on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

the existence of multiple equilibrium points in a global attractor for some p laplacian equation abstract in this paper we are mainly concerned with some properties of the global attractor for some p laplacian equation with a lyapunov function in a banach space under some suitable assumptions we prove the existence of multiple equilibrium points in a global attractor for some p laplacian equation

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

on nodal solutions for nonlinear elliptic equations with a nonhomogeneous differential operator abstract we consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator our main result shows the existence of infinitely many nodal solutions in the case where we do not impose any restriction on the behavior of the reaction term f at infinity this result completes some recent papers the studied differential operators incorporate as special cases the p laplacian the p q differential operator and the generalized p mean curvature differential operator our approach is based on variational methods combining suitable truncation and comparison techniques

some qualitative questions on the equation div a x u u f x u abstract in this article we establish several applications of picone s identity for the operator of the form d i v a x u u such as hardy type inequality sturmian comparison theorem monotonicity property of the first eigenvalue nonexistence of positive supersolutions and caccioppoli inequality under certain conditions on a

nonuniformly elliptic equations of p laplacian type abstract this paper deals with the existence of a generalized solution in w 0 1 p to a nonuniformly nonlinear elliptic equation of the form div a x u f x u in a bounded domain of r n here a satisfies a x c 0 h 0 x h 1 x p 1 for any in r n a e x h 0 l p p 1 and h 1 l loc 1

a multiplicity theorem for hemivariational inequalities with a p laplacian like differential operator abstract we consider a parametric nonlinear elliptic inclusion with a multivalued p laplacian like differential operator and a nonsmooth potential hemivariational inequality using a variational approach based on the nonsmooth critical point theory we show that for all the values of the parameter in an open half line the problem admits at least two nontrivial solutions our result extends a recent one by kristaly lisei and varga a kristaly h lisei c varga multiple solutions for p laplacian type operator nonlinear anal 68 5 2008 1375 1381

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

the monge problem for supercritical mane potentials on compact manifolds abstract we prove the existence of optimal transport maps for the monge problem when the cost is a finsler distance on a compact manifold our point of view consists in considering the distance as a mane potential and to rely on recent developments in the theory of viscosity solutions of the hamilton jacobi equation

variational evolution problems and nonlocal geometric motion we consider two variational evolution problems related to monge kantorovich mass transfer these problems provide models for collapsing sandpiles and for compression molding we prove the following connection between these problems and nonlocal geometric curvature motion the distance functions to surfaces moving according to certain nonlocal geometric laws are solutions of the variational evolution problems thus we do the first step of the proof of heuristics developed in earlier works the main techniques we use are differential equation methods in the monge kantorovich theory

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

variational approximation for fokker planck equation on riemannian manifold under the bounded geometry assumption on riemannian manifold m a variational approximation for fokker planck equation on m is constructed by the scheme of jordan et al in siam j math anal 29 1 1 17 1998 moreover the uniqueness and global l p estimate of the solution for 1 p dim m dim m 1 are obtained for a broad class of potential

multiple solutions for p harmonic type equations abstract in this paper we establish the existence of multiple solutions for an equation involving a p harmonic operator subject to dirichlet boundary conditions in a bounded smooth open domain of r n while the nonlinearity has a p 1 sublinear growth at infinity using a three critical points theorem we prove the existence of at least three distinct weak solutions in w 0 2 p to this problem

multiplicity results for magnetic fractional problems abstract the paper deals with the existence of multiple solutions for a boundary value problem driven by the magnetic fractional laplacian a s that is a s u f u u in u 0 in r n where is a real parameter f is a continuous function and is an open bounded subset of r n with lipschitz boundary we prove that the problem admits at least two nontrivial weak solutions under two different sets of conditions on the nonlinear term f which are dual in a suitable sense

the monge problem for distance cost in geodesic spaces we address the monge problem in metric spaces with a geodesic distance x d is a polish space and dl is a geodesic borel distance which makes x dl a non branching geodesic space we show that under the assumption that geodesics are d continuous and locally compact we can reduce the transport problem to 1 dimensional transport problems along geodesics we introduce two assumptions on the transport problem which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w r t the 1 dimensional hausdorff distance induced by dl it is known that this regularity is sufficient for the construction of a transport map we study also the dynamics of transport along the geodesic the stability of our conditions and show that in this setting dl cyclical monotonicity is not sufficient for optimality

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

monge problem in metric measure spaces with riemannian curvature dimension condition abstract we prove the existence of solutions for the monge minimization problem addressed in a metric measure space x d m enjoying the riemannian curvature dimension condition rcd k n with n for the first marginal measure we assume that 0 m as a corollary we obtain that the monge problem and its relaxed version the monge kantorovich problem attain the same minimal value moreover we prove a structure theorem for d cyclically monotone sets neglecting a set of zero m measure they do not contain any branching structures that is they can be written as the disjoint union of the image of a disjoint family of geodesics

ground states of degenerate quasilinear schrodinger equation with vanishing potentials abstract in this paper we study the existence of nontrivial ground state solutions for the following class of p laplacian type equation div a x u v x x p u p 2 u k x x p f u in r n where 1 p n n 3 n p p e 1 d 1 e p p e n p n d p critical hardy sobolev exponent f has a quasicritical growth v and k are nonnegative potentials the function a satisfies a x u c 0 x p h 0 x u p 1 c 0 1 x p h 1 x u p 1 for any r n a e x r n for any two positive functions h 1 l l o c r n h 0 l p p 1 r n with p p

