landau de gennes theory of isotropic nematic smectic liquid crystal transitions we propose a landau de gennes variational theory fit to simultaneously describe isotropic nematic smectic a and smectic c phases of a liquid crystal the unified description allows us to deal with systems in which one or all of the order parameters develop because of the influence of defects external fields and or boundary conditions we derive the complete phase diagram of the system that is we characterize how the homogeneous minimizers depend on the value of the constitutive parameters the coupling between the nematic order tensor and the complex smectic order parameter generates an elastic potential which is a nonconvex function of the gradient of the smectic order parameter this lack of convexity yields in turn a loss of regularity of the free energy minimizers we then consider the effect on an infinitesimal second order regularization term in the free energy functional which fixes the optimal number of defects in the singular configurations

regularity of subelliptic monge ampere equations in the plane we establish a c regularity result for c 1 1 solutions of degenerate monge ampere equation in r 2 under the assumption that the trace of the hessian is bounded from below

a relaxation model for numerical approximations of the multidimensional pressureless gas dynamics system abstract relaxation models for the pressureless gas dynamics pgd equations attempt to satisfy the strictly hyperbolic conservation law in order to employ the well posed approximated riemann solvers in this study a new type of relaxation model is proposed to resolve two shortcomings of the existing relaxation models the constant propagation speed of sound and the collapse of delta shock waves in multidimensional problems the proposed model seeks a strictly hyperbolic system of equations without any special consideration for the proper values of the propagation speed of sound numerical tests showed that the proposed model can accurately describe the behavior of the pgd equations in particular the occurrence of delta shock waves and vacuum states in a multidimensional problem

on the dirichlet problem for degenerate monge ampere equations we prove global second derivative estimates for the dirichlet problem for degenerate monge ampere equations which yield corresponding existence and regularity results our conditions are essentially optimal and our techniques while drawing on previous investigations are substantially new

minimal coupling model of the biaxial nematic phase a minimal coupling model exhibiting isotropic uniaxial and biaxial nematic phases is analyzed in detail and its relation to existing models known in the literature is clarified its intrinsic symmetry properties are exploited to restrict the relevant ranges of coupling constants further on properties of the model are thoroughly investigated by means of bifurcation theory as proposed by kayser and raveche phys rev a 17 2067 1978 and mulder phys rev a 39 360 1989 as a first step toward this goal the bifurcation theory is applied to a general formulation of density functional theory in terms of direct correlation functions on a general formal level the theory is then analyzed to show that the bifurcation points from the reference high symmetry equilibrium phase to a low symmetry structure depend only on the properties of the one particle distribution function and the direct pair correlation function of the reference phase the character of the bifurcation whether spinodal critical tricritical isolated landau point etc depends in addition on a few higher order direct correlation functions explicit analytical results are derived for the case when only the leading l 2 terms of the potential mean field analysis or of the direct pair correlation function expansion in the symmetry adapted basis are retained formulas are compared with the numerical calculations for the mean field momentum l 2 potential model in which case they are exact in particular bifurcations from the isotropic and uniaxial nematic to the biaxial nematic phases are discussed the possibility of the recently reported nematic uniaxial nematic biaxial tricritical point a m sonnet e g virga and g e durand phys rev e 67 061701 2003 is analyzed as well

regularity of subelliptic monge ampere equations in dimension n 3 for k x 2m that can be written as a sum of squares of smooth functions we prove that a c2 convex solution u to a subelliptic monge ampere equation detd2u k x u du is itself smooth if the elementary n 1 st symmetric curvature kn 1 of u is positive the case m 2 uses an additional nondegeneracy condition on the sum of squares our proof uses the partial legendre transform calabi s identity for uij ij where is the square of the third order derivatives of u the campanato method xu and zuily use to obtain regularity for systems of sums of squares of hormander vector fields and our earlier work using guan s subelliptic methods

