectifiable and depending on the precise assumptions c 1 alpha smooth under these hypotheses any n 1 affinely independent components of the target measure select at most one point from the source measure where the transport divides between all n 1 specified target components

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

polar factorization and pseudo rearrangements applications to polya szego type inequalities abstract we are interested in the polar factorization of a function f defined in an open bounded set r n it is well known that there exists a measure preserving map such that f f where f is the decreasing rearrangement of f we prove that under suitable assumptions besides the classical polar factorization of f we have f f u where f u is a pseudo rearrangement of f with respect to the measurable function u and is the measure preserving map such that u u as an application we characterize those functions that realize equality in the polya szego inequality

on cournot nash equilibrium distributions for games with a nonmetrizable action space and upper semi continuous payoffs we report results on the existence of a cournot nash equilibrium distribution for games in which the action space is not necessarily metrizable and separable and the payoff functions are not necessarily continuous our work relies on the theory of radon measures as developed by schwartz topsoe and on the epitopology as developed by dolecki salinetti wets

inequalities for quermassintegrals on k convex domains abstract in this paper we study the aleksandrov fenchel inequalities for quermassintegrals on a class of non convex domains our proof uses optimal transport maps as a tool to relate curvature quantities of different orders defined on the boundary of the domain

large free linear algebras of real and complex functions let x be a set of cardinality such that we prove that the linear algebra rx or cx contains a free linear algebra with 2 generators using this we prove several algebrability results for spaces cc and rr in particular we show that the set of all perfectly everywhere surjective functions f c c is strongly 2c algebrable we also show that the set of all functions f r r whose sets of continuity points equals some fixed g set g is strongly 2c algebrable if and only if r g is c dense in itself

weak p points in cech stone compactifications let x be a nonpseudocompact space which is either nowhere ccc or nowhere of weight 2 then fix x contains a point x which is a weak p point of 3x i e if f c f3x fx is countable then x a f in addition under ma if xis any nonpseudocompact space then fix x contains a point x such that whenever f c fix x3 is countable and nowhere dense then x a f 0 introduction all spaces are completely regular and x denotes 3x x frolik s f proof that the cech stone remainder of a nonpseudocompact space is not homogeneous is elegant and ingenious but does not give points which are topologically distinct by an obvious reason when kunen k1 proved that there are rudin keisler incomparable points in fc frolik s ideas were used by comfort c and van douwen vd2 to show that respectively no infinite compact space in which countable discrete subspaces are c embedded is homogeneous and that 3x is not homogeneous for any nonpseudocompact space x these results showed that certain spaces are not homogeneous but not why they are not homogeneous this suggests an obvious question which has been considered by several authors during the last years the first promising partial answers to this question were obtained by van douwen vd3 who showed that each nonpseudocompact space of countable r weight has a remote point independently this was also shown by chae and smith cs and that remote points can be used to show that certain cech stone remainders are not homogeneous unfortunately it was soon clear by examples in van douwen and van mill vdvm1 that this line of attack did not solve the entire problem since many spaces do not have remote points earlier van douwen vd1 had introduced far points and c far points and showed that these points exist in certain cech stone remainders and that they also could be used to show that a restrictive class of cech stone remainders is not homogeneous each remote point is a far point and each far point is an co far point when we restrict our attention to spaces without isolated points when it was shown that not every nonpseudocompact space has a remote point van douwen s vd1 question whether every nonpseudocompact space without isolated points has an w far point again became interesting the examples in vdvml did not clarify this question since they all have far points from vdvm2 it received by the editors november 12 1980 and in revised form june 24 1981 1980 mathematics subject classification primary 54d35

asymptotics of best sobolev constants on thin manifolds abstract we study the asymptotic behaviour of best sobolev constants on a compact manifold with boundary that we contract in k directions or to a point we find in the limit best sobolev constants for weighted sobolev spaces of the limit manifold

