viscosity versus vorticity stretching global well posedness for a family of navier stokes alpha like models abstract we study global well posedness and regularity of solutions for a family of incompressible three dimensional navier stokes alpha like models that employ fractional laplacian operators this family of equations depends on two parameters 1 and 2 which affect the strength of non linearity vorticity stretching and the degree of viscous smoothing varying 1 and 2 interpolates between the incompressible navier stokes equations and the incompressible lagrangian averaged navier stokes model our main result which contains previously established results of j l lions and others provides a relationship between 1 and 2 that is sufficient to guarantee global existence uniqueness and regularity of solutions

the oberbeck boussinesq problem modified by a thermo absorption term abstract we consider the oberbeck boussinesq problem with an extra coupling establishing a suitable relation between the velocity and the temperature our model involves a system of equations given by the transient navier stokes equations modified by introducing the thermo absorption term the model involves also the transient temperature equation with nonlinear diffusion for the obtained problem we prove the existence of weak solutions for any n 2 and its uniqueness if n 2 then considering a low range of temperature but upper than the phase changing one we study several properties related with vanishing in time of the velocity component of the weak solutions first assuming the buoyancy forces field extinct after a finite time we prove the velocity component will extinct in a later finite time provided the thermo absorption term is sublinear in this case considering a suitable buoyancy forces field which vanishes at some instant of time we prove the velocity component extinct at the same instant we prove also that for non zero buoyancy forces but decaying at a power time rate the velocity component decay at analogous power time rates provided the thermo absorption term is superlinear at last we prove that for a general non zero bounded buoyancy force the velocity component exponentially decay in time whether the thermo absorption term is sub or superlinear

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

sharp constants for moser trudinger inequalities on spheres in complex space ℂn the main results of this paper concern sharp constants for the moser trudinger inequalities on spheres in complex space c n we derive moser trudinger inequalities for smooth functions and holomorphic functions with different sharp constants see theorem 1 1 the sharp moser trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the cr setting though there is a close connection in spirit between inequalities proven here on complex spheres and those on the heisenberg group for functions with compact support in any finite domain proven earlier by the same authors 17 derivation of the sharp constants for moser trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the heisenberg group variants of moser onofri type inequalities are also given on complex spheres as applications of our sharp inequalities see theorems 1 2 and 1 3 one of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients see theorem 1 4 c 2004 wiley periodicals inc

degenerate first order inverse problems in banach spaces abstract we are concerned with an inverse problem for a degenerate linear evolution equation of first order both hyperbolic and parabolic cases will be considered we indicate sufficient conditions for the existence and uniqueness of a solution all the results can be applied to inverse problems for equations from mathematical physics as a possible application of the abstract theorems some examples of partial differential equations are given

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

extinction properties of solutions for a class of fast diffusive p laplacian equations abstract we consider the extinction properties of solutions for the homogeneous dirichlet boundary value problem for the p laplacian equation u t div u p 2 u u q u r with 1 p 2 q 1 and r 0 for 0 it is known that r p 1 is the critical extinction exponent for the weak solution for 0 we show that r p 1 is still the critical extinction exponent when q 1 moreover the precise decay estimates of solutions before the occurrence of the extinction are derived however extinction can always occur when 0 q r 1

kelvin voigt equations perturbed by anisotropic relaxation diffusion and damping abstract the purpose of this work is the analysis of the existence and uniqueness of weak solutions to a kelvin voigt problem wherein the viscous and relaxation parts of the stress tensor are given by distinct power laws we assume that the viscous and relaxation terms may be fully anisotropic and that the momentum equation is perturbed by a damping term which may also be fully anisotropic in the particular case of considering this problem with a linear and isotropic relaxation term we prove the existence of global and local weak solutions the uniqueness of weak solutions is established in this case as well for the full anisotropic problem we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients must interact in order to be established global and local in time a priori estimates

