the regularization method for an obstacle problem we give a relatively complete analysis for the regularization method which is usually used in solving non differentiable minimization problems the model problem considered in the paper is an obstacle problem in addition to the usual convergence result and a priori error estimates we provide a posteriori error estimates which are highly desired for practical implementation of the regularization method

qualitative and numerical analysis of quasi static problems in elastoplasticity the quasi static problem of elastoplasticity with combined kinematic isotropic hardening is formulated as a time dependent variational inequality vi of the mixed kind that is it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space this vi differs from the standard parabolic vi in that time derivatives of the unknown variable occur in all of its terms the problem is shown to possess a unique solution we consider two types of approximations to the vi corresponding to the quasi static problem of elastoplasticity semidiscrete approximations in which only the spatial domain is discretized by finite elements and fully discrete approximations in which the spatial domain is again discretized by finite elements and the temporal domain is discretized and the time derivative appearing in the vi is approximated in an appropriate way estimates of the errors inherent in the above two types of approximations in suitable sobolev norms are obtained for the quasi static problem of elastoplasticity in particular these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and where appropriate of the time step size k a major difficulty in solving the problems is caused by the presence of the nondifferentiable terms we consider some regularization techniques for overcoming the difficulty besides the usual convergence estimates we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

reactive trajectories and the transition path process we study the trajectories of a solution x t to an ito stochastic differential equation in mathbb r d as the process passes between two disjoint open sets a and b these segments of the trajectory are called transition paths or reactive trajectories and they are of interest in the study of chemical reactions and thermally activated processes in that context the sets a and b represent reactant and product states our main results describe the probability law of these transition paths in terms of a transition path process y t which is a strong solution to an auxiliary sde having a singular drift term we also show that statistics of the transition path process may be recovered by empirical sampling of the original process x t as an application of these ideas we prove various representation formulas for statistics of the transition paths we also identify the density and current of transition paths our results fit into the framework of the transition path theory by weinan and vanden eijnden

laplacian flow of homogeneous g2 structures and its solitons we use the bracket flow algebraic soliton approach to study the laplacian flow of g2 structures and its solitons in the homogeneous case we prove that any homogeneous laplacian soliton is equivalent to a semi algebraic soliton that is a g invariant g2 structure on a homogeneous space g k that flows by pull back of automorphisms of g up to scaling algebraic solitons are geometrically characterized among laplacian solitons as those with a diagonal evolution unlike the ricci flow case where any homogeneous ricci soliton is isometric to an algebraic soliton we have found as an application of the above characterization an example of a left invariant closed semi algebraic soliton on a nilpotent lie group which is not equivalent to any algebraic soliton the normalized bracket flow evolution of such a soliton is periodic in the context of solvable lie groups with a codimension one abelian normal subgroup we obtain long time existence for any closed laplacian flow solution furthermore the norm of the torsion is strictly decreasing and converges to zero we also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding laplacian solitons

variational principles and convergence of finite element approximations of a holonomic elastic plastic problem it is shown that a boundary value problem based on a holonomic elastic plastic constitutive law may be formulated equivalently as a variational inequality of the second kind a regularised form of the problem is analysed and finite element approximations are considered it is shown that solutions based on finite element approximation of the regularised problem converge

on the moment map for the variety of lie algebras we consider the moment map m pvn iu n for the action of gl n on vn 2 cn cn the critical points of the functional fn m 2 pvn r are studied in order to understand the stratification of ln pvn defined by the negative gradient flow of fn where ln is the algebraic subset of all lie algebras we obtain a description of the critical points which lie in ln in terms of those which are nilpotent as well as the minima and maxima of fn ln r a characterization of the critical points modulo isomorphism as the union of categorical quotients of suitable actions is considered and some applications to the study of ln are given

finding einstein solvmanifolds by a variational method we use a variational approach to prove that any nilpotent lie algebra having a codimension one abelian ideal and anyone of dimension leq 5 admits a rank one solvable extension which can be endowed with an einstein left invariant riemannian metric a curve of 8 dimensional einstein solvmanifolds is also given