degenerate sobolev spaces and regularity of subelliptic equations we develop a notion of degenerate sobolev spaces naturally associated with nonnegative quadratic forms that arise from a large class of linear subelliptic equations with rough coefficients these sobolev spaces allow us to make the widest possible definition of a weak solution that leads to local holder continuity of solutions extending our results in an earlier work where we studied regularity of classical weak solutions in cases when the quadratic forms arise from collections of rough vector fields we study containment relations between the degenerate sobolev spaces and the corresponding spaces defined in terms of weak derivatives relative to the vector fields

regularity of a class of quasilinear degenerate elliptic equations which is in a form of eq 1 in general convex solution u of 2 is at most in c 1 if k is only assumed to be smooth and nonnegative e g see ckns and g1 in g1 a sharp sufficient condition was introduced to establish c 1 regularity of 2 a basic question left is when is a solution u of 2 smooth or better than c we remark that if k is positive and smooth by elliptic theory u is smooth one may expect u to be smooth if the decay of k near its null set is under control say of finite type unfortunately this is not true the function u x x 2 2 n provides an example that the solution u of eq 2 article no ai971677

weighted sobolev poincare inequalities and pointwise estimates for a class of degenerate elliptic equations in this paper we prove a sobolev poincare inequality for a class of function spaces associated with some degenerate elliptic equations these estimates provide us with the basic tool to prove an invariant harnack inequality for weak positive solutions in addition holder regularity of the weak solutions follows in a standard way let y z 7 ai aij9 a be a second order degenerate elliptic operator in divergence form with measurable coefficients in this paper we shall obtain pointwise estimates for the weak solutions of su 0 h61der continuity of the weak solutions and harnack inequality for nonnegative solutions let us recall that the original results for elliptic operators were obtained by de giorgi nash and moser an extensive bibliography about the degenerate case can be found in fli fl2 fs to introduce the results of the present paper let us recall some recent results in fl1 fl2 a suitable metric d is associated with the differential operator y in such a way that we obtain a new geometry which is natural for the degenerate operator as the euclidean geometry is natural for the laplace operator or more precisely as a suitable riemannian geometry is natural for a secondorder elliptic operator in the smooth case this idea is contained in many papers we refer to fp nsw the basic results in fl 1 fl2 are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the hormander hypoellipticity condition for sum of squares operators we note that the same idea is used in nsw s j v to obtain pointwise estimates for sum of squares operators on the other hand a different class of degenerate elliptic operators is considered in fks instead of a geometrical degeneracy a measure degeneracy is allowed a typical example of this class is given by yu div wo x vu where cl is a weight function belonging to the a2 class of muckenhoupt unified results for a class containing both the operators in fl 1 and in fks have received by the editors august 1 1989 1980 mathematics subject classification 1985 revision primary 46e35 35j70 partially supported by g n a f a of c n r and m u r s t italy 1991 american mathematical society 0002 9947 91 1 00 25 per page

evans krylov estimates for a nonconvex monge ampere equation we establish evans krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial legendre transformations the equations under consideration arise in part from the study of the pluriclosed flow introduced by streets and tian int math res not 16 3101 3133 2010

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

minimum principle for indefinite mean field free energies the extension of bogoliubov jr s minimax principle to biaxial nematic liquid crystals has recently made it possible to derive the universal mean field phase diagram for all quadrupolar interactions between nematogenic biaxial molecules including those rendering the mean field free energy fancyscript f 0 indefinite to justify this extension so far largely based on heuristic arguments we prove here a minimum principle that also applies when fancyscript f 0 is indefinite bogoliubov jr s minimax principle then emerges as a characterization of this minimum principle the theory presented here is more general than its application to biaxial nematic liquid crystals could reveal it essentially builds upon a concavity property of fancyscript f 0 in the order tensor associated with the repulsive component of the molecular interaction

