stability of the difference type methods for linear volterra equations in hilbert spaces we study the stability of backward difference timestepping method for linear volterra equations of scalar type in a hilbert space framework the results and methods extend and simulate numerically those introduced by pruss for integrability with respect to continuous solutions

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

a morse theoretical approach for the boundary mean curvature problem on b4 abstract in this paper we investigate existence as well as multiplicity of scalar flat metric of prescribed boundary mean curvature on the standard 4 dimensional ball due to the existence of critical point at infinity the standard variational methods cannot be applied to overcome this difficulty we prove that in a neighborhood of critical points at infinity a morse lemmas at infinity reduction holds then develop a whole morse theory of this noncompact variational problem in particular we establish under generic boundary condition morse inequalities at infinity which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated euler lagrange functional as further application of this morse theoretical approach we prove more existence results and extend a topological invariant introduced by a bahri

yamabe equations on half spaces where n 2 n 2 is the critical sobolev exponent n n 2 is the critical sobolev exponent on the boundary is the unit outer normal of and h x is the mean curvature this problem is related to the yamabe problem on manifolds with boundary see 7 8 it was proved in 7 8 that if either k x 1 h x 0 or k x 0 h x 1 then eq 1 1 has a classical solution which minimizes the yamabe ratio when k x 6 0 h x 6 0 it is natural to seek for the nehari minimizers this idea works when is not conformal to a ball see 9 however when is a ball the situation is very di erent in fact for certain functions k x and h x eq 1 1 has no positive solution at all therefore solvability conditions on k x and h x are needed to make it more clear we consider the yamabe equation

a convex concave problem with a nonlinear boundary condition in this paper we study the existence of nontrivial solutions of the problem u u u p 2uin u u q 2uon with 1 q 2 n 1 n 2 and 1 p 2n n 2 in the concave convex case i e 1 q 2 p if is small there exist two positive solutions while for large there is no positive solution when p is critical and q subcritical we obtain existence results using the concentration compactness method finally we apply the implicit function theorem to obtain solutions for small near u0 1

blow up phenomena for scalar flat metrics on manifolds with boundary abstract let m n g be a compact riemannian manifold with boundary m this article is concerned with the set of scalar flat metrics on m which are in the conformal class of g and have m as a constant mean curvature hypersurface we construct examples of metrics on the unit ball b n in dimensions n 25 for which this set is non compact these manifolds have umbilic boundary but they are not conformally equivalent to b n

the prescribed boundary mean curvature problem on the standard n dimensional ball abstract in this paper we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball b n n 3 using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem we prove the existence of a metric conformally equivalent to the euclidean metric with zero scalar curvature and prescribed mean curvature on the boundary

multiple solutions of an inhomogeneous neumann problem for an elliptic system with critical sobolev exponent abstract in this paper we prove the existence of two solutions for the inhomogeneous neumann problem with critical sobolev exponent

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

a compactness theorem for scalar flat metrics on 3 manifolds with boundary abstract let m g be a compact riemannian three dimensional manifold with boundary we prove the compactness of the set of scalar flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface this involves a blow up analysis of a yamabe type equation with critical sobolev exponent on the boundary

the han li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds abstract the han li conjecture states that let m g 0 be an n dimensional n 3 smooth compact riemannian manifold with boundary having positive generalized yamabe constant and c be any real number then there exists a conformal metric of g 0 with scalar curvature 1 and boundary mean curvature c combining with z c han and y y li s results we answer this conjecture affirmatively except for the case that n 8 the boundary is umbilic the weyl tensor of m vanishes on the boundary and has an interior non zero point

on the information rate of secret sharing schemes we derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs we give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1 2 e where e is an arbitrary positive constant we also provide several general lower bounds on information rate and average information rate of graphs in particular we show that any graph with n vertices admits a secret sharing scheme with information rate log n n

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

a poincare sobolev type inequality on compact riemannian manifolds with boundary in this paper we study a poincare sobolev type inequality on compact riemannian n manifolds with boundary where the exponent growth is critical two constants have to be determined we show that contrary to the classical sobolev inequality the first best constant in this inequality does not depend on the dimension only but depends on the geometry it can be represented as the minimum mu of a given energy functional we study the nonlinear pde associated to this functional which involves the geometry of the boundary for a star shaped domain d in mathbb r n whose boundary has positive ricci curvature we give explicitly two sobolev constants corresponding to the embedding h 1 d in l frac 2 n 1 n 2 partial d this result is used to obtain an explicit geometrical lower bound for mu

compactness results in conformal deformations of riemannian metrics on manifolds with boundaries this paper is devoted to the study of a problem arising from a geometric context namely the conformal deformation of a riemannian metric to a scalar flat one having constant mean curvature on the boundary by means of blow up analysis techniques and the positive mass theorem we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the c2 norm and the total leray schauder degree of all solutions is equal to 1 then we deduce from this compactness result the existence of at least one solution to our problem

conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary abstract this paper considers the prescribed zero scalar curvature and mean curvature problem on the n dimensional euclidean ball for n 3 we consider the limits of solutions of the regularization obtained by decreasing the critical exponent we characterize those subcritical solutions which blow up at the least possible energy level determining the points at which they can concentrate and their morse indices we show that when n 3 this is the only blow up which can occur for solutions we use this in combination with the morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three dimensional euclidean ball in the higher dimensional case n 4 we give conditions on the function h to guarantee there is only one simple blow up point

sharp weighted sobolev trace inequalities and fractional powers of the laplacian abstract we establish a family of sharp sobolev trace inequalities involving the w k 2 r n 1 y a norm these inequalities are closely related to the realization of fractional powers of the laplacian on r n r n 1 as generalized dirichlet to neumann operators associated to powers of the weighted laplacian in upper half space generalizing observations of caffarelli silvestre and of yang

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

the best sobolev trace constant as limit of the usual sobolev constant for small strips near the boundary in this paper we prove that the best constant in the sobolev trace embedding h1 lq in a bounded smooth domain can be obtained as the limit as 0 of the best constant of the usual sobolev embedding h1 lq dx where x 2 dist x is a small neighborhood of the boundary we also analyze symmetry properties of extremals of this last embedding when is a ball

lichnerowicz type equations with sign changing nonlinearities on complete manifolds with boundary abstract we prove an existence theorem for positive solutions to lichnerowicz type equations on complete manifolds with boundary m m and nonlinear neumann conditions this kind of nonlinear problems arise quite naturally in the study of solutions for the einstein scalar field equations of general relativity in the framework of the so called conformal method