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

numerical verification of solutions for obstacle problems this article is an extension of the previous paper numer math 81 1998 305 by the same authors we propose a method to prove the existence of solutions for more general obstacle problems by numerical computations the main task in this paper consists of the numerical determination of some constants which appear in a priori error estimations for the finite element approximation of a simple variational inequality we present a numerical example for verification

the structure of the free boundary for lower dimensional obstacle problems we study the regularity of the free surface in boundary obstacle problems we show that near a non degenerate point the free boundary is a c 1 a n 2 dimensional surface in r n 1

einstein solvmanifolds existence and non existence questions the aim of this paper is to study the problem of which solvable lie groups admit an einstein left invariant metric the space mathcal n of all nilpotent lie brackets on mathbb r n parametrizes a set of n 1 dimensional rank one solvmanifolds s mu mu in mathcal n containing the set of all those which are einstein in that dimension the moment map for the natural gl n action on mathcal n evaluated at mu in mathcal n encodes geometric information on s and suggests the use of strong results from geometric invariant theory for instance the functional on mathcal n whose critical points are precisely the einstein s s is the square norm of this moment map we use a gl n invariant stratification for the space mathcal n and show that there is a strong interplay between the strata and the einstein condition on the solvmanifolds s as an application we obtain criteria to decide whether a given nilpotent lie algebra can be the nilradical of a rank one einstein solvmanifold or not we find several examples of mathbb n graded even 2 step nilpotent lie algebras which are not a classification in the 7 dimensional 6 step case and an existence result for certain 2 step algebras associated to graphs are also given

behavior near the boundary of positive solutions of second order parabolic equations ii a boundary backward harnack inequality is proved for positive solutions of second order parabolic equations in non divergence form in a bounded cylinder q q x 0 t which vanish on o q oq x 0 t where q is a bounded lipschitz domain in r this inequality is applied to the proof of the holder continuity of the quotient of two positive solutions vanishirlg on a portion of o q

immortal homogeneous ricci flows we show that for an immortal homogeneous ricci flow solution any sequence of parabolic blow downs subconverges to a homogeneous expanding ricci soliton this is established by constructing a new lyapunov function based on curvature estimates which come from real geometric invariant theory

a new boundary harnack principle equations with right hand side we introduce a new boundary harnack principle in lipschitz domains for equations with a right hand side our approach which uses comparisons and blow ups will adapt to more general domains as well as other types of operators we prove the principle for divergence form elliptic equations with lower order terms including zero order terms the inclusion of a zero order term appears to be new even in the absence of a right hand side

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

noncompact homogeneous einstein manifolds attached to graded lie algebras in this paper we study the nilradicals of parabolic subalgebras of semisimple lie algebras and the natural one dimensional solvable extensions of them we investigate the structures curvatures and einstein conditions of the associated nilmanifolds and solvmanifolds we show that our solvmanifold is einstein if the nilradical is two step new examples of einstein solvmanifolds with three step and four step nilradicals are also given

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

inequalities defining orbit spaces the orbit space of a representation of a compact lie group has a natural semialgebraic structure we describe explicit ways of finding the inequalities defining this structure and we give some applications

numerical verification of solutions for variational inequalities in this paper we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities this technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates for obstacle problems we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example for an obstacle problem is presented

on characterizations and regularity of the solution of bilateral obstacle problems in this paper we introduce a new reformulation of bilateral obstacle problems which allows us to characterize the domain of noncontact with the obstacles and give a new proof for h2 regularity of the solution

path following and augmented lagrangian methods for contact problems in linear elasticity a certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite dimensional semismooth newton methods or in this case equivalently primal dual active set strategies we present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem first order augmented lagrangian and path following methods the first strategy is based on a multiplier update while path following with respect to the regularization parameter uses theoretical results about the path value function to increase the regularization parameter appropriately comprehensive numerical tests investigate the performance of the proposed strategies for both a 2d as well as a 3d contact problem

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

a step towards the alekseevskii conjecture we refine existing structure results for non compact homogeneous einstein manifolds and provide a reduction in the classification problem of such spaces using this work we verify the generalized alekseevskii conjecture for a large class of homogeneous spaces