universal mean field phase diagram for biaxial nematics obtained from a minimax principle we study a class of quadratic hamiltonians which describe both fully attractive and partly repulsive molecular interactions characteristic of biaxial liquid crystal molecules to treat the partly repulsive interactions we establish a minimax principle for the associated mean field free energy we show that the phase diagram described by sonnet et al phys rev e 67 061701 2003 is universal our predictions are in good agreement with the recent observations on both v shaped and tetrapodal molecules

renormalisation group study of a lattice model of the isotropic nematic transition the isotropic nematic transition is studied using the lebwohl lasher lattice model of liquid crystal behaviour the details of the transition are investigated using the migdal kadanoff potential moving version of the real space renormalisation group the method gives good results when applied to the closely related heisenberg model of ferromagnetism a second order transition is predicted in disagreement with mean field and some computer simulation results

computer simulation study of a simple tetrahedratic mesogenic lattice model over the last 12 years the possible existence of a tetrahedratic mesophase involving a third rank orientational order parameter and no positional order has been addressed theoretically and predicted in some cases no experimental realizations of a purely tetrahedratic phase are known at the time being but various pieces of evidence suggest that interactions of tetrahedral symmetry do play a significant role in the macroscopic properties of mesophases resulting from banana shaped bent core mesogens we address a very simple tetrahedratic mesogenic lattice model involving continuous interactions we consider particles possessing t d symmetry whose centers of mass are associated with a three dimensional simple cubic lattice the pair potential is taken to be isotropic in orientation space and restricted to nearest neighboring sites we let the two orthonormal triads u alpha alpha 1 2 3 and v gamma gamma 1 2 3 define the orientations of a pair of interacting particles we let the unit vectors u alpha be combined to yield four unit vectors e j j 1 2 3 4 arranged in a tetrahedral fashion we let the unit vectors v gamma be similarly combined to yield the four unit vectors f k k 1 2 3 4 and finally we let h jk e j f k the interaction model studied here is defined by the simplest nontrivial cubic polynomial in the scalar products h jk consistent with the assumed symmetry and favoring orientational order it is so to speak the tetrahedratic counterpart of the lebwohl lasher model for uniaxial nematics the model was investigated by molecular field mf theory and monte carlo simulations mf theory predicts a low temperature tetrahedrically ordered phase undergoing a second order transition to the isotropic phase at higher temperature on the other hand available theoretical treatments point to the transition being driven first order by thermal fluctuations simulations showed evidence of a first order transition

macroscopic and molecular symmetries of unconventional nematic phases the effects of the molecular symmetry on macroscopic properties of conventional and unconventional nematic phases are investigated theoretically these effects concern 1 the form of the molecular orientation fluctuations 2 the thermodynamic behavior and the list of stable ordered phases 3 the effective symmetry of the molecules in the ordered phases the order parameter consists of tensors forming the main harmonics of the fluctuation distribution in conventional models valid for uniaxial molecules a single tensor is sufficient while unconventional models valid for less symmetric molecules need several tensors with the same rank we analyze the qualitative differences arising when the number of equivalent tensors varies we show how to work out complete models in the general case and to calculate the sequences of stable phases and the corresponding effective molecular symmetries this yields for each molecular group and each tensor rank a complete classification and a deep insight into the structure of thermotropic nematics this work generalizes the approach we have applied to polar nematics recently observed in polyester compounds and to unconventional uniaxial and biaxial phases of bent core materials

mesogenic lattice models with partly antinematic interactions producing uniaxial nematic phases the present paper considers nematogenic lattice models involving particles of d 2h symmetry whose centers of mass are associated with a three dimensional simple cubic lattice the pair potential is isotropic in orientation space and restricted to nearest neighbors let two orthonormal triads define orientations of a pair of interacting particles the simplest potential models proposed in the literature can be reduced to a linear combination involving the squares of the scalar products between corresponding unit vectors only and depending on three parameters by now various sets of potential parameters have been proposed and studied in the literature some of which capable of producing biaxial orientational order at sufficiently low temperature on the other hand in experimental terms mesogenic biaxial molecules mostly produce uniaxial mesophases thus we address here two very simple cases involving a nematic calamitic term as well as one model p0m or two model ppm antinematic ones whose coefficients are set equal in magnitude when only one antinematic coefficient is used the third one is set to zero the calamitic term favors the alignment of two corresponding molecular axes whereas antinematic terms or geometric constraints tend to keep two other pairs of axes mutually orthogonal the models were investigated by molecular field treatments and monte carlo simulation and found to predict a first or second order transitions between uniaxial nematic and isotropic phases the molecular field treatments yielded results in reasonable agreement with simulation