blow up of solutions to parabolic equations with nonstandard growth conditions we study the phenomenon of finite time blow up in solutions of the homogeneous dirichlet problem for the parabolic equation u t div a x t u p x email protected u b x t u s x t 2u with variable exponents of nonlinearity p x s x t 1 two different cases are studied in the case of semilinear equation with p x 2 a x t 1 b x t b 0 we show that the finite time blow up happens if the initial function is sufficiently large and either min email protected x t email protected t 2 for all t 0 or s t 2 s t 2 as t and 1 e s 2 s s ds b 0 a t x t 0 b t x t 0 email protected x 2 and maxp x email protected x

semi linear liouville theorems in the heisenberg group via vector field methods we consider a global liouville type theorem for semi linear elliptic equation on heisenberg group our main idea is the method of integration by parts for a vector field analogous to the quantity in gidas and spruck 1981 5 c 2009 elsevier inc all rights reserved

on some modification navier stokes equations abstract this article investigates some modification of the navier stokes equations of type as the modification that was suggested by lions quelques methodes de resolution des problemes aux limites non lineares dunod gauthier villaris paris 1969 in the form u j t i 1 n d i u j p i 2 d i u j i 1 n u i d i u j p x j h j t x t x q r n 1 div u 0 t x q 0 t t 0 0 u j 0 x 0 x u 0 0 t under p i p i 1 i n n 2 in this we prove the existence theorem for the different pi max 2 3 2 n and on some additional conditions i e of the pi p 4 in the isotropic nonlinearity case we prove the uniqueness theorem for the considered problem

improved moser trudinger type inequalities in the hyperbolic space hn we establish an improved version of the moser trudinger inequality in the hyperbolic space hnhn n 2n 2 namely we prove the following result for any 0 n 1nn then we have supu c0 hn hn gu gndvolg hn u ndvolg 1 hn n n u nn 1 dvolg where n n n 11n 1 n 1 n 1 denotes the surface area of the unit sphere in rnrn and n t et j 0n 2tjj this improves the moser trudinger inequality in hyperbolic spaces obtained recently by mancini and sandeep 2010 by mancini et al 2013 and by adimurthi and tintarev 2010 for 0 0 in the limiting case n 1n n we prove a moser trudinger inequality with exact growth in hnhn supu c0 hn hn gu gndvolg n 1n n hn u ndvolg 11 hn u ndvolg hn n n u nn 1 1 u nn 1dvolg this improves the moser trudinger inequality with exact growth in hnhn established by lu and tang 2016 these inequalities are achieved from the comparison of the symmetric non increasing rearrangement of a function both in the hyperbolic and the euclidean spaces and the same inequalities in the euclidean space this approach seems to be new comparing with the previous ones

kelvin voight equation with p laplacian and damping term existence uniqueness and blow up abstract in this paper we consider the initial boundary value problem for a generalized kelvin voight equation with p laplacian and a damping term v t v v p x t div v t div v p 2 v v m 2 v div v 0 here v x t is the velocity field p x t is the pressure is the viscosity kinematic coefficient and is the viscosity relaxation coefficient is a length scale parameter characterizing the elasticity of the fluid the coefficient and the exponents p m are given constants under appropriate conditions on the data we prove the existence and uniqueness of the global and local weak solutions under several assumptions on the exponents p m the coefficients and specified initial data a finite time blow up and the behavior of the solutions for large times are also established

existence uniqueness and blowup for hyperbolic equations with nonstandard growth conditions in this paper we consider the following problem utt u b x t u x t 2u f x t x t qt 0 t u x 0 u0 x ut x 0 u1 x 0 x u t 0 t 0 t under conditions 0 b t b x t b t 0 b t b x t b t 1 t x t t 1 t x t t we study the existence and uniqueness of weak and strong solutions we prove the blowup of nonnegative solutions also we consider the blowup effect for a more generalized problem utt lu f x t u inqt u x 0 u0 x in ut x 0 u1 x in u 0on t where ll is the linear elliptic operator lu di aij x dju ai x u a0 x u lu di aij x dju ai x u a0 x u and the right hand side has the form f x t u k 1nbk x t u k x t 1 i nkci x t di s t u i s t 1ds with bk 0bk 0 ci 0ci 0 di 0di 0

