on an inequality by n trudinger and j moser and related elliptic equations it has been shown by trudinger and moser that for normalized functions u of the sobolev space w 1 n whereis a bounded domain in r n one has r exp njuj n n 1 dx cn where n is an explicit constant depending only on n and cn is a constant depending only on n and carleson and chang proved that there exists a corresponding extremal function in the case that is the unit ball in r n in this paper we give a new proof a generalization and a new interpretation of this result in particular we give an explicit sequence that is maximizing for the above integral among all normalized concentrating sequences as an application the existence of a nontrivial solution for a related elliptic equation with trudinger moser growth is proved c 2002 john wiley sons inc

global existence of solutions for compressible navier stokes equations with vacuum abstract in this paper we will investigate the global existence of solutions for the one dimensional compressible navier stokes equations when the density is in contact with vacuum continuously more precisely the viscosity coefficient is assumed to be a power function of density i e a where a and are positive constants new global existence result is established for 0 1 when the initial density appears vacuum in the interior of the gas which is the novelty of the presentation

existence of solutions to a model for sparse one dimensional fluids abstract we prove the global existence of solutions to a model for a viscous compressible barotropic fluid initially occupying a general open subset of a finite one dimensional interval the fluid equations are applied only on the support of the density understood in the sense of distributions this support must be tracked and accommodation must be made for the possibly infinite number of collisions of fluid packets occurring on a possibly dense set of collision times our approach avoids certain nonphysical properties of solutions which are constructed as limits of solutions in which artificial mass has been inserted

time analyticity and backward uniqueness of weak solutions of the navier stokes equations of multidimensional compressible flow abstract we prove that solutions of the navier stokes equations of three dimensional compressible flow restricted to fluid particle trajectories can be extended as analytic functions of complex time as consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher order lagrangean time derivatives the solutions under consideration are in a reasonably broad regularity class corresponding to small energy initial data with a small degree of regularity the latter being required for conversion to the lagrangean coordinate system in which the analysis is carried out

a rearrangement based proof for the existence of extremal functions for the sobolev poincare inequality on bn abstract existence of extremals for 1 p n 1 2 is shown for the sobolev poincare inequality on b n using a mean value zero preserving rearrangement

a symmetry problem related to wirtinger s and poincare s inequality j u 1 1 u p u q among periodic functions in w 1 1 with mean value zero and for p q 1 the minimizers are scaled in such a way that their maximum is 1 and their minimum is m 0 1 they raise the question if m 1 since this implies symmetries for the minimizers their answer is positive for q 2p and negative for q 1 practically the same question is raised in 2 and 3 if m 1 minimizers must be odd or symmetric in the sense that u x u x it was shown in 3 that for p q 2 the minimizer is symmetric and that for p 2 q it is not symmetric moreover in 2 one can find that the minimizer is symmetric for q 2p and nonsymmetric for for q 4p 1 the nonsymmetry result was derived by egorov in the following way restricted to the following set a

the best constant for the sobolev trace embedding from w1 1 ω into l1 ω abstract in this paper we study the best constant 1 for the trace map from w 1 1 into l 1 we show that this constant is attained in bv when 1 1 moreover we prove that this constant can be obtained as limit when p 1 of the best constant of w 1 p l p to perform the proofs we will look at neumann problems involving the 1 laplacian 1 u div du du

sharp hardy inequalities in the half space with trace remainder term abstract in this paper we deal with a class of inequalities which interpolate the kato inequality and the hardy inequality in the half space starting from the classical hardy s inequality in the half space r n r n 1 0 we show that if we replace the optimal constant n 2 2 4 with a smaller one 2 2 4 2 n then we can add an extra trace term similar to that one that appears in the kato inequality the constant in the trace remainder term is optimal and it tends to zero when goes to n while it is equal to the optimal constant in kato s inequality when 2 the approach is based on a very classical method of calculus of variation due to weierstrass and developed by hilbert that usually is considered to prove that the solutions of the euler lagrange equation associated to a functional are in fact extremals in this paper we will show how this method is well suited also to functionals that have no extremals

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

extreme points and support points of subordination families this paper surveys the main results and open questions in the theory of extreme points and support points of subordination families of analytic functions an effort has been made to make the list of references as complete as possible

on extreme points of subordination families with a convex majorant abstract let z c z b 0 denote the set of functions analytic in and satisfying z 0 0 suppose f is analytic and univalent in and maps it onto a convex domain d other than a strip a wedge or a half plane let s f fo b 0 let es f denote the set of extreme points of s f d j hallenbeck and t h macgregor asked in 3 whether the set es f ever lies strictly between the minimal fo b 0 lim r 1 re i 1 a e and the maxima fc is an extreme point of b 0 sets we prove that this always happens when the boundary of d contains a line segment let d f be any convex domain and let f be subordinate to f we show that if f is an extreme point of s f then the closure of the boundary values f of f f lim r 1 f re i a e has a nonempty intersection with the set of extreme points of the boundary of d we also completely determine extreme points of s f when the boundary of d consists of finitely many arcs wih positive curvature