extension and generalization of the gay berne potential in this paper we report a generalized form for the range parameter governing the pair interaction between soft ellipsoidal particles for nonequivalent uniaxial particles we extend the berne pechukas gaussian overlap formalism to obtain an explicit expression for this range parameter we confirm that this result is identical to that given by an approach that is not widely recognized based on an approximation to the perram wertheim hard ellipsoid contact function we further illustrate the power of the latter route by using it to write down the range parameter for the interaction between two nonequivalent biaxial particles an explicit interaction potential for nonequivalent uniaxial particles is obtained by importing the uniaxial range parameter result into the standard gay berne form a parametrization of this potential is investigated for a rod disk interaction

numerical simulations of the euler system with congestion constraint in this paper we study the numerical simulations for euler system with maximal density constraint this model is developed in 6 17 with the constraint introduced into the system by a singular pressure law which causes the transition of different asymptotic dynamics between different regions to overcome these difficulties we adapt and implement two asymptotic preserving ap schemes originally designed for low mach number limit 16 18 to our model these schemes work for the different dynamics and capture the transitions well several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes

the mixed two qubit system and the structure of its ring of local invariants the local invariants of a mixed two qubit system are discussed these invariants are polynomials in the elements of the corresponding density matrix they are counted by means of group theoretic branching rules which relate this problem to one arising in spin isospin nuclear shell models the corresponding molien series and a refinement in the form of a four parameter generating function are determined a graphical approach is then used to construct explicitly a fundamental set of 21 invariants relations between them are found in the form of syzygies by using these the structure of the ring of local invariants is determined and complete sets of primary and secondary invariants are identified there are 10 of the former and 15 of the latter

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

traceless cartesian tensor forms for spherical harmonic functions new theorems and applications to electrostatics of dielectric media a solid spherical harmonic of degree n at a point r takes the tensor contraction form a n n rn where a n is a totally symmetric and traceless nth rank cartesian tensor the utility of this form rests on the properties of a detracer operator fn which transforms any totally symmetric nth rank tensor to a totally traceless form in particular the components of the tensor fnrn which is related to the nth gradient of r 1 are spherical harmonics with properties analogous to those of the tesseral harmonics ynm theta phi including adherence to an addition theorem and an unsold theorem these properties lead to new formulae for the legendre polynomials and their derivatives in terms of cartesian tensors the traceless cartesian tensor forms are used to treat problems in electrostatics of a dielectric medium requiring spherical harmonic expansions of the potential these include the potentials arising from an arbitrary charge distribution in a spherical dielectric cavity the reaction field gradients in the cavity the response of a dielectric sphere embedded in a dielectric medium to an arbitrary external field and the gradients of the lorentz internal field in a homogeneous dielectric expressions are obtained for nth order field gradients and induced multipole moments in cartesian tensor form

