non existence of extremals for the adimurthi druet inequality abstract the adimurthi druet 1 inequality is an improvement of the standard moser trudinger inequality by adding a l 2 type perturbation quantified by 0 1 where 1 is the first dirichlet eigenvalue of on a smooth bounded domain it is known 3 10 14 19 that this inequality admits extremal functions when the perturbation parameter is small by contrast we prove here that the adimurthi druet inequality does not admit any extremal when the perturbation parameter approaches 1 our result is based on sharp expansions of the dirichlet energy for blowing sequences of solutions of the corresponding euler lagrange equation which take into account the fact that the problem becomes singular as 1

on a liouville type equation with sign changing weight in this paper we study the existence nonexistence and multiplicity of non negative solutions for the family of problems u a x e f x u u h 0 where is a bounded domain in r2 and 0 is a parameter the coefficient a x is allowed to change sign the techniques used in the proofs are a combination of upper and lower solutions the trudingermoser inequality and variational methods note that when f x u 0 the equation is of liouville type

a nonlinear elliptic equation arising from gauge field theory and cosmology we study radially symmetric solutions of a nonlinear elliptic partial differential equation in r 2 with critical sobolev growth i e the nonlinearity is of exponential type this problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology our results lead to many interesting implications for the physical problems considered for example for the self dual chern simons theory we are able to conclude that the electric charge magnetic flux or energy of a non topological n vortex solution may assume any prescribed value above an explicit lower bound for the einstein matter gauge equations we find a necessary and sufficient condition for the existence of a self dual cosmic string solution such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant

an asymptotic preserving all speed scheme for the euler and navier stokes equations we present an asymptotic preserving all speed scheme for the simulation of compressible flows valid at all mach numbers ranging from very small to order unity the scheme is based on a semi implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly this discretization which is the key to the asymptotic preserving property provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime the divergence free condition on the velocity in the incompressible regime is respected and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations the implicit treatment of the acoustic part allows the time step to be independent of the mach number the scheme is conservative and applies to steady or unsteady flows and to general equations of state one and two dimensional numerical results provide a validation of the asymptotic preserving all speed properties

on the existence and profile of nodal solutions for a two dimensional elliptic problem with large exponent in nonlinearity we study the existence of nodal solutions to the boundary value problem u u p 1u in a bounded smooth domain in ir with homogeneous dirichlet boundary condition when p is a large exponent we prove that for p large enough there exist at least two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of

well balanced schemes for the euler equations with gravitation well balanced high order finite volume schemes are designed to approximate the euler equations with gravitation the schemes preserve discrete equilibria corresponding to a large class of physically stable hydrostatic steady states based on a novel local hydrostatic reconstruction the derived schemes are well balanced for any consistent numerical flux for the euler equations the form of the hydrostatic reconstruction is both very simple and computationally efficient as it requires no analytical or numerical integration moreover as required by many interesting astrophysical scenarios the schemes are applicable beyond the ideal gas law both first and second order accurate versions of the schemes and their extension to multi dimensional equilibria are presented several numerical experiments demonstrating the superior performance of the well balanced schemes as compared to standard finite volume schemes are also presented

an all speed second order well balanced imex relaxation scheme for the euler equations with gravity abstract we present an implicit explicit well balanced finite volume scheme for the euler equations with a gravitational source term which is able to deal also with low mach flows to visualize the different scales we use the non dimensionalized equations on which we apply a pressure splitting and a suliciu relaxation on the resulting model we apply a splitting of the flux into a linear implicit and an non linear explicit part that leads to a scale independent time step the explicit step consists of a godunov type method based on an approximative riemann solver where the source term is included in the flux formulation we develop the method for a first order scheme and give an extension to second order both schemes are designed to be well balanced preserve the positivity of density and internal energy and have a scale independent diffusion we give the low mach limit equations for well prepared data and show that the scheme is asymptotic preserving these properties are numerically validated by various test cases

global compactness properties of semilinear elliptic equations with critical exponential growth abstract sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the sobolev h 1 topology or concentrate at isolated points of the domain for energies allowing at most single point blow up we establish a universal blow up pattern near the concentration point and uniquely characterize the blow up energy in terms of a geometric limiting problem

