sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

on schwarz type smoothers for saddle point problems in this paper we consider additive schwarz type iteration methods for saddle point problems as smoothers in a multigrid method each iteration step of the additive schwarz method requires the solutions of several small local saddle point problems this method can be viewed as an additive version of a multiplicative vanka type iteration well known as a smoother for multigrid methods in computational fluid dynamics it is shown that under suitable conditions the iteration can be interpreted as a symmetric inexact uzawa method in the case of symmetric saddle point problems the smoothing property an important part in a multigrid convergence proof is analyzed for symmetric inexact uzawa methods including the special case of the additive schwarz type iterations as an example the theory is applied to the crouzeix raviart mixed finite element for the stokes equations and some numerical experiments are presented

solutions to the master equations governing fractional vortices abstract by means of variational methods in this paper we establish sharp existence results for solutions of the master equations governing fractional multiple vortices in the doubly periodic situation the conditions for existence are both necessary and sufficient and give the upper bounds for the vortex numbers in terms of the size of the periodic cell domain in the planar situation there is no restriction on the vortex numbers in both situations the solutions are uniquely determined by the prescribed locations and the local winding numbers of the vortices

convergence of nonconforming multigrid methods without full elliptic regularity we consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity we prove that there is a bound 1 for the contraction number of the w cycle algorithm which is independent of mesh level provided that the number of smoothing steps is sufficiently large we also show that the symmetric variable v cycle algorithm is an optimal preconditioner

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

maximal solution of the liouville equation in doubly connected domains abstract in this paper we consider the liouville equation u 2 e u 0 with dirichlet boundary conditions in a two dimensional doubly connected domain we show that there exists a simple closed curve such that for a sequence n 0 and a sequence of solutions u n it holds u n log 1 n h where h is a harmonic function in and n 2 log 1 n e u n d x 8 c where c is a constant depending on the conformal class of only

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

removable sets in elliptic equations with musielak orlicz growth abstract in this study in terms of the intrinsic hausdorff measures we characterize the size of removable sets for holder continuous solutions to elliptic equations with musielak orlicz growth in the general case we provide a result on a new scale that is more relevant and that captures the classical results as special cases the results slightly refine the known ones provided for problems stated in the variable exponent and double phase spaces and they essentially improve the one known in the orlicz case

multigrid methods for saddle point problems stokes and lame systems we develop new multigrid methods for a class of saddle point problems that include the stokes system in fluid flow and the lame system in linear elasticity as special cases the new smoothers in the multigrid methods involve optimal preconditioners for the discrete laplace operator we prove uniform convergence of the w w cycle algorithm in the energy norm and present numerical results for w w cycle and v v cycle algorithms

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

fast iterative solvers for discrete stokes equations we consider saddle point problems that result from the finite element discretization of stationary and nonstationary stokes equations three efficient iterative solvers for these problems are treated namely the preconditioned conjugate gradient method introduced by bramble and pasciak the preconditioned minres method and a method due to bank et al we give a detailed overview of algorithmic aspects and theoretical convergence results for the method of bank et al a new convergence analysis is presented a comparative study of the three methods for a three dimensional stokes problem discretized by the hood taylor p2 p1 finite element pair is given

the scalar curvature flow on s n perturbation theorem revisited for the problem of finding a geometry on s n for n 3 with a prescribed scalar curvature there is a well known result which is called the perturbation theorem it is due to chang and yang duke math j 64 27 69 1991 their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm it is important to know how large that difference in sup norm can possibly be here we consider prescribing scalar curvature problem using the scalar curvature flow

on the convergence of multi grid methods with transforming smoothers in the present paper we give a convergence theory for multi grid methods with transforming smoothers as introduced in 31 applied to a general system of partial differential equations the theory follows hackbusch s approach for scalar pde and allows a convergence proof for some well known multi grid methods for stokes and navier stokes equations as dgs by brandt dinar 5 tilu from 31 and the simple methods by patankar spalding 23