bridging the gap between phenomenology and microscopic theory asymptotes in nematic colloids the ornstein zernike equation is applied to nematic colloids with up down symmetry to determine how the electrostatic analogy and other phenomenological results appear in molecular theory in contrast to phenomenological approaches the molecular theory does not assume particular boundary conditions anchoring at colloidal surfaces for our molecular parameters the resulting anchoring appears to be realistic neither rigid nor infinitely weak for this case the effective force between a colloidal pair at large separation remains essentially constant over the entire region of nematic stability we show that a simple van der waals approximation gives a potential of mean force that in some important aspects is similar to the phenomenological results obtained in the limit of weak anchoring at large separations the potential varies as sigma8 where sigma is the colloidal diameter in contrast the more sophisticated mean spherical approximation yields a sigma6 dependence consistent with phenomenological calculations employing rigid boundary conditions we show that taking proper account of the correlation or magnetic coherence length xi inherent in the nematic sample is essential in an analysis of the sigma dependence at infinite xi the leading sigma dependence is sigma6 but this shifts to sigma8 when xi is finite the correlation length also influences the orientational behavior of the effective interaction the so called quadrupole interaction that determines the long range behavior at infinite xi transforms into a superposition of screened multipoles when xi is finite the basic approach employed in this paper can be readily applied to a broad range of physically interesting systems these include patterned and nonspherical colloids colloids trapped at interfaces and nematic fluids in confined geometries such as droplets

biaxial and uniaxial phases produced by partly repulsive mesogenic models involving d2h molecular symmetries the present paper considers biaxial nematogenic lattice models involving particles of d 2h symmetry whose centers of mass are associated with a three dimensional simple cubic lattice the pair potential is isotropic in orientation space and restricted to nearest neighbors let two orthonormal triads define orientations of a pair of interacting particles the investigated potential models are quadratic with respect to the nine scalar products between the two sets of unit vectors actually based on available geometric identities these expressions can be reduced to diagonal form containing only the scalar products between corresponding unit vectors and depending on three parameters over the years this comparatively simple functional form has also proven to be rather versatile by now various sets of potential parameters capable of producing mesogenic behavior of some kind have been proposed and studied in the literature a new and simplified form was recently proposed and investigated by sonnet virga durand and de matteis a m sonnet e g virga and g e durand phys rev e 67 061701 2003 g de matteis and e g virga phys rev e 71 061703 2005 and is known to support a biaxial phase at sufficiently low temperature following the idea of the above authors we have studied a more extended range of parameters including cases where biaxiality cannot be sustained in the pair ground state in cases where a biaxial phase survives an appropriate mean field analysis may predict the existence of a direct second order transition to the isotropic phase as well as a second order sequence isotropic to uniaxial to biaxial a second order phase transition is also predicted which involves isotropic and uniaxial phases only monte carlo simulations have been carried out as well for a few points in the parameter space and found to produce results which partly confirm mean field predictions

an efficient numerical scheme for the euler equations a finite difference scheme based on flux difference splitting is presented for the solution of the euler equations for the compressible flow of an ideal gas a linearised riemann problem is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem an average of the flow variables across the interface between cells is required and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging this is in contrast to the usual square root averages found in this type of riemann solver where the computational expense can be prohibitive the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids nonphysical spurious oscillations the scheme is applied to a shock tube problem and a blast wave problem each approximate solution compares well with those given by other schemes and for the shock tube problem is in agreement with the exact solution

mathematical modeling of wide range compressible two phase flows abstract the paper considers flow modeling of two phase heterogeneous medium each phase of the medium is considered as continuum which is described by the compressible euler equations the phases are separated by the contact surface interface and are not mixed on the molecular level examples of such medium are mixtures of solid particles and gas in dense or dilute concentration of particles liquids with small bubbles solid porous materials filled with gas or liquid the phase can be of connected structure dense particles porous solid gas between dilute particles etc or of non connected structure separated inclusions as gas bubbles dilute particles closed pores in solid etc the connectivity of the phase is closely related to the propagation of acoustic perturbations an attempt was made to consider all the cases of the phase connectivity in the framework of a unique approach the paper presents an approach that couples the models designed for different cases of phase connectivity to the generalized hyperbolic and thermodynamically consistent form the proposed model is applicable for simulating flows with change of the phase connectivity e g dense to dilute two phase flows the model is verified on several problems of gas solid granular medium flows in numerical simulations the godunov method with the hllem flux approximation on arbitrary moving euler grids is used