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

existence and uniqueness of low energy weak solutions to the compressible 3d magnetohydrodynamics equations abstract we prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics mhd equations we first obtain the existence of weak solutions with small l 2 norm which may display codimension one discontinuities in density pressure magnetic field and velocity gradient the weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible mhd equations our results generalise and extend those for the intermediate weak solutions of compressible navier stokes equations

a sharp trace inequality for functions of bounded variation in the ball the best constant in a mean value trace inequality for functions of bounded variation on admissible domains n is shown to agree with an isoperimetric constant associated with the existence and form of extremals is also discussed this result is exploited to compute the best constant in the relevant trace inequality when is a ball the existence and the form of extremals in this special case turn out to depend on the dimension n in particular the best constant is not achieved when is a disc in 2

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

sobolev embeddings into spaces of campanato morrey and holder type abstract necessary and sufficient conditions on a rearrangement invariant banach function space x q on a cube q in r n n 2 are given for the corresponding sobolev space w1x q to be continuously embedded into generalized campanato morrey or holder spaces the optimal such r i spaces x q are found as a by product sharp inclusion relations are proved among campanato morrey and holder type spaces

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

existence and uniqueness of small energy weak solution to multi dimensional compressible navier stokes equations with large external potential force we study the 3 d compressible navier stokes equations with an external potential force and a general non decreasing pressure we prove the global in time existence of weak solutions with small energy initial data and with densities being non negative and essentially bounded a solution may have large oscillations and contain vacuum states no smallness assumption is made on the external force nor the initial perturbation in l for density initial velocity u0 is taken to be bounded in lq for some q 6 and no further regularity assumption is imposed on u0 finally we discuss the uniqueness of weak solutions

fundamental solution for the q laplacian and sharp moser trudinger inequality in carnot groups abstract for a general carnot group g with homogeneous dimension q we prove the existence of a fundamental solution of the q laplacian u q and a constant a q 0 such that exp a q u q is a homogeneous norm on g this implies a representation formula for smooth functions on g which is used to prove the sharp carnot group version of the celebrated moser trudinger inequality

sharp exponential integrability for critical riesz potentials and fractional laplacians on rn abstract we derive sharp adams inequalities for the riesz and more general riesz like potentials on the whole of r n as a consequence we obtain sharp moser trudinger inequalities for the critical sobolev spaces w n r n 0 n these inequalities involve fractional laplacians higher order gradients general homogeneous elliptic operators with constant coefficients and general trace type borel measures

global solutions of the navier stokes equations for viscous compressible flows the compressible navier stokes equations for viscous flows with general large continuous initial data as well as with large discontinuous initial data are studied both a homogeneous free boundary problem with zero outer pressure and a fixed boundary problem are considered for the large initial data in h1 the existence uniqueness and regularity of global solutions in h1 for real viscous flows are established and it is showed that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time for the large discontinuous data the global existence of large weak solutions for the perfect gases is also established using a different argument and it is indicated that the solutions do not develop vacuum or concentration although the solutions have large discontinuity for the free boundary problem the interface separating the flows from the zero outer pressure expands at a finite speed

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

support points of families of analytic functions described by subordination we determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination the families we study include the functions with a positive real part the typically real functions and the functions which are subordinate to a given majorant if the majorant f is univalent then each support point has the form f 0 where p is a finite blaschke product and o 0 o this completely characterizes the set of support points when f is convex the set of support points is found for some specific majorants including f z 1 z l z p where p 1 let k and st denote the set of normalized convex and starlike mappings respectively we find the support points of the families k and st defined by the property of being subordinate to some member of k or st respectively introduction let a z e c i z i 1 and let e denote the set of functions analytic in a then 6 is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of la by a continuous linear functional on e we mean a complex valued functional defined on 6f that is linear and continuous in other words if j is such a functional then j af bg aj f bj g whenever a and b belong to c and f and g belong to b also if fn e 6b n 1 2 and fn f then j fn j f each continuous linear funcn tional j on ef is given by a sequence bn n 0 1 which satisfies limn n i n i 1 and