a nonlinear free boundary problem with a self driven bernoulli condition abstract we study a bernoulli type free boundary problem with two phases j u u x 2 d x m u m u u u w 0 1 2 where u w 1 2 is a given boundary datum here m 1 and m 2 are weighted volumes of u 0 and u 0 respectively and is a nonnegative function of two real variables we show that for this problem the bernoulli constant which determines the gradient jump condition across the free boundary is of global type and it is indeed determined by the weighted volumes of the phases in particular the bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself and therefore it depends on the minimizer itself and not only on the structural constants of the problem another property of this type of problems is that the minimizer in is not necessarily a minimizer in a smaller subdomain due to the nonlinear structure of the problem due to these features this problem is highly unstable as opposed to the classical case studied by alt caffarelli and friedman it also interpolates the classical case in the sense that the blow up limits of u are minimizers of the alt caffarelli friedman functional namely the energy of the problem somehow linearizes in the blow up limit as a special case we can deal with the energy levels generated by the volume term 0 r 2 r 2 n 1 n which interpolates the athanasopoulos caffarelli kenig salsa energy thanks to the isoperimetric inequality in particular we develop a detailed optimal regularity theory for the minimizers and for their free boundaries

concerning a type of sobolev inequality on the sphere in chang and yang 1 5 a type of sobolev inequality was proved for a class of functions on s 2 it played an important role in the proof of the existence by the min max method of a pointwise conformal metric on s 2 which has some given function as its scalar curvature in this paper we prove an analogous inequality for functions on s n n 3 from now on denotes the integral average on s i e f s fdx vol s for functions on s and x j j 1 2 n 1 denote orthogonal coordinates in r 1

a multigrid homotopy method for nonlinear inverse problems abstract in the present contribution we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems in which the forward problems are modeled by some forms of partial differential equations the method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest and then carries out the inversion on the coarsest grid by the homotopy method the strategy may give a rapidly and globally convergent method as a practical application this method is used to solve the nonlinear inverse problem of a nonlinear convection diffusion equation which is the saturation equation within the two phase porous media flow we demonstrate the effectiveness and merits of the multigrid homotopy method on two actual model problems

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

an optimization problem with volume constraint for a degenerate quasilinear operator abstract we consider the optimization problem of minimizing u p d x with a constraint on the volume of u 0 we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

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

on a galerkin lagrange multiplier method for the stationary navier stokes equations a galerkin lagrange multiplier formulation is used for the numerical solution of the stationary navier stokes equations in order to avoid the construction of zero divergence elements the formulation is based on different approximating spaces for the velocity field and the pressure optimal rate of convergence estimates are derived moreover a galerkin newton scheme for the solution of the nonlinear equations is shown to be quadratically locally convergent another scheme is shown to be linearly globally convergent

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

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

a multilevel algorithm for mixed problems we describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise e g from mixed finite element approximations of the stokes equation the main difficulty besides the indefiniteness is the lack of regularity of the solution of the corresponding continuous problem this is overcome by introducing a scale of mesh dependent norms the convergence rate of the described algorithm is bounded independently of the meshsize for convenience we only discuss jacobi relaxation as smoothing operator in detail in the last section we comment on lanczos type smoothing procedures

regularity of the free boundary in an optimization problem related to the best sobolev trace constant in this paper we study the regularity properties of a free boundary problem arising in the optimization of the best sobolev trace constant in the immersion h 1 omega hookrightarrow l q partial omega for functions that vanish in a subset of omega this problem is also related to a minimization problem for steklov eigenvalues

multigrid analysis for the time dependent stokes problem certain implicit time stepping procedures for the incompressible stokes or navier stokes equations lead to a singular perturbed stokes type problem at each time step the paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method several smoothing iterative methods are considered a smoother based on distributive iterations the braess sarazin and inexact uzawa smoother convergence analysis is based on smoothing and approximation properties in special norms a robust independent of time step and mesh parameter estimate is proved for the two grid and multigrid w cycle convergence factors

the asymptotic behavior of a class of φ harmonic functions in orlicz sobolev spaces abstract the asymptotic behavior of the sequence u n of solutions for a class of inhomogeneous problems with prescribed dirichlet data on the boundary is studied in the setting of orlicz sobolev spaces we prove that u n u uniformly in as n where u is an harmonic function satisfying the prescribed dirichlet data on the boundary

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

multigrid methods for saddle point problems oseen system abstract we develop and analyze multigrid methods for the oseen system in fluid flow we show that the w cycle algorithm is a uniform contraction if the number of smoothing steps is sufficiently large numerical results that illustrate the performance of the methods are also presented

a minimum problem with free boundary in orlicz spaces abstract we consider the optimization problem of minimizing g u u 0 d x in the class of functions w 1 g with u 0 w 0 1 g for a given 0 0 and bounded w 1 g is the class of weakly differentiable functions with g u d x the conditions on the function g allow for a different behavior at 0 and at we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface also we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the lipschitz regularity of the weak solutions and the c 1 regularity of their free boundaries near flat free boundary points