independent families in complete boolean algebras we present a proof without any set theoretical assumptions that every infinite complete boolean algebra includes a free subalgebra of the same cardinality it follows that the set of all ultrafilters on an infinite complete boolean algebra b has power 2 1 problems and results let b be a boolean algebra and st b its stone space i e the set of all ultrafilters on b with the usual topology if b is finite then obviously i st b i log2 1 b i i see ma the classical result due to hausdorff h and pospisil pi states that for a power set boolean algebra b 6t k for any infinite cardinal k i st b i 22 2ibi the key role in their proof is played by the notion of an independent family of sets and the fact that there is such a family of subsets of k of size 2k the very natural problem of whether this can be generalized for all infinite complete boolean algebras was first formulated as far as we know by efimov e partial positive solutions were known from works of kesl yakov k koppelberg ko monk m and blaszczyk bla a short historical survey of this problem can be found in bla independent systems in complete boolean algebras are studied in vladimirov s book we shall present proofs in zfc only of the following theorems theorem a in every infinite complete boolean algebra b there is an independent family 6yc b so that i yl i b i theorem b if b is an infinite complete boolean algebra homogeneous in sat then b is well semifree for precise definitions see 3 although many partial results concerning theorem a have been achieved e g monk proved theorem a under a hypothesis slightly weaker than gch we are offering a general approach covering all results obtained so far it is not surprising that notions and results about filters and ultrafilters on power set algebras can be translated to ones on complete boolean algebras we shall give one such example concerning good ultrafilters kunen s proof see kul ch k of the existence of good ultrafilters on a power set algebra can be pushed through in this more general situation as well received by the editors february 10 1981 1980 mathematics subject classification primary 06e10 54a25 secondary 03e05 06e15 54g05 otherwise i bi i st b i 2jb 1982 american mathematical society 0002 9947 82 0000 0229 03 75

addendum conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature is finite on the other hand the argument at the beginning of proposition 2 1 in e shows that if al b is finite then q m om is finite jin zhiren pointed out to me that the sobolev quotient q m o9m can be oo this is the case if we delete a small geodesic ball on a compact manifold without boundary with negative scalar curvature more generally the sobolev quotient is oo if the first eigenvalue for the conformal laplacian with respect to dirichlet boundary condition is negative in order to see that let p1 be the first eigenfunction for the problem

the b conjecture for the gaussian measure of dilates of symmetric convex sets and related problems we show that given a symmetric convex set k rd the function t etk is log concave on r where denotes the standard d dimensional gaussian measure we also comment on the extension of this property to unconditional log concave measures and sets and on the complex case

on the jordan kinderlehrer otto scheme abstract in this paper we prove that the jordan kinderlehrer otto scheme for a family of linear parabolic equations on the flat torus converges uniformly in space

optimal boundary holes for the sobolev trace constant in this paper we study the problem of minimizing the sobolev trace rayleigh quotient u w1 p p u lq p among functions that vanish in a set contained on the boundary of given boundary measure we prove existence of extremals for this problem and analyze some particular cases where information about the location of the optimal boundary set can be given moreover we further study the shape derivative of the sobolev trace constant under regular perturbations of the boundary set

fredholm alternative for the p laplacian in higher dimensions abstract in this paper we characterize the set of all right hand sides h c formula for which the boundary value problem p u 1 u p 2 u h in u 0 on has at least one weak solution u w01 p here 1 0 is the first eigenvalue of the p laplacian in particular we prove that for h 1 0 this problem is solvable and the energy functional e h u 1 p u p 1 p u p hu is unbounded from below

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

discretization of functionals involving the monge ampere operator gradient flows in the wasserstein space have become a powerful tool in the analysis of diffusion equations following the seminal work of jordan kinderlehrer and otto jko the numerical applications of this formulation have been limited by the difficulty to compute the wasserstein distance in dimension geqslant 2 one step of the jko scheme is equivalent to a variational problem on the space of convex functions which involves the monge ampere operator convexity constraints are notably difficult to handle numerically but in our setting the internal energy plays the role of a barrier for these constraints this enables us to introduce a consistent discretization which inherits convexity properties of the continuous variational problem we show the effectiveness of our approach on nonlinear diffusion and crowd motion models

decomposition and representation of coalitional games a coalitional game is a real valued set function defined on an algebra f of subsets of a space x such that o 0 we prove the existence of a one to one correspondence between coalitional games bounded with respect to the composition norm and countably additive measures defined on an appropriate space

existence results for the p laplacian with nonlinear boundary conditions in this paper we study the existence of nontrivial solutions for the problem pu u p 2u in a bounded smooth domain rn with a nonlinear boundary condition given by u p 2 u f u on the boundary of the domain the proofs are based on variational and topological arguments

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

best constant in sobolev trace inequalities on the half space using a mass transportation method we study optimal sobolev trace inequalities on the half space and prove a conjecture made by escobar in 1988 about the minimizers

set relations and set systems induced by some families of integral domains abstract in this paper given an integral domain u we investigate the main properties of a relation m o d which is based on the interrelation between subdomains of u and finitely generated unitary submodules of u we shall characterize it in terms of a second relation between n tuples u 1 u n of elements of u and subdomains d of u defined by the vanishing in u 1 u n of some polynomial p z 1 z n belonging to a specific subset of the polynomial ring in several variables d z 1 z n such an equivalence shall be used in order to introduce three specific collections of subdomains x u b u and p u whose algebraic properties present a close connection with geometrical and combinatorial properties induced by m o d on the other hand the characterization of the subdomains of x u leads to the more general problem of finding a map associating with a subdomain d of u a collection d of subdomains of k u such that the intersection of some or of any member of d gives d in this perspective in the present paper we shall study two further collections of subdomains of u denoted respectively by e u and l u whose main properties are related to those of the families p u and b u finally our investigation of all the aforementioned subdomain families shall be also related to the study of pairs e where e u 0 and is an idempotent ring endomorphism of u whose kernel agrees with the ideal of u generated by e we shall exhibit several results concerning the membership of u and of u e to the above subdomain families