molecular field theory approach to the landau theory of liquid crystals uniaxial and biaxial nematics nematic liquid crystal phase diagrams in temperature biaxiality space are usually complex we construct a landau theory based on the analogous molecular field theory for orthorhombic biaxial nematic fluids a formal procedure yields coefficients some of which unusually can be tensorial in this landau expansion correctly predicts the complete set of invariants formed from the ordering tensors and avoids ad hoc parametrization of the molecular biaxiality by regularizing the landau expansion to avoid unwanted order parameter divergences at low temperatures we predict phase behavior over the whole range of biaxiality the resulting phase diagrams have the same topology as those of molecular field theory

molien series and low degree invariants for a natural action of rm so 3 wr rm z _ 2 we investigate the invariants of the 25 dimensional real representation of the group so 3 wr2 given by the left and right actions of so 3 on 5 5 matrices together with matrix transposition the action on column vectors is the irreducible five dimensional representation of so 3 the 25 dimensional representation arises naturally in the study of nematic liquid crystals where the second rank orientational order parameters of a molecule are represented by a symmetric 3 3 traceless symmetric matrix and where a rigid rotation in r3 induces a linear transformation of this space of matrices the entropy contribution to a free energy density function in this context turns out to have so 3 wrz2 symmetry although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants we are able to calculate as a rational function the molien series that gives the number of linearly independent invariants at each homogeneous degree the form of the function indicates a basis of 19 primary invariants and suggests there are n 1453926048 linearly independent secondary invariants we prove that their number is an integer multiple of n 4 the algebraic structure of invariants up to degree 4 is investigated in detail

hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematic liquid crystals the paradigm of spontaneous symmetry breaking encompasses the breaking of the rotational symmetries text o 3 of isotropic space to a discrete subgroup i e a three dimensional point group the subgroups form a rich hierarchy and allow for many different phases of matter with orientational order such spontaneous symmetry breaking occurs in nematic liquid crystals and a highlight of such anisotropic liquids is the uniaxial and biaxial nematics generalizing the familiar uniaxial and biaxial nematics to phases characterized by an arbitrary point group symmetry referred to as generalized nematics leads to a large hierarchy of phases and possible orientational phase transitions we discuss how a particular class of nematic phase transitions related to axial point groups can be efficiently captured within a recently proposed gauge theoretical formulation of generalized nematics k liu j nissinen r j slager k wu and j zaanen phys rev x 6 041025 2016 these transitions can be introduced in the model by considering anisotropic couplings that do not break any additional symmetries by and large this generalizes the well known uniaxial biaxial nematic phase transition to any arbitrary axial point group in three dimensions we find in particular that the generalized axial transitions are distinguished by two types of phase diagrams with intermediate vestigial orientational phases and that the window of the vestigial phase is intimately related to the amount of symmetry of the defining point group due to inherently growing fluctuations of the order parameter this might explain the stability of the observed uniaxial biaxial phases as compared to the yet to be observed other possible forms of generalized nematic order with higher point group symmetries

numerical approximations of pressureless and isothermal gas dynamics we study several schemes of first or second order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations the pressureless gas system is weakly hyperbolic giving rise to the formation of density concentrations known as delta shocks for the isothermal gas system the infinite speed of expansion into vacuum leads to zero timestep in the godunov method based on exact riemann solver the schemes we consider are able to bypass these difficulties they are proved to satisfy positiveness of density and discrete entropy inequalities to capture the delta shocks and to treat data with vacuum

a catastrophe theoretic approach to tricritical points with application to liquid crystals a criterion to locate tricritical points in phase diagrams is proposed the criterion is formulated in the framework of elementary catastrophe theory and encompasses all the existing criteria in that it applies to systems described by a generally nonsymmetric free energy which can depend on one or more order parameters we show that tricritical points are necessarily given by free energies which are not 4 determined an application to smectic c liquid crystals is briefly discussed