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

classification of isolated singularities for nonhomogeneous operators in divergence form abstract consider the equation div u u u 0 on the punctured unit ball from r n n 2 where is an odd increasing homeomorphism from r onto r of class c 1 under reasonable assumptions on we prove that if u is a non negative solution of our equation then either 0 is a removable singularity of u or u behaves near 0 as the fundamental solution of the equation investigated here in particular our result complements to the case on nonhomogeneous operators in divergence form bocher s theorem and some classical results by serrin

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

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

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

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

a free boundary problem for the p x laplacian abstract we consider the optimization problem of minimizing 1 p x u p x x u 0 d x in the class of functions w 1 p with u 0 w 0 1 p for a given 0 0 and bounded w 1 p is the class of weakly differentiable functions with u p x d x we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface

an optimization problem with volume constraint in orlicz spaces we consider the optimization problem of minimizing g u dx in the class of functions w1 g with a constraint on the volume of u 0 the conditions on the function g allow for a different behavior at 0 and at we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

multigrid methods for saddle point problems optimality systems abstract we develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter we prove the uniform convergence of the w cycle algorithm and demonstrate the performance of v cycle and w cycle algorithms through numerical experiments

on quasilinear elliptic problems without the ambrosetti rabinowitz condition abstract using variational arguments we establish the existence of multiple solutions for quasilinear elliptic problems driven by the laplacian operator a major point is that we ensure compactness without the well known ambrosetti rabinowitz superlinearity condition

fractional order orlicz sobolev spaces abstract in this paper we define the fractional order orlicz sobolev spaces and prove its convergence to the classical orlicz sobolev spaces when the fractional parameter s 1 in the spirit of the celebrated result of bourgain brezis mironescu we then deduce some consequences such as convergence of the modulars and convergence of solutions for some fractional versions of the g operator as the fractional parameter s 1

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

multilevel algorithms for mixed problems ii treatment of the mini element we present a multilevel algorithm for the mixed finite element approximation of the stokes problem by the mini element a local l 2 projection of the bubble functions and a strengthened cauchy inequality are essential tools for the convergence analysis of the multigrid algorithm

inhomogeneous hopf oleĭnik lemma and regularity of semiconvex supersolutions via new barriers for the pucci extremal operators abstract in this paper we construct new barriers for the pucci extremal operators with unbounded rhs the geometry of these barriers is given by a harnack inequality up to the boundary type estimate under the possession of these barriers we prove a new quantitative version of the hopf ole nik lemma for quasilinear elliptic equations with g laplace type growth finally we prove sharp regularity for semiconvex supersolutions for some nonlinear pdes these results are new even for second order linear elliptic equations in nondivergence form moreover these estimates extend and improve a classical a priori estimate proven by l caffarelli j j kohn j spruck and l nirenberg in 13 in 1985 as well as a more recent result on the c 1 1 regularity for convex supersolutions obtained by c imbert in 33 in 2006

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

an overlapping domain decomposition preconditioner for a class of discontinuous galerkin approximations of advection diffusion problems we consider a scalar advection diffusion problem and a recently proposed discontinuous galerkin approximation which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency for the corresponding sparse non symmetric linear system we propose and study an additive two level overlapping schwarz preconditioner consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains this is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces related to the lack of continuity of our approximation spaces some interesting new features arise in our generalization which have no analog in the conforming case we prove an upper bound for the number of iterations obtained by using this preconditioner with gmres which is independent of the number of degrees of freedom of the original problem and the number of subdomains the performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions

an optimization problem with volume constraint we consider the optimization problem of minimizing dirichlet integral int omega nabla u 2 with volume constraint on the set u 0 we use a penalization method which for small values of the penalization parameter leads to a solution of the original problem

new real variable characterizations of musielak orlicz hardy spaces abstract let r n 0 0 be such that x is an orlicz function and t is a muckenhoupt a r n weight the musielak orlicz hardy space h r n is defined to be the space of all f s r n such that the grand maximal function f belongs to the musielak orlicz space l r n luong dang ky established its atomic characterization in this paper the authors establish some new real variable characterizations of h r n in terms of the vertical or the non tangential maximal functions or the littlewood paley g function or g function via first establishing a musielak orlicz fefferman stein vector valued inequality moreover the range of in the g function characterization of h r n coincides with the known best results when h r n is the classical hardy space h p r n with p 0 1 or its weighted variant