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

on a conformally invariant variational problem on differential forms abstract we consider a conformally invariant constrained variational problem on p differential forms for p dim m 2 2 coming from conformal geometry and the associated yamabe type system of partial differential equations it is shown that the infimum p m of the functional is not bigger than p r n and if p m p r n then p m is achieved by a c 2 p form for some 0

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

concentration compactness principle for the sharp adams inequalities in bounded domains and whole space rn abstract we prove an improvement of the sharp adams inequality in w 0 m n m where is a bounded domain in r n inspired by lions concentration compactness principle for the sharp moser trudinger inequality our method gives an alternative approach to a concentration compactness principle in w 0 m n m recently established by do o and macedo furthermore we obtain a sharp threshold for m odd improving the one of do o and macedo our approach is also successfully applied to whole space r n to establish the improvements of the sharp adams inequalities in w m n m r n this type of improvement is still unknown in general except the case m 1 due to do o de souza de medeiros and severo our method is a further development of the one of cerny cianchi and hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

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

fundamental solutions for non divergence form operators on stratified groups we construct the fundamental solutions and for the non divergence form operators i j a i j x t x i x j t and i j a i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we also provide gaussian estimates of r and its derivatives and some results for the relevant cauchy problem suitable long time estimates of allow us to construct using both t saturation and approximation arguments

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

on the moser trudinger inequality in complex space abstract in this paper we prove the pluricomplex counterpart of the moser trudinger and sobolev inequalities in complex space we consider these inequalities for plurisubharmonic functions with finite pluricomplex energy and we estimate the concerned constants

generalized kelvin voigt equations with p laplacian and source absorption terms abstract this study considers the blow up in finite time and large time behavioral properties of solutions to the initial boundary value problem for generalized kelvin voigt equations with p laplacian and source absorption terms v t v v p div d v q 2 d v t d v p 2 d v v m 2 v div v x t 0 x t q t where d v 1 2 v v t is the rate of the strain tensor v x t is the velocity field p x t is the pressure is the viscosity kinematic coefficient and is the viscosity relaxation coefficient the coefficient and the exponents p m are given constants two different cases are considered in the case where 0 with an absorption term we prove that the solutions of the associated problem exhibit exponential and power decay and in the case where 0 with nonlinear source term the solutions blow up in a finite time under suitable assumptions on the exponents p q m and the coefficients with certain initial data

isocapacity estimates for hessian operators abstract through a new powerful potential theoretic analysis this paper is devoted to discovering the geometrically equivalent isocapacity forms of chou wang s sobolev type inequality and tian wang s moser trudinger type inequality for the fully nonlinear 1 k n 2 hessian operators

cr yamabe constant cr yamabe flow and its soliton abstract on a compact strictly pseudoconvex cr manifold m we consider the cr yamabe constant of its infinite conformal covering by using the maximum principles we then prove a uniqueness theorem for the cr yamabe flow on a complete noncompact cr manifold finally we obtain some properties of the cr yamabe soliton on complete noncompact cr manifolds

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

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

adams inequalities on measure spaces abstract in 1988 adams obtained sharp moser trudinger inequalities on bounded domains of r n the main step was a sharp exponential integral inequality for convolutions with the riesz potential in this paper we extend and improve adams results to functions defined on arbitrary measure spaces with finite measure the riesz fractional integral is replaced by general integral operators whose kernels satisfy suitable and explicit growth conditions given in terms of their distribution functions natural conditions for sharpness are also given most of the known results about moser trudinger inequalities can be easily adapted to our unified scheme we give some new applications of our theorems including sharp higher order moser trudinger trace inequalities sharp adams moser trudinger inequalities for general elliptic differential operators scalar and vector valued for sums of weighted potentials and for operators in the cr setting

concentration compactness principle for trudinger moser inequalities with logarithmic weights and their applications abstract in this paper we establish a sharp concentration compactness principle associated with the trudinger moser inequality on sobolev spaces with logarithmic weights as applications we establish the existence of ground state solutions to the following equation with critical double exponential nonlinearity d i v u n 2 u x f x u in b 1 0 u 0 in b 1 0 u 0 on b 1 0 where x log e x n 1 is the logarithmic weight the nonlinear term f x u is continuous radial in x b 1 0 and has critical double exponential growth which behaves like exp e e n u n