semi implicit extension of a godunov type scheme based on low mach number asymptotics i one dimensional flow a single scale multiple space scale asymptotic analysis provides detailed insight into the low mach number limit behavior of solutions of the compressible euler equations we use the asymptotics as a guideline for developing a low mach number extension of an explicit higher order shock capturing scheme this semi implicit scheme involves multiple pressure variables large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step this equation is a pressure correction equation for incompressible flows when the mach number is zero in the low mach number limit the time step is restricted by a courant number based essentially on the maximum flow velocity for moderate and large mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm

a weakly asymptotic preserving low mach number scheme for the euler equations of gas dynamics we propose a low mach number godunov type finite volume scheme for the numerical solution of the compressible euler equations of gas dynamics the scheme combines klein s non stiff stiff decomposition of the fluxes j comput phys 121 1995 pp 213 237 with an explicit implicit time discretization f cordier p degond and a kumbaro j comput phys 231 2012 pp 5685 5704 for the split fluxes this results in a scalar second order partial differential equation pde for the pressure which we solve by an iterative approximation due to our choice of a crucial reference pressure the stiff subsystem is hyperbolic and the second order pde for the pressure is elliptic the scheme is also uniformly asymptotically consistent numerical experiments show that the scheme needs to be stabilized for low mach numbers unfortunately this affects the asymptotic consistency which becomes nonuniform in the mach number and requires an unduly fine grid in the small mach number limit on the other han

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

asymptotic preserving imex finite volume schemes for low mach number euler equations with gravitation in this paper we will present and analyze a new class of the imex finite volume schemes for the euler equations with a gravity source term we will in particular concentrate on a singular limit of weakly compressible flows when the mach number m1 in order to efficiently resolve slow dynamics we split the whole nonlinear system in a stiff linear part governing the acoustic and gravity waves and a non stiff nonlinear part that models nonlinear advection effects for time discretization we use a special class of the so called globally stiffly accurate imex schemes and approximate the stiff linear operator implicitly and the non stiff nonlinear operator explicitly for spatial discretization the finite volume approximation is used with the central and rusanov laxfriedrichs numerical fluxes for the linear and nonlinear subsystem respectively in the case of a constant background potential temperature we prove theoretically that the method is asymptotically consistent and asymptotically stable uniformly with respect to small mach number we also analyze experimentally convergence rates in the singular limit when the mach number tends to zero

the existence of chern simons vortices a new type of vortices called chern simons vortices is considered and the existence is established

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 all speed relaxation scheme for gases and compressible materials abstract we present a novel relaxation scheme for the simulation of compressible material flows at all speeds an eulerian model describing gases fluids and elastic solids with the same system of conservation laws is adopted the proposed numerical scheme is based on a fully implicit time discretization easily implemented thanks to the linearity of the transport operator in the relaxation system the spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in all different mach regimes the scheme is validated by simulating gas and liquid flows with different mach numbers the simulation of the deformation of compressible solids is also approached assessing the ability of the scheme in approximating material waves in hyperelastic media

topological multivortex solutions of the self dual maxwell chern simons higgs system abstract we study the existence and various behaviors of topological multivortex solutions of the relativistic self dual maxwell chern simons higgs system we first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional then by an iteration method we prove the existence of topological solutions satisfying some extra conditions which we call admissible solutions we establish asymptotic exponential decay estimates for these topological solutions we also investigate the limiting behavior of the admissible solutions as parameters in our system go to some limits for the abelian higgs limit we obtain strong convergence result while for the chern simons limit we only obtained that our admissible solutions weakly approximate one of the chern simons solutions

new solutions for trudinger moser critical equations in r2 abstract let be a bounded smooth domain in r 2 we consider critical points of the trudinger moser type functional j u 1 2 u 2 2 e u 2 in h 0 1 namely solutions of the boundary value problem u u e u 2 0 with homogeneous dirichlet boundary conditions where 0 is a small parameter given k 1 we find conditions under which there exists a solution u which blows up at exactly k points in as 0 and j u 2 k we find that at least one such solution always exists if k 2 and is not simply connected if has d 1 holes in addition d 1 bubbling solutions with k 1 exist these results are existence counterparts of one by druet in o druet multibump analysis in dimension 2 quantification of blow up levels duke math j 132 2 2006 217 269 which classifies asymptotic bounded energy levels of blow up solutions for a class of nonlinearities of critical exponential growth including this one as a prototype case

weak solutions to a parabolic elliptic system of chemotaxis abstract we study a parabolic elliptic system of partial differential equations which describes the chemotactic feature of slime molds it is known that the blowup solution forms singularities such as delta functions referred to as the collapses here we study the case that the domain is a flat torus and show that the post blowup continuation of the solution is possible only when those collapses are quantized with the mass 8