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

on cournot nash equilibria in generalized qualitative games with a continuum of players in recent years the nash debreu theorem 28 8 of 1951 52 has been the subject of two particularly far reaching generalizations the first is a neglected contribution of ma 24 and is a result on the existence of cournot nash equilibria in games with an arbitrary set of players each of whom has strategy sets in an arbitrary hausdorff topological vector space the second is a result of toussaint 32 and it can be viewed as a generalization of ma s theorem to a setting where each player s pay off is generated by a nonordered relation both of these results are elegant and simple enough that they can be fully stated the following result was proved in 1969

equilibrium points of nonatomic games over a banach space schmeidler s results on the equilibrium points of nonatomic games with strategy sets in euclidean space are generalized to nonatomic games with strategy sets in a banach space our results also extend previous work of the author which assumed the banach space to be separable and its dual to possess the radon nikodym property our proofs use recent results in functional analysis 1 introduction in this paper we study cournot nash equilibria of games with a continuum of players each of whom has a strategy set in a banach space the importance of such games for economic theory has been recently underscored by dubey mas colell and shubik 7 see their example 2 in particular however 7 is primarily concerned with the equivalence of walrasian and cournot nash equilibria our focus here is on the existence theory the results reported here can be seen as a continuation of research initiated by schmeidler 19 who formulated and studied nonatomic games over zz dimensional euclidean space they also extend earlier work of the author in at least three important respects to begin with 14 assumed that each player s strategy set lies in a separable banach space whose dual has the radon nikodym property this condition is not satisfied for the infinite dimensional spaces typically studied in economics see bewley 2 and mas colell 18 it is of interest that such an assumption can be relaxed at the cost of a rather mild uniformity assumption on the distribution of strategy sets secondly it seems desirable to have results for con jugate banach spaces which are not separable and for which the strategy sets are not weakly compact but only norm bounded and weak closed we also present such results here they are especially relevant for commodity spaces studied in 2 and 18 finally unlike 14 our results on the existence of mixed strategy equilibria rely only on preference orderings rather than on payoff functions it is not known to us whether our conditions on preferences are sufficient for them to be represented as jointly measurable payoff functions such theorems are difficult to obtain even for a finite dimensional setting see wesley 21

convergence of functions equi semicontinuity the ever increasing complexity of the systems to be modeled and analyzed taxes the existing mathematical and numerical techniques far beyond our present day capabilities by their intrinsic nature some problems are so difficult to solve that at best we may hope to find a solution to an approximation of the original problem stochastic optimization problems except in a few special cases are typical examples of this class this however raises the question of what is a valid approximate to the original problem the design of the approximation must be such that i the solution to the approximate provides approximate solutions to the original problem and ii a refinement of the approximation yields a better approximate solution the classical techniques for approximating functions are of little use in this setting in fact very simple examples show that classical approximation techniques dramatically fail in meeting the objectives laid out above what is needed at least at a theoretical level is to design the approximates to the original problem in such a way that they satisfy an epi convergence criterion the convergence of the functions defining the problem is to be replaced by the convergence of the sets defined by these functions that type of convergence has many properties but for our purpose the main one is that it implies the convergence of the optimal solutions this article is devoted to the relationship between the epi convergence and the classical notion of pointwise convergence a strong semicontinuity condition is introduced and it is shown to be the link between these two types of convergences it provides a number of useful criteria which can be used in the design of approximates to difficult problems

optimal transport and cournot nash equilibria we study a class of games with a continuum of players for which a cournot nash equilibria can be obtained by the minimisation of some cost related to optimal transport this cost is not convex in the usual sense in general but it turns out to have hidden strict convexity properties in many relevant cases this enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

the behavior of the best sobolev trace constant and extremals in thin domains in this paper we study the asymptotic behavior of the best sobolev trace constant and extremals for the immersion w 1 p o l q o in a bounded smooth domain when it is contracted in one direction we find that the limit problem when rescaled in a suitable way is a sobolev type immersion in weighted spaces over a projection of o w 1 p p o a l q p o b for the special case p q this problem leads to an eigenvalue problem with a nonlinear boundary condition we also study the convergence of the eigenvalues and eigenvectors in this case