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

blow up of solutions of pseudoparabolic equations we obtain sufficient conditions for the blow up of solutions of the initial boundary value problem for nonlinear pseudoparabolic equation involving nonlinear convective term

existence of extremal functions for the stein weiss inequalities on the heisenberg group abstract in this paper we establish the existence of extremals for two kinds of stein weiss inequalities on the heisenberg group more precisely we prove the existence of extremals for the stein weiss inequalities with full weights in theorem 1 1 and the stein weiss inequalities with horizontal weights in theorem 1 4 different from the proof of the analogous inequality in euclidean spaces given by lieb 26 using riesz rearrangement inequality which is not available on the heisenberg group we employ the concentration compactness principle to obtain the existence of the maximizers on the heisenberg group our result is also new even in the euclidean case because we don t assume that the exponents of the double weights in the stein weiss inequality 1 1 are both nonnegative see theorem 1 3 and more generally theorem 1 5 therefore we extend lieb s celebrated result of the existence of extremal functions of the stein weiss inequality in the euclidean space to the case where the exponents are not necessarily both nonnegative see theorem 1 3 furthermore since the absence of translation invariance of the stein weiss inequalities additional difficulty presents and one cannot simply follow the same line of lions idea to obtain our desired result our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities theorem 1 5

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

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

sharp subcritical moser trudinger inequalities on heisenberg groups and subelliptic pdes the aim of this paper is to prove a sharp subcritical moser trudinger inequality on the whole heisenberg group let h cn rh cn r be the n n dimensional heisenberg group q 2n 2q 2n 2 be the homogeneous dimension of hh q qq 1 and z 4 t2 14 be the homogeneous norm of z t h z t h then we establish the following inequality on hh theorem 1 1 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q0 q 0 q 1 q there exists a constant 0 c c 0 c c such that the following inequality holds sup hu lq h 11 u lq h q h1 exp u q q 1 k 0q 2 kk u kq q 1 c the above result is the best possible in the sense when q 1 q the integral is still finite for any u w1 q h u w1 q h but the supremum is infinite in contrast to the analogous inequality in euclidean spaces proved in adachi and tanaka 1999 6 using symmetrization our argument is completely different and avoids the symmetrization method which is not available on the heisenberg group in an optimal way moreover our restriction on the norm hu lq h 1 hu lq h 1 of the function uu is much weaker than hu lq h u lq h 1 which was assumed in lam and lu 2012 16 as a consequence our inequality fails at q 1 q in contrast to the one in 16 as an application of this inequality we will prove that the following nonlinear subelliptic equation of qq laplacian type without perturbation equation 0 1 qu v u q 2u f u in h has a nontrivial weak solution where the nonlinear term ff has the critical exponential growth e u qq 1 as u u but does not satisfy the ambrosetti rabinowitz condition

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

solvability and long time behavior of nonlinear reaction diffusion equations with robin boundary condition abstract in this paper we investigate the long time behavior and solvability of the reaction diffusion equation which has a polynomial growth nonlinearity of arbitrary order with robin boundary condition we begin this paper with the existence and uniqueness of the solution to the problem for the long time behavior we firstly prove the existence of an absorbing set in two different spaces secondly for the autonomous case the existence of a global attractor is obtained in w 2 1 l 2

the dirichlet problem for the kohn laplacian on the heisenberg group ii for x y t rn rn r denote xj xj 2yj t yj yj 2xj t and l 14 j 1nxj2 yj2 t when n 2q la represents the action of the kohn laplacian b on q forms on the heisenberg group for n n we construct a parametrix for the dirichlet problem in smooth domains d near non characteristic points of d a point w of d is non characteristic if one of x1 xn y1 yn is transverse to d at w this yields sharp local estimates in the dirichlet problem in the appropriate non isotropic lipschitz classes the main new tool is a convolution calculus of pseudo differential operators that can be applied to the relevant layer potentials for which the usual asymptotic composition formula is false characteristic points are treated in part ii

a cr poincare inequality on the complex sphere abstract the main purpose of this paper is to prove a cr poincare inequality with sharp exponent on the sphere in complex space we use the complex tangential gradient on the sphere instead of the usual laplace beltrami gradient on the sphere