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

equivalent moser type inequalities in r2 and the zero mass case abstract we first investigate concentration and vanishing phenomena concerning moser type inequalities in the whole plane which involve complete and reduced sobolev norms in particular we show that the critical ruf inequality is equivalent to an improved version of the subcritical adachi tanaka inequality which we prove to be attained then we consider smooth compactly supported functions with respect to the dirichlet norm 2 and we prove an optimal lorentz zygmund type inequality with explicit extremals and from which can be derived classical inequalities in h 1 r 2 such as the adachi tanaka inequality and a version of ruf s inequality

concentrating solutions for an exponential nonlinearity with robin boundary condition abstract we study the following robin boundary value problem 0 1 u 2 u p 1 e u p 0 u 0 in u n u 0 on where is a bounded domain in r 2 with smooth boundary as 0 and simultaneously 0 p 2 by using a lyapunov schmidt reduction procedure we extend for the whole range of exponents 0 p 2 the results of davila and topp 2012 who constructed bubbling solutions to problem 0 1 for p 1

bubbling solutions for an exponential nonlinearity in r2 abstract we study the following boundary value problem 0 1 u u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 is a small parameter and 0 p 2 we construct bubbling solutions to problem 0 1 using a lyapunov schmidt reduction procedure

local exact boundary controllability of the boussinesq equation we study the local exact boundary controllability problem for the boussinesq equations that describe an incompressible fluid flow coupled to thermal dynamics the result that we get in this paper is as follows suppose that hat y t x is a given solution of the boussinesq equation where t in 0 t x in omega omega is a bounded domain with c infty boundary partial omega let y 0 x be a given initial condition and vert hat y 0 cdot y 0 vert epsilon where epsilon epsilon hat y is small enough then there exists boundary control u such that the solution y t x of the boussinesq equations satisfying y vert 0 t times partial omega u qquad y vert t 0 y 0 coincides with hat y t x at the instant t y t x equiv hat y t x

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

bubbling nodal solutions for a large perturbation of the moser trudinger equation on planar domains in this work we study the existence of nodal solutions for the problem begin aligned delta u lambda u e u 2 u p quad text in quad omega quad u 0 quad text on partial omega end aligned where omega subseteq mathbb r 2 is a bounded smooth domain and p rightarrow 1 if omega is a ball it is known that the case p 1 defines a critical threshold between the existence and the non existence of radially symmetric sign changing solutions in this work we construct a blowing up family of nodal solutions to such problem as p rightarrow 1 when omega is an arbitrary domain and lambda is small enough as far as we know this is the first construction of sign changing solutions for a moser trudinger critical equation on a non symmetric domain

blow up solutions for a liouville equation with henon term abstract we study the following boundary value problem 0 1 u x 2 u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 n 0 is a small parameter and 0 p 2 we construct the bubbling solution to problem 0 1 which blows up at the origin and at some other m points

mixed interior and boundary bubbling solutions for neumann problem in r 2 let be a bounded domain in r2 with smooth boundary we consider the following problem u u up 1eup u 0 in with neumann boundary condition u 0 on where 0 is a small parameter 0 p 2 and denotes the outer normal vector to we construct solutions of this problem with k interior bubbling points and l boundary bubbling points for any k 1 and l 1

phenomena of critical exponent in ℝ2 in this paper we make an attempt to explain the critical phenomena in r 2 we do this by exhibiting a class of functions having e u 2 growth and for which u u in b r u 0 u 0 on b r do not admit a solution when r is sufficiently small where b r denotes the ball of radius r in r 2

uniqueness of the topological multivortex solution in the self dual chern simons theory we establish a uniqueness result for the topological multivortex solution to the self dual equations of the abelian relativistic self dual chern simons higgs model we prove that the topological multivortex solution is unique if the chern simons coupling parameter 0 is sufficiently small we also establish a uniqueness result for 0 sufficiently large

multiple blow up solutions for an anisotropic 2 dimensional nonlinear neumann problem we study the following boundary value problem begin aligned left begin array ll div a x nabla u a x u 0 u 0 quad mathrm in omega frac partial u partial nu lambda u p 1 e u p quad mathrm on partial omega end array right end aligned 0 1 where omega is a bounded domain in mathbb r 2 with smooth boundary nu is the unit outer normal vector of partial omega and a x is a positive smooth function in overline omega lambda 0 is a small parameter and 0 p 2 we construct bubbling solutions to problem 0 1 which blow up at points near a local maximum of a x on partial omega