classification of point group symmetric orientational ordering tensors the concept of symmetry breaking has been a propelling force in understanding phases of matter while rotational symmetry breaking is one of the most prevalent examples the rich landscape of orientational orders breaking the rotational symmetries of isotropic space i e o 3 to a three dimensional point group remain largely unexplored apart from simple examples such as ferromagnetic or uniaxial nematic ordering here we provide an explicit construction utilizing a recently introduced gauge theoretical framework to address the three dimensional point group symmetric orientational orders on a general footing this unified approach allows us to enlist order parameter tensors for all three dimensional point groups by construction these tensor order parameters are the minimal set of simplest tensors allowed by the symmetries that uniquely characterize the orientational order we explicitly give these for the point groups c n d n t o i ensuremath subset text so 3 and c nv s 2n c nh d nh d nd t h t d o h i h ensuremath subset text o 3 for n 2n ensuremath in 1 2 3 4 6 ensuremath infty this central result may be perceived as a road map for identifying exotic orientational orders that may become more and more in reach in view of rapid experimental progress in e g nanocolloidal systems and novel magnets

global bifurcation and convex solutions for the monge ampere equation abstract we study the following monge ampere equation det d 2 u 1 n f u in u 0 on by bifurcation technique we establish some results about the existence nonexistence uniqueness and multiplicity of convex solutions for this problem our results generalize and improve many important known results from previous literature

on the cartesian definition of orientational order parameters orientational order parameters can be effectively and economically defined using spherical tensors however their definition in terms of cartesian tensors can sometimes provide a clearer physical intuition we show that it is possible to build a fully cartesian theory of the orientational order parameters which is consistent with the traditional spherical tensor approach the key idea is to build a generalised multi pole expansion of the orientational probability distribution function in terms of outer products of rotation matrices furthermore we show that the saupe ordering super matrix as found for example in the text by de gennes and prost the physics of liquid crystals 2nd ed oxford university press oxford uk 1995 and which is used to define the cartesian second rank orientational order parameters is not consistent with its spherical tensor counterpart we then propose a symmetric version of the saupe super matrix which is fully consistent with the spherical tensor definition the proposed definition is important for a correct description of liquid crystal materials composed of low symmetry molecules

a higher dimensional partial legendre transform and regularity of degenerate monge ampere equations in dimension n 3 we define a generalization of the classical two dimensional partial legendre transform that reduces interior regularity of the generalized monge ampere equation detd2u k x u du to regularity of a divergence form quasilinear system of special form this is then used to obtain smoothness of c2 1 solutions having n 1 nonvanishing principal curvatures to certain subelliptic monge ampere equations in dimension n 3 a corollary is that if k 0 vanishes only at nondegenerate critical points then a c2 1 convex solution u is smooth if and only if the symmetric function of degree n 1 of the principal curvatures of u is positive and moreover u fails to be c3 1 2n when not smooth

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

molecular field theory of nematics density functional approach i bulk effects the molecular field theory of nematics is rephrased in density functional language the free energy in the nematic state is expressed as a functional expansion around the isotropic state in powers of the nematic order parameters pl and the fractional density change eta at the isotropic nematic transition the crucial quantities in the expansion are spherical harmonics of the direct correlation functions in the isotropic state sensible approximations to these direct correlation functions lead to other well known formulations of nematic field theory an exact in the authors approximation result for eta is obtained in terms of the potential parameters ul and the order parameters pl at the phase transition the authors discuss the relationship of their approach to other microscopic theories of nematics the theory can be extended to discuss inhomogeneous systems

gevrey regularity of subelliptic monge ampere equations in the plane in this paper we establish the gevrey regularity of solutions for a class of degenerate monge ampere equations in the plane under the assumptions that one principal entry of the hessian is strictly positive and the coefficient of the equation is degenerate with appropriately finite type degeneracy we prove that the solution of the degenerate monge ampere equation will be smooth in gevrey classes