the evolution fractional p laplacian equation in rn fundamental solution and asymptotic behaviour abstract we consider the natural time dependent fractional p laplacian equation posed in the whole euclidean space with parameters p 2 and s 0 1 fractional exponent we show that the cauchy problem for data in the lebesgue l q spaces is well posed and show that the solutions form a family of non expansive semigroups with regularity and other interesting properties as main results we construct the self similar fundamental solution for every mass value m and prove that general finite mass solutions converge towards that fundamental solution having the same mass and convergence holds in all l q spaces a number of additional properties and estimates complete the picture

weak solutions of the porous medium equation in a cylinder we show that if d r n is a bounded domain with smooth boundary and u l m d e t u 0 solves u t u m m 1 in the sense of distributions on d 0 t and vanishes on d 0 t in a suitable weak sense then u is holder continuous in d 0 t

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

non negative solutions of the evolution p laplacian equation initial traces and cauchy problem when 1 p 2 from the introduction we continue here the investigation initiated earlier by us trans amer math soc 314 1989 no 1 187 224 concerning the solvability of the cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation 1 ut div du p 2 du 0 in st rn 0 t 0 2 both in terms of results and techniques the main difference stems from the fact that unlike solutions in the case p 2 solutions of 1 are not in general locally bounded

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

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

the porous medium equation on riemannian manifolds with negative curvature the superquadratic case we study the long time behaviour of nonnegative solutions of the porous medium equation posed on cartan hadamard manifolds having very large negative curvature more precisely when the sectional or ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole we find an unexpected separate variable behaviour that reminds one of dirichlet problems on bounded euclidean domains as a crucial step we prove existence of solutions to a related sublinear elliptic problem a result of independent interest uniqueness of solutions vanishing at infinity is also shown along with comparison principles both in the parabolic and in the elliptic case our results complete previous analyses of the porous medium equation flow on negatively curved riemannian manifolds which were carried out first for the hyperbolic space and then for general cartan hadamard manifolds with a negative curvature having at most quadratic growth we point out that no similar analysis seems to exist for the linear heat flow we also translate such results into some weighted porous medium equations in the euclidean space having special weights

phase diagram of onsager crosses onsager crosses are hard nonconvex bodies formed by rigidly connecting three elongated rods equally thick but not necessarily equally long to form perpendicular crosses we study the phase behavior of systems of such particles focusing on their ability to form spatially homogeneous orientationally ordered phases with a symmetry lower than that of the standard uniaxial nematic we treat these systems in the onsager second virial coefficient approximation we apply bifurcation analysis to build up a global picture of the phase diagram which is then refined using approximate numerical calculations finally we generalize the gaussian approximation for the nematic orientational distribution function to deal with the more ordered phases encountered here and compare with the results from the previous techniques to see whether it is feasible to reliably predict the phase diagrams from a computationally cheaper technique

fractional nonlinear degenerate diffusion equations on bounded domains part i existence uniqueness and upper bounds abstract we investigate quantitative properties of nonnegative solutions u t x 0 to the nonlinear fractional diffusion equation t u l f u 0 posed in a bounded domain x r n with appropriate homogeneous dirichlet boundary conditions as l we can use a quite general class of linear operators that includes the two most common versions of the fractional laplacian s 0 s 1 in a bounded domain with zero dirichlet boundary conditions but it also includes many other examples since our theory only needs some basic properties that are typical of linear heat semigroups the nonlinearity f is assumed to be increasing and is allowed to be degenerate the prototype is the power case f u u m 1 u with m 1 in this paper we propose a suitable class of solutions of the equation and cover the basic theory we prove existence uniqueness of such solutions and we establish upper bounds of two forms absolute bounds and smoothing effects as well as weighted l 1 estimates the class of solutions is very well suited for that work the standard laplacian case s 1 is included and the linear case m 1 can be recovered in the limit in a companion paper bonforte and vazquez in preparation we will complete the study with more advanced estimates like the upper and lower boundary behaviour and harnack inequalities for which the results of this paper are needed

large systems of interacting particles and the porous medium equation abstract we study the empirical processes of systems of interacting particles whose time evolution is given by coupled ordinary differential equations in particular as the population size tends to infinity we derive different versions of the porous medium equation as limit dynamics

global solutions of the fast diffusion equation with gradient absorption terms abstract we investigate the existence of nonnegative weak solutions to the problem u t u m u p in r n 0 with 1 2 n m 1 it will be proved that i when 1 p 2 if the initial datum u 0 d r n then there exists a solution ii when 1 p 2 m n n 1 if the initial datum u 0 x is a bounded and nonnegative measure then the solution exists iii when 2 m n n 1 p 2 if the initial datum is a dirac mass then the solution does not exist we also study the large time behavior of the l 1 norm of solutions for 1 p 2 m n n 1 and the large time behavior of t 1 u t e c t l for 2 m n n 1 p 2

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

uniqueness of solutions of degenerate diffusion equations with measures as initial conditions on considere le probleme de cauchy u t u m u p sur r n 0 t u x 0 x sur r n ou m 1 p 1 n 1 0 1 p m 2 n et une mesure finie

a fatou theorem for the solution of the heat equation at the corner points of a cylinder in this paper the author proves existence and uniqueness of the initial dirichlet problem for the heat equation in a cylindrical domain d 0 where d is a bounded smooth domain in r n with zero lateral values a unique representation of the strong solution is given in terms of measures on d and on d we also show that the strong solution u x t of the heat equation in a cylinder converges a e x 0 d 0 as x t converges to points on d 0 along certain nontangential paths

bmo spaces related to schrodinger operators with potentials satisfying a reverse holder inequality we identify the dual space of the hardy type space related to the time independent schrodinger operator v with v a potential satisfying a reverse holder inequality as a bmo type space we prove the boundedness in this space of the versions of some classical operators associated to hardy littlewood semigroup and poisson maximal functions square function fractional integral operator we also get a characterization of in terms of carlesson measures

hp theory on a smooth domain in rn and elliptic boundary value problems in this paper we answer the following two questions question 1 let be an appropriate domain inrn what are the possible natural notions of hp that generalize the usual hardy spaces hp rn we shall see that several versions are possible we concentrate on the two most relevant here which may be considered to be the largest version hpr and the smallest version hpz for the second question consider the dirichlet and neumann problems for the former consists in finding the u g which solves u in and u 0 the latter consists in finding the u g which solves u in and u n 0 question 2 in the context of the relevant hp can one obtain the boundedness of 2g xi xj and g xi xj

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

on the wave equation with a large rough potential we prove an optimal dispersive l decay estimate for a three dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular kato class the proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator

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

the asymptotic behavior of solutions for a class of doubly degenerate nonlinear parabolic equations abstract by using the morse interaction technique supposing that the uniqueness of the barenblatt type solution is true the paper studies the large time asymptotic behavior of solutions for the doubly degenerate parabolic equation u t div d u m p 2 d u m d u m p 1 u q with initial condition u x 0 u 0 x here the exponents m p p 1 q satisfy p p 1 p 1 q m p 1 1 p 1 m 1

nonnegative solutions of the initial dirichlet problem for generalized porous medium equations in cylinders in this paper we study nonnegative solutions u of a class of nonlinear evolutions au at a u whose best known example is the porous medium equation u u m 1 see 9 for a recent survey on these equations we study our solutions u subject to dirichlet boundary conditions in d x 0 t where d is a bounded smooth domain in rn i e uiadx 0 t 0 the corresponding theory for d rn was studied by the authors in 3 we find that there is one special solution x t the friendly giant which tends to infinity as t 1 0 and that all other solutions u x t are in one to one correspondence with suitable pairs of measures u on d and a ou ad we also show that any solution on d x 0 t extends to a solution in d x 0 oo and that the friendly giant f x t governs the asymptotic behavior as t t oo the nonlinearity is assumed to be continuous increasing with p 0 0 and also verifies the growth conditions

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

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

besov spaces with non doubling measures suppose that is a radon measure on r d which may be non doubling the only condition on is the growth condition namely there is a constant c 0 0 such that for all x supp and r 0 b x r c 0 r n where 0 0 is a real number which depends on the non doubling measure c 0 n and d the method used to define these spaces is new even for the classical case as applications the lifting properties of these spaces by using the riesz potential operators and the dual spaces are obtained

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

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

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

feynman kac functionals and positive solutions of 1 2δu qu 0 let d be a bounded c 2 domain in d and let q be a bounded borel function in d for x d and z d suppose x t under the law p x z is brownian motion in d starting at x and conditioned to converge to z let be the lifetime of x t we show that if the quantity e x z left exp left int limits 0 t q x s ds right right is finite for one x d and one z d then this quantity remains bounded as x varies over d and z varies over d this may be considered one quantitative expression of the qualitative statement that no matter where brownian motion in d eventually hits d it goes all over d before it gets there we apply this result to show that if the equation 1 2 u qu 0 admits a non negative solution in d which is strictly positive on a subset of d of positive harmonic measure then for any non negative bounded borel function f on d it admits a unique bounded solution u satisfying u f on d and this solution u is non negative

moser nash kernel estimates for degenerate parabolic equations abstract in this paper we deal with the cauchy problem associated to a class of nonlinear degenerate parabolic equations whose prototype is the parabolic p laplacian 2 p in his seminal paper after stating the harnack estimates moser proved almost optimal estimates for the parabolic kernel by using the so called harnack chain method in the linear case sharp estimates come by using nash s approach fabes and stroock proved that gaussian estimates are equivalent to a parabolic harnack inequality in this paper by using the dibenedetto degiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations to obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

asymptotic behavior of the solutions to the damped compressible euler equations with vacuum abstract the asymptotic behavior of solutions of the damped compressible euler equations is conjectured to obey to the famous porous media equations pmes the previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic in present paper we prove that the l entropy weak solution with vacuum obtained by the compensated compactness theory converges strongly in l loc p 1 p space to the unique similarity solution of the related pme as time goes to infinity

on the flows associated to selfadjoint operators on metric measure spaces let x be a metric space with a doubling measure satisfying mu b gtrsim r b n for any ball b with any radius r b 0 let l be a non negative selfadjoint operator on l 2 x we assume that e tl satisfies a gaussian upper bound and that the flow e itl satisfies a typical l 1 l infty dispersive estimate of the form begin aligned vert e itl vert l 1 rightarrow l infty lesssim t frac n 2 end aligned then we prove a similar l 1 l infty dispersive estimate for a general class of flows e it phi l with phi r of power type near 0 and near infty in the case of fractional powers phi l l nu nu in 0 1 we deduce dispersive estimates for e itl nu with data in sobolev besov or hardy spaces h p l with p in 0 1 associated to the operator l

besov class via heat semigroup on dirichlet spaces i sobolev type inequalities abstract we introduce heat semigroup based besov classes in the general framework of dirichlet spaces general properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained as a highlight of the paper we obtain a far reaching l p analogue p 1 of the sobolev inequality that was proved for p 2 by n varopoulos under the assumption of ultracontractivity for the heat semigroup the case p 1 is of special interest since it yields isoperimetric type inequalities

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

measure valued solutions and the phenomenon of blow down in logarithmic diffusion abstract the existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory a prominent example being the fundamental solutions of the linear theory in nonlinear equations the existence of such solutions may find special obstacles that can be either essential or otherwise they may lead to more general concepts of solution we give a particular review of results in the field of nonlinear diffusion as a new contribution we study in detail the case of logarithmic diffusion associated with ricci flow in the plane where we can prove existence of measure valued solutions the surprising thing is that these solutions become classical after a finite time in that general setting the standard concept of weak solution is not adequate but we can solve the initial value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions we prove that the problem is well posed the phenomenon of blow down in finite time is precisely described initial point masses diffuse into the medium and eventually disappear after a finite time t i m i 4

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

a remark on infinite initial values for quasilinear parabolic equations abstract we study the possibility of prescribing infinite initial values for solutions of the evolutionary p laplace equation in the fast diffusion case p 2 this expository note has been extracted from our previous work when infinite values are prescribed on the whole initial surface such solutions can exist only if the domain is a space time cylinder

holder regularity up to the boundary for critical sqg on bounded domains we consider the dissipative sqg equation in bounded domains first introduced by constantin and ignatova in 2016 we show global holder regularity up to the boundary of the solution with a method based on the de giorgi techniques the boundary introduces several difficulties in particular the dirichlet laplacian is not translation invariant near the boundary which leads to complications involving the riesz transform

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

universal decay in nonlinear parabolic equations on considere le probleme aux valeurs initiales pour une equation d evolution du dt a u 0 t 0 u 0 u 0 on deduit des taux de decroissance universel pour des problemes d evolution de la forme u t u f u x t 0 u x t 0 x t 0 u x 0 u 0 x x ou r n est borne et et f sont des non linearites satisfaisant certaines conditions de structure

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

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

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

calderon reproducing formulas and new besov spaces associated with operators abstract let l be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of poisson type acting on l 2 x where x is a possibly non doubling space of polynomial upper bound on volume growth the aim of this paper is to introduce a new class of besov spaces associated with the operator l so that when l is the laplace operator or its square root acting on the euclidean space r n the new besov spaces are equivalent to the classical besov spaces depending on the choice of l the new besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers l since our approach does not require the doubling volume property of the underlying space it is applicable to any subset of euclidean spaces without any smoothness requirement on the boundary we will also develop a number of calderon reproducing formulas which play an important role in the theory of function spaces and are of independent interest as an application we study besov spaces associated with schrodinger operators with non negative potentials satisfying reverse holder estimates

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

uniqueness of entropy solutions to fractional conservation laws with fully infinite speed of propagation our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non lipschitz convection term and a diffusion term of nonlocal porous medium type the nonlocality is given by a fractional power of the laplace operator for a wide class of nonlinearities the l 1 contraction principle is established despite the fact that the finite infinite speed of propagation alibaud jee 2007 cannot be exploited in our framework existence is deduced with perturbation arguments the method of proof adapted from andreianov maliki nodea 2010 requires a careful analysis of the action of the fractional laplacian on truncations of radial powers

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

heat kernels on metric measure spaces and an application to semilinear elliptic equations we consider a metric measure space m d p and a heat kernel p t x v on m satisfying certain upper and lower estimates which depend on two parameters a and we show that under additional mild assumptions these parameters are determined by the intrinsic properties of the space m d namely a is the hausdorff dimension of this space whereas called the walk dimension is determined via the properties of the family of besov spaces w 2 on m moreover the parameters a and are related by the inequalities 2 a 1 we prove also the embedding theorems for the space w 2 2 and use them to obtain the existence results for weak solutions to semilinear elliptic equations on m of the form u f x u g x where is the generator of the semigroup associated with p t the framework in this paper is applicable for a large class of fractal domains including the generalized sierpinski carpet in r n

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

regularity of weak solutions and supersolutions to the porous medium equation abstract we investigate the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation in particular we establish the equivalence of the conditions u m l loc 2 0 t h loc 1 and u m 1 2 l loc 2 0 t h loc 1 in the definition of weak solutions our proof is based on approximation by solutions to obstacle problems

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

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

weak solutions of the porous medium equation we show that if u 0 u l loc m r n 1 solves u t u m m 1 in the sense of distributions then u is locally holder continuous in

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

contractive properties for the heat equation in sobolev spaces abstract consider the heat equation r u x u 0 in a cylinder 0 t r n 1 smooth lateral boundary under zero neumann or dirichlet conditions geometric conditions for are given that guarantee that for a given p x u t l p will be non increasing for any solution decay rates are also given for arbitrary and p it is shown how to construct an equivalent l p norm such that x t is non increasing in this norm

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

the hele shaw problem as a mesa limit of stefan problems existence uniqueness and regularity of the free boundary we study a hele shaw problem with a mushy region obtained as a mesa type limit of one phase stefan problems in exterior domains we deal with both neumann and dirichlet data and show pointwise convergence of the stefan solutions to the hele shaw solution we make no assumptions on the geometry topology or connectivity of the injection slot

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

asymptotic expansion of solutions to the cauchy problem for doubly degenerate parabolic equations with measurable coefficients abstract we study the asymptotic behaviour of nonnegative solutions of the cauchy problem for doubly degenerate parabolic equations with variable coefficients when the initial datum has a finite mass the asymptotic expansion of the solution for a large time uniformly in whole space is established

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

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

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