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

nonuniformly nonlinear elliptic equations of p biharmonic type this paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p biharmonic operator in particular p biharmonic operator under dirichlet boundary conditions or navier boundary conditions our method is mainly based on variational arguments

embedding sn into rn 1 with given integral gauss curvature and optimal mass transport on sn in a d aleksandrov convex polyhedra gittl moscow ussr 1950 in russian english translation a d aleksandrov convex polyhedra springer berlin new york 2005 a d aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data the first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in euclidean space with prescribed integral gauss curvature our solution includes the case of a convex polytope this problem was also first considered by aleksandrov and below it is referred to as aleksandrov s problem the second goal of this paper is to show that in variational form the aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

the disintegration of the lebesgue measure on the faces of a convex function we consider the disintegration of the lebesgue measure on the graph of a convex function f rn r w r t the partition into its faces which are convex sets and therefore have a well defined linear dimension and we prove that each conditional measure is equivalent to the k dimensional hausdorff measure on the k dimensional face on which it is concentrated the remarkable fact is that a priori the directions of the faces are just borel and no lipschitz regularity is known notwithstanding that we also prove that a green gauss formula for these directions holds on special sets

on the structure of covariant dynamical semigroups abstract dynamical semigroups in b h without the norm continuity assumption are investigated by introducing the notion of form generator giving the mathematical expression for the idea of markovian master equation it is shown that any form generator on b h admits a standard representation and the non uniqueness of this representation is described the equations for the components of the standard representation of a covariant form generator are deduced in terms of low order cohomology of the symmetry group in particular noncommutative levy khinchin type formulae for the shift covariant form generators are obtained relations with previous works on characterisation of the unbounded generator of a dynamical semigroup are discussed

quantitative isoperimetry a la levy gromov on a riemannian manifold with a positive lower bound on the ricci tensor the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius the deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise these results are actually obtained in the more general context of possibly non smooth metric measure spaces with curvature dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method

optimal transport maps for monge kantorovich problem on loop groups let g be a compact lie group and consider the loop group leg l c 0 1 g l 0 l 1 e let be the heat kernel measure at the time 1 for any density function f on leg such that ent f we shall prove that there exists a unique optimal transportation map t leg leg which pushes forward to f

ground state solutions for singular p laplacian equation in rn abstract in this work we study the ground state solution for the class of singular quasilinear elliptic problem of the form div x a p u p 2 u h x u m 2 u h x u q 2 u x r n where 1 p n a n p p and h r h r 0 we prove the compactly embedding results of the weighted z d a r 1 p r n l h r m r n space of radially symmetric functions and then obtain the existence of nontrivial ground state solution

optimal transport with convex obstacle we consider the monge transportation problem when the cost is the squared geodesic distance around a convex obstacle we show that there exists at least one and in general infinitely many optimal transport maps

monge s transport problem on a riemannian manifold monge s problem refers to the classical problem of optimally transporting mass given borel probability measures find the measurepreserving map s m m between them which minimizes the average distance transported set on a complete connected riemannian manifold m and assuming absolute continuity of an optimal map will be shown to exist aspects of its uniqueness are also established

the existence of three positive solutions of a singular p laplacian problem abstract we establish the existence of three positive solutions of the p laplacian problem which involves a singular nonlinearity three solutions are obtained by using the cutoff argument and the three critical points theorem proved by ricceri 2000 in 15 and bonanno 2003 in 12 provided the nonlinear term is a p 1 sublinear growth at infinity

mountain pass solutions to equations of p laplacian type abstract this work is devoted to study the existence of solutions to equations of p laplacian type we prove the existence of at least one solution and under further assumptions the existence of infinitely many solutions in order to apply mountain pass results we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm

non differentiable functionals and applications to elliptic problems with discontinuous nonlinearities abstract multiple critical points theorems for non differentiable functionals are established applications both to elliptic variational hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented

three solutions for a navier boundary value problem involving the p biharmonic the existence of at least three weak solutions is established for a class of navier boundary value problem involving the p biharmonic our technical approach is based on the three critical points theorem obtained by b ricceri

triple solutions for nonlinear elliptic problems driven by a non homogeneous operator abstract some multiplicity results for a parametric nonlinear dirichlet problem involving a nonhomogeneous differential operator of p laplacian type are given via variational methods the article furnishes new contributions and completes some previous results obtained for problems considering other types of differential operators and or nonlinear terms satisfying different asymptotic conditions

fast transport optimization for monge costs on the circle consider the problem of optimally matching two measures on the circle or equivalently two periodic measures on the real line and suppose the cost of matching two points satisfies the monge condition we introduce a notion of locally optimal transport plan motivated by the weak kam aubry mather theory and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter this theory is applied to a transportation problem arising in image processing for two sets of point masses on the circle both of which have the same total mass find an optimal transport plan with respect to a given cost function satisfying the monge condition in the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations for the case of n real valued point masses we present an o n log epsilon algorithm that approximates the optimal cost within epsilon when all masses are integer multiples of 1 m the algorithm gives an exact solution in o n log m operations

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

generators of quantum markov semigroups quantum markov semigroups qmss originally arose in the study of the evolutions of irreversible open quantum systems mathematically they are a generalization of classical markov semigroups where the underlying function space is replaced by a non commutative operator algebra in the case when the qms is uniformly continuous theorems due to the works of lindblad commun math phys 48 119 130 1976 stinespring proc am math soc 6 211 216 1955 and kraus ann phys 64 311 335 1970 imply that the generator of the semigroup has the form l a n 1 vn avn ga ag where vn and g are elements of the underlying operator algebra in the present paper we investigate the form of the generators of qmss which are not necessarily uniformly continuous and act on the bounded operators of a hilbert space we prove that the generators of such semigroups have forms that reflect the results of lindblad and stinespring we also make some progress towards forms reflecting kraus result finally we look at