the existence of non topological multivortex solutions in the relativistic self dual chern simons theory we construct a general type of multivortex solutions of the self duality equations the bogomol nyi equations of 2 1 dimensional relativistic chern simons model with the non topological boundary condition near infinity for such construction we use a perturbation argument around the explicit solutions of the liouville equation

on oscillatory solutions to the complete euler system abstract the euler system in fluid dynamics is a model of a compressible inviscid fluid incorporating the three basic physical principles conservation of mass momentum and energy we show that the cauchy problem is basically ill posed for the l initial data in the class of weak entropy solutions as a consequence there are infinitely many measure valued solutions for a vast set of initial data finally using the concept of relative energy we discuss a singular limit problem for the measure valued solutions where the mach and froude number are proportional to a small parameter

existence and non existence of radial solutions for elliptic equations with critical exponent in ℝ2 where 2 2 n n 2 is the critical sobolev exponent for the embedding h 52 c l2 0 criticality and subcriticality play important rbles concerning the solvability of equation 1 1 using by now classical variational methods one sees that equation 1 1 is solvable i f f has subcritical growth and satisfies some additional hypotheses while for f u iuip u the well known pohoiaev identity shows that the solvability of equation 1 1 is lost if p reaches 2 and r is starshaped in their important paper 6 brezis nirenberg showed that for n e 4 solvability is regained if one adds a lower order linear term to the critical term more precisely they showed that for f r ar lr12 2r one has a denotes the 1st eigenvalue of a on ha r

multiple blow up solutions for an exponential nonlinearity with potential in r 2 abstract we study the following boundary value problem 0 1 u a x u p 1 e u p 0 u 0 in u 0 on where is a bounded domain in r 2 with smooth boundary 0 is a small parameter the function a x 0 is a smooth potential and the exponent p satisfies 0 p 2 we construct a family of solutions to problem 0 1 which blows up as 0 at some points of which stay outside the zero set of a x we relate the number of possible blow up points with the zero set of a x

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

second order implicit explicit total variation diminishing schemes for the euler system in the low mach regime abstract in this work we consider the development of implicit explicit total variation diminishing tvd methods also termed ssp strong stability preserving for the compressible isentropic euler system in the low mach number regime the proposed scheme is asymptotically stable with a cfl condition independent of the mach number in addition it degenerates in the low mach number regime to a consistent discretization of the incompressible system since it has been proved that implicit schemes of order higher than one cannot be tvd ssp 30 we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing tvd methods in space for this particular class of schemes the tvd property is first proved on a linear model advection equation and then extended to the isentropic euler case the result is a method which interpolates from the first to the second order both in space and time it preserves the monotonicity of the solution and is highly accurate for all choices of the mach number moreover the time step is only restricted by the non stiff part of the system we finally show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties

glueing a peak to a non zero limiting profile for a critical moser trudinger equation abstract druet 6 proved that if f is a sequence of moser trudinger type nonlinearities with critical growth and if u solves 1 and converges weakly in h 0 1 to some u then the dirichlet energy is quantified namely there exists an integer n 0 such that the energy of u converges to 4 n plus the dirichlet energy of u as a crucial step to get the general existence results of 7 it was more recently proved in 8 that for a specific class of nonlinearities see 2 the loss of compactness i e n 0 implies that u 0 in contrast we prove here that there exist sequences f of moser trudinger type nonlinearities which admit a noncompact sequence u of solutions of 1 having a nontrivial weak limit

concentrating solutions in a two dimensional elliptic problem with exponential neumann data abstract we consider the elliptic equation u u 0 in a bounded smooth domain in r 2 subject to the nonlinear neumann boundary condition u e u here 0 is a small parameter we prove that any family of solutions u for which e u is bounded develops up to subsequences a finite number m of peaks i in the sense that e u 2 k 1 m i as 0 reciprocally we establish that at least two such families indeed exist for any given m 1

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

the solution structure of the o 3 sigma model in a maxwell chern simons theory in this paper a system of semilinear elliptic equations arising from a relativistic self dual maxwell chern simons o 3 sigma model is considered we reveal the uniqueness aspect of the topological solutions for the model the uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case we locate each solution set denoted by a planar diagram