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained

sharp hardy littlewood sobolev inequalities on quaternionic heisenberg groups this paper is the second one following christ et al nonlinear anal 130 361 395 2016 in a series considering sharp hardy littlewood sobolev inequalities on groups of heisenberg type the first important breakthrough was made in frank et al ann math 176 349 381 2012 in this paper analogous results are obtained for the octonionic heisenberg group

projection method for solving degenerate first order identification problem in this paper we study an inverse problem for a degenerate differential equation on a banach space a projection method will be used to reduce the problem to a regular abstract inverse problem to this end some conditions on the operators are assumed to avoid the calculations of some inverses of the involved operators

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

some nonlinear equations of the nonstable filtration type and embedding theorems abstract in this article we consider some mixed problems with implicit degenerate equations for their investigation we study certain properties of a class of pn spaces connected with nonlinear boundary value problems further using the results obtained we prove the solvability of some nonlinear boundary value problems with various elliptic parabolic equations and also a boundary problem for an equation of the nonstable filtration type

identification problems for singular integro differential equations of parabolic type ii abstract we recover unknown kernels depending on time only in linear singular first order integro differential cauchy problems in banach spaces singular means here that the integro differential equation is not in normal form nor it can be reduced to such a form for this class of problems we prove an existence and uniqueness theorem in the framework of general banach spaces under the condition that the resolvent operator cf 1 5 admits a polar singularity at 0 see section 3 moreover when the banach space under consideration is reflexive we can prove a local in time existence and uniqueness result when the resolvent operator decays as 1 1 finally we give a few applications to explicit singular partial integro differential equations of parabolic type

sharp hardy adams inequalities for bi laplacian on hyperbolic space of dimension four abstract we establish sharp hardy adams inequalities on hyperbolic space b 4 of dimension four namely we will show that for any 0 there exists a constant c 0 such that b 4 e 32 2 u 2 1 32 2 u 2 d v 16 b 4 e 32 2 u 2 1 32 2 u 2 1 x 2 4 d x c for any u c 0 b 4 with b 4 h 9 4 h u u d v 1 as applications we obtain a sharpened adams inequality on hyperbolic space b 4 and an inequality which improves the classical adams inequality and the hardy inequality simultaneously the later inequality is in the spirit of the hardy trudinger moser inequality on a disk in dimension two given by wang and ye in 46 and on any convex planar domain by lu and yang in 33 the fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work

perturbation of the mapping and solvability theorems in banach space abstract here we consider perturbations of continuous mappings on banach spaces and investigate their images under various conditions consequently we study the solvability of some classes of equations and inclusions for these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings moreover we study different mixed problems

existence and nonexistence of extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2 abstract though much progress has been made with respect to the existence of extremals of the critical first order trudinger moser inequalities in w 1 n r n and higher order adams inequalities on finite domain r n whether there exists an extremal function for the critical higher order adams inequalities on the entire space r n still remains open the current paper represents the first attempt in this direction by considering the critical second order adams inequality in the entire space r 4 the classical blow up procedure cannot apply to solving the existence of critical adams type inequality because of the absence of the polya szego type inequality in this paper we develop some new ideas and approaches based on a sharp fourier rearrangement principle see 31 sharp constants of the higher order gagliardo nirenberg inequalities and optimal poly harmonic truncations to study the existence and nonexistence of the maximizers for the adams inequalities in r 4 of the form s sup u h 2 1 r 4 exp 32 2 u 2 1 u 2 d x where 32 2 we establish the existence of the threshold where 32 2 2 b 2 2 and b 2 1 24 2 such that s is attained if 32 2 and is not attained if 32 2 this phenomenon has not been observed before even in the case of first order trudinger moser inequality therefore we also establish the existence and non existence of an extremal function for the trudinger moser inequality on r 2 furthermore the symmetry of the extremal functions can also be deduced through the fourier rearrangement principle

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds