potential theory for the sum of the 1 laplacian and p laplacian abstract the variational problem of minimizing the functional u d u 1 p d u p a u on a domain r n under zero boundary values which appears for example in the theory of bingham fluids shows interesting phenomena such as the formation of a subset of with positive measure where the minimizer is constant a plateau the corresponding euler lagrange equation should be 1 u p u a but the 1 laplacian is not defined in points where d u 0 we show that it is nevertheless possible to define subsolutions and supersolutions for this equation we show that the comparison principle is valid and we begin developing a potential theory analogous to what is known about the p laplacian in addition we apply those notions to rotationally invariant solutions and we discuss properties of the plateau and a possible connection to the theory

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

the application of the preconditioned biconjugate gradient algorithm to nlte rate matrix equations abstract this paper reports the success of the preconditioned biconjugate gradient pbcg and the conjugate gradient square cgs algorithms in solving the matrix equations resulting from the discretization of systems of population rate equations which arise in nonequilibrium kinetics modeling the success of the pbcg and cgs can be attributed to two main ideas first the singularity of the rate matrix resulting from population conservation requirement was removed through a reduction of matrix order so as to improve the condition number of the matrix second an efficient preconditioner was found to reduce the reduce the eigenvalue spread of the rate matrix the preconditioning matrix was selected on the basis of retaining the largest few rates in each column of the well conditioned rate matrix this preconditioner along with the reduced rate matrix enabled the algorithms to converge very rapidly so as to make them an attractive alternative to standard direct methods

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

the use of a preconditioned bi conjugate gradient method for hybrid plasma stability analysis abstract the numerical stability analysis of compact toroidal plasmas using implicit time differencing requires the solution of a set of coupled 2 dimensional elliptic partial differential equations for the field quantities at every timestep when the equations are spatially finite differenced and written in matrix form the resulting matrix is large sparse complex non hermitian and indefinite the use of the preconditioned bi conjugate gradient method for solving these equations is discussed the effect of block diagonal preconditioning and of incomplete block lu preconditioning on the convergence of the method is investigated for typical matrices arising in our studies the eigenvalue spectra of the original and preconditioned matrices are calculated as an illustration of the effectiveness of the preconditioning we show that the preconditioned bi conjugate gradient method converges more rapidly than the conjugate gradient method applied to the normal equations and that it is an effective iterative method for the class of non hermitian indefinite problems of interest

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

on the nash point equilibria in the calculus of variations abstract the existence and partial regularity of the nash point equilibria for a pair of multiple integrals j u v f x u v u v dx k u v g x u v u v dx are studied the conditions as well as the results are similar to those for local minima obtained by acerbi fusco and giaquinta giusti

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

stability for the solutions of div u p 2 u f with varying p abstract elliptic partial differential equations with principal part div u p 2 u are applied in physics e g for the description of phenomena in glaceology the objective of our note is a natural question of stability for solutions as p varies

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

liouville theorems partial regularity and holder continuity of weak solutions to quasilinear elliptic systems this paper describes the connections between liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth it is shown that equivalence does in general hold only in some restricted sense a complete correspondence can be established in certain cases e g for small solutions and for minima of quadratic variational integrals

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

second order elliptic equations with variably partially vmo coefficients abstract the solvability in w p 2 r d spaces is proved for second order elliptic equations with coefficients which are measurable in one direction and vmo in the orthogonal directions in each small ball with the direction depending on the ball this generalizes to a very large extent the case of equations with continuous or vmo coefficients

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

a proof of lusztig s conjectures for affine type g2 with arbitrary parameters we prove lusztig s conjectures bf p1 bf p15 for the affine weyl group of type tilde g 2 for all choices of parameters our approach to compute lusztig s mathbf a function is based on the notion of a balanced system of cell representations for the hecke algebra we show that for arbitrary coxeter type the existence of balanced system of cell representations is sufficient to compute the mathbf a function and we explicitly construct such a system in type tilde g 2 for arbitrary parameters we then investigate the connection between kazhdan lusztig cells and the plancherel theorem in type tilde g 2 allowing us to prove bf p1 and determine the set of duflo involutions from there the proof of the remaining conjectures follows very naturally essentially from the combinatorics of weyl characters of types g 2 and a 1 along with some explicit computations for the finite cells

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

affine cellularity of affine hecke algebras of rank two we show that affine hecke algebras of rank two with generic parameters are affine cellular in the sense of koenig xi

schauder estimates for parabolic nondivergence operators of hormander type let x1 x2 xq be a system of real smooth vector fields satisfying hormander s rank condition in a bounded domain of rn let a aij t x i j 1q be a symmetric uniformly positive definite matrix of real functions defined in a domain u r for operators of kind h t i j 1qaij t x xixj i 1qbi t x xi c t x we prove local a priori estimates of schauder type in the natural parabolic ck u spaces defined by the vector fields xi and the distance induced by them namely for aij bi c ck u and u u we prove u ck 2 u c hu ck u u l u

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

vanishing viscosity limit for homogeneous axisymmetric no swirl solutions of stationary navier stokes equations abstract 1 homogeneous axisymmetric no swirl solutions of three dimensional incompressible stationary navier stokes equations which are smooth on the unit sphere minus the north and south poles have been classified in this paper we study the vanishing viscosity limit of sequences of these solutions as the viscosity tends to zero some sequences of solutions c l o c m converge to solutions of euler equations on the sphere minus the poles while for other sequences of solutions transition layer behaviors occur for every latitude circle there are sequences which c l o c m converge respectively to different solutions of the euler equations on the spherical caps above and below the latitude circle we give detailed analysis of these convergence and transition layer behaviors

interior w2 p x regularity for elliptic equations of nondivergence form abstract in this paper we consider the following elliptic equations of nondivergence form i j 1 n a i j 2 u x i x j f in r n where the coefficients a i j are real valued bounded measurable functions defined in r n belong to vmo loc see definition 2 5 and satisfy the uniformly elliptic condition under the assumption that p satisfies the log holder continuity condition we will establish the interior w 2 p x regularity for strong solutions to the elliptic equations of nondivergence form

the based ring of the lowest generalized two sided cell of an extended affine weyl group abstract let c 0 be the lowest generalized two sided cell of an extended affine weyl group w with unequal parameters we first prove that certain conjectures of lusztig called p1 p15 hold for c 0 which implies the existence of the based ring of c 0 and then we determine the structure of the based ring as an application we use the structure of the based ring to study certain simple modules of the hecke algebra with unequal parameters associated to w namely those attached to c 0 further we give a set of prime ideals p of the center z of the affine hecke algebra h such that the reduced algebra k p h is simple over k p where k p frac z p is the residue field of z at p in particular we show that the algebra h z frac z is split simple over the field frac z

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

some regularity results for p harmonic mappings between riemannian manifolds abstract let m be a c 2 smooth riemannian manifold with boundary and n a complete c 2 smooth riemannian manifold we show that each stationary p harmonic mapping u m n whose image lies in a compact subset of n is locally c 1 for some 0 1 provided that n is simply connected and has non positive sectional curvature we also prove similar results for minimizing p harmonic mappings with image being contained in a regular geodesic ball moreover when m has non negative ricci curvature and n is simply connected with non positive sectional curvature we deduce a gradient estimate for c 1 smooth weakly p harmonic mappings from which follows a liouville type theorem in the same setting

parabolic equations with bmo coefficients in lipschitz domains abstract in this paper we are concerned with certain natural sobolev type estimates for weak solutions of inhomogeneous problems for second order parabolic equations in divergence form the geometric setting is that of time independent cylinders having a space intersection assumed to be locally given by graphs with small lipschitz coefficients the constants of the operator being uniformly parabolic we prove the relevant l p estimates assuming that the coefficients are in parabolic bounded mean oscillation bmo and that their parabolic bmo semi norms are small enough

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

the weighted universal coxeter group and some related conjectures of lusztig abstract lusztig proposed 15 conjectures for any weighted coxeter groups in this study we explicitly describe kazhdan lusztig cells distinguished involutions and the function a for any weighted universal coxeter group we also verify these conjectures for this group

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

the regularity of harmonic maps into spheres and applications to bernstein problems we show the regularity of and derive a priori estimates for weakly harmonic maps from a riemannian manifold into a euclidean sphere under the assumption that the image avoids some neighborhood of a half equator the proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems we apply these results to the spherical and euclidean bernstein problems for minimal hypersurfaces obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in euclidean spaces are trivial

w 1 p regularity for elliptic equations with measurable coefficients in nonsmooth domains we establish global w 1 p estimates for second order elliptic equations in divergence form under the natural assumption that p is log holder continuous to this end we assume that the coefficients are measurable in one variable and have small bmo semi norms in the other variables and the boundary of the domain is reifenberg flat our work is an optimal and natural extension of w 1 p regularity for such equations with merely measurable coefficients beyond lipschitz domains

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

interior w 1 p x t regularity for parabolic divergence equations of hormander s type abstract let x 1 x 2 x q be a system of real smooth vector fields satisfying hormander s condition in a bounded domain r n we consider the following parabolic equations u t x i a i j x t x j u x i f i in t where x i is the formal adjoint of x i the coefficients a i j x t are real valued measurable functions defined in t 0 t satisfying the uniform parabolic condition 0 1 2 i j 1 q a i j x t i j 1 2 for almost every x t r n r every r q and some constant we prove the interior w 1 p x t regularity for weak solutions to the parabolic equations under the assumptions that p x t satisfies the strong log holder continuity condition and the coefficients a i j x t belong to the space v m o l o c l our method relies on a gagliardo nirenberg inequality constituted on hormander s vector fields and a certain vitali covering lemma

parabolic equations in reifenberg domains natural sobolev type estimates are proved for weak solutions of inhomogeneous parabolic equations in divergence form in a bounded cylinder 0 t which is reifenberg flat in the space direction the principal coefficients of the operator are assumed to be in bmo space with their bmo semi norms small enough

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

interior hw 1 p estimates for divergence degenerate elliptic systems in carnot groups abstract let x 1 x q be the basis of the space of horizontal vector fields on a homogeneous carnot group g r n q n we consider the following divergence degenerate elliptic system 1 n i j 1 q x i a i j x x j u i 1 q x i f i 1 2 n where the coefficients a i j are real valued bounded measurable functions defined in g satisfying the strong legendre condition and belonging to the space v m o l o c defined by the carnot caratheodory distance induced by the x i s we prove interior h w 1 p estimates 2 p for weak solutions to the system