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

l bounds of steklov eigenfunctions and spectrum stability under domain variation abstract we give a practical tool to control the l norm of the steklov eigenfunctions in a lipschitz domain in terms of the norm of the bv trace operator the norm of this operator has the advantage to be characterized by purely geometric quantities as a consequence we give a spectral stability result for the steklov eigenproblem under geometric domain perturbations and several examples where stability occurs in particular we deal with geometric domains which are not equi lipschitz like vanishing holes merging sets approximations of inner peaks

the longest shortest fence and sharp poincare sobolev inequalities we prove a long standing conjecture concerning the fencing problem in the plane among planar convex sets of given area the disc and only the disc maximizes the length of the shortest area bisecting curve although it may look intuitive the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so called auerbach triangle

the failure on continuous dependence on initial data for the navier stokes equations of compressible flow it is shown that physical solutions of the navier stokes equations for one dimensional compressible flow need not depend continuously on their initial data at least when vacuum states are allowed specifically two fluid regions initially separated by a third region of very low density 6 are considered it is shown that as delta to 0 the unique solutions corresponding to delta 0 do not in fact converge to a physical solution but rather to a nonphysical weak solution in which the two fluids cannot collide independent of their initial velocities and whose separate momenta need not be conserved a particular consequence is that solutions of the cavity problem delta 0 are not unique

a vacuum problem for the one dimensional compressible navier stokes equations with density dependent viscosity this paper is concerned with the free boundary problem for the one dimensional compressible navier stokes equations with density dependent viscosity a local in time existence result is established when the initial density is of compact support and connects to the vacuum continuously c 2002 elsevier science usa

global low energy weak solutions of the equations of three dimensional compressible magnetohydrodynamics we prove the global in time existence of weak solutions of the equations of compressible magnetohydrodynamics in three space dimensions with initial data small in l2 and initial density positive and essentially bounded a great deal of information concerning partial regularity is obtained velocity vorticity and magnetic field become relatively smooth in positive time h1 but not h2 and singularities in the pressure cancel those in a certain multiple of the divergence of the velocity thus giving concrete expression to conclusions obtained formally from the rankine hugoniot conditions

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

construction of solutions for compressible isentropic navier stokes equations in one space dimension with nonsmooth initial data we prove the global existence of weak solutions for the cauchy problem for the navier stokes equations for one dimensional isentropic flow when the initial velocity is in l 2 and the initial density is in l 2 bv solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi discrete difference scheme this allows for a rather simple analysis in which pointwise control is achieved through piecewise h 1 and total variation estimates

an isoperimetric inequality and the first steklov eigenvalue let mn g be a compact riemannian manifold with boundary in this paper we give upper and lower estimates for the first nonzero steklov eigenvalue 0inm 1 on m where 1 is a positive real number the estimate from below is for a star shaped domain on a manifold whose ricci curvature is bounded from below the upper estimate is for a convex manifold with nonnegative ricci curvature and is given in terms of the first nonzero eigenvalue for the laplacian on the boundary we prove a comparison theorem for simply connected domains in a simply connected manifold we exhibit annuli domains for which the comparison theorem fails to be true in j f escobar j funct anal 60 1997 544 556 we introduced the isoperimetric constant i m defined asi m inf mvol min vol 1 vol 2 where 1 m is a nonempty domain with boundary in the manifold m 2 m 1 and int m where int m is the interior of m we proved a cheeger s type inequality for 1 using the constant i m in this paper we give upper and lower estimates for the constant i in terms of isoperimetric constants of the boundary of m

the shape of extremal functions for poincare sobolev type inequalities in a ball we study extremal functions for a family of poincare sobolev type inequalities these functions minimize for subcritical or critical p 2 p 2 the quotient u 2 u p u 2 u p among all u h 1 b 0 u h 1 b 0 with b u 0 b u 0 here b is the unit ball in rn r n we show that the minimizers are axially symmetric with respect to a line passing through the origin we also show that they are strictly monotone in the direction of this line in particular they take their maximum and minimum precisely at two antipodal points on the boundary of b we also prove that for p close to 2 minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis and that once the symmetry axis is fixed they are unique up to multiplication by a constant in space dimension two we prove that minimizers are not antisymmetric for large p

classes of analytic functions subordinate to convex functions and extreme points in this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping f when the image of the unit disk under f is a convex domain other than a half plane a strip or an infinite wedge

the sharp affine l 2 sobolev trace inequality and variants we establish a sharp affine l p sobolev trace inequality by using the l p busemann petty centroid inequality for p 2 our affine version is stronger than the famous sharp l 2 sobolev trace inequality proved independently by escobar and beckner our approach allows also to characterize all extremizers in this case for this new inequality no euclidean geometric structure is needed

an estimate for the morse index of a stokes wave stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two dimensional flow under gravity without surface tension they can be described in terms of solutions of the euler lagrange equation of a certain functional this allows one to define the morse index of a stokes wave it is well known that if the morse indices of the elements of a set of non singular stokes waves are bounded then none of them is close to a singular one the paper presents a quantitative variant of this result