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

the ricci flow on solvmanifolds of real type abstract we show that for any solvable lie group of real type any homogeneous ricci flow solution converges in cheeger gromov topology to a unique non flat solvsoliton which is independent of the initial left invariant metric as an application we obtain results on the isometry groups of non flat solvsoliton metrics and einstein solvmanifolds

regularity of one phase stefan problem near lipschitz initial data in this paper we show that starting from lipschitz initial free boundary with small lipschitz constant the solution of the one phase stefan problem st immediately regularizes and is smooth in space and time for a small positive time

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

finite element analysis of a holonomic elastic plastic problem a thorough analysis of the finite element method is given for a holonomic elastic plastic problem an inequality of the cea s lemma type is proved which is the basis of error estimates for various finite element solutions difficulty caused by a non differentiable term in the problem can be overcome by using two convergent procedures an iterative procedure and a regularization procedure an a posteriori quantitative error estimate is derived for the regularized solution

analysis of a mixed formulation of a bilateral obstacle problem abstract in this paper we present a reformulation of a bilateral obstacle problem as a mixed formulation problem based on subdifferential of a continuous function of which the subdifferential can characterize the non contact domain then we present the analysis of the discrete problem we prove the convergence of the approximate solution to the exact one and we provide an error estimate this formulation was established in an abstract way then the theoretical results was applied to a bilateral obstacle problem

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

caloric functions in lipschitz domains and the regularity of solutions to phase transition problems in some domain d c rn l that vanishes locally on some distinguished part of od the term refers to the fact that in some sense along that part of od u resembles the fundamental solution of the heat equation in d this is particularly the case when the domain under consideration d is the intersection of some n 1 dimensional cube q with q one side of a lipschitz graph i e q fx f x t that is d q n q and the distinguished part of od is precisely q n q this is the main first area of study of this paper relying on work of fabes garofalo salsa fgs on backward harnack type inequalities for such domains lipschitz regularity in time versus lipschitz regularity in space is not of course the natural homogeneity balance for the study of parabolic equations but it is so for the study of phase transition relations of the form

parabolic subgroups of semisimple lie groups and einstein solvmanifolds in this paper we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple lie algebras these solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type we show that the ricci curvatures of our solvmanifolds coincide with the restrictions of the ricci curvatures of the ambient symmetric spaces consequently all of our solvmanifolds are einstein which provide a large number of new examples of noncompact homogeneous einstein manifolds we also show that our solvmanifolds are minimal but not totally geodesic submanifolds of symmetric spaces

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

hyperpolar homogeneous foliations on symmetric spaces of noncompact type a foliation f on a riemannian manifold m is homogeneous if its leaves coincide with the orbits of an isometric action on m a foliation f is polar if it admits a section that is a connected closed totally geodesic submanifold of m which intersects each leaf of f and intersects orthogonally at each point of intersection a foliation f is hyperpolar if it admits a flat section these notes are related to joint work with jose carlos diaz ramos and hiroshi tamaru about hyperpolar homogeneous foliations on riemannian symmetric spaces of noncompact type apart from the classification result which we proved in 1 we present here in more detail some relevant material about symmetric spaces of noncompact type and discuss the classification in more detail for the special case m slr 1 r sor 1 1 isometric actions on the real hyperbolic plane the special linear group g sl2 r a b c d a b c d r ad bc 1 acts on the upper half plane h z c z 0 by linear fractional transformations of the form a b c d z az b cz d by equipping h with the riemannian metric ds dx dy y2 dzdz z 2 z x iy

on solvable lie groups of negative ricci curvature we study solvable lie groups which admit a left invariant metric of strictly negative ricci curvature we obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions

regularity properties of a free boundary near contact points with the fixed boundary in the upper half of the unit ball b x 0 let u and omega a domain in r n x is an element of r n x 1 0 solve the following overdetermined problem deltau chi omega in b u

existence and non existence of homogeneous einstein metrics on demontre un theoreme d existence general pour des metriques d einstein homogenes et on presente des espaces homogenes compacts simplement connexes qui ne portent pas une metrique d einstein homogene

the sub elliptic obstacle problem c1 a regularity of the free boundary in carnot groups of step two the sub elliptic obstacle problem arises in various branches of the applied sciences e g in mechanical engineering and robotics mathematical finance image reconstruction and neurophysiology in the recent paper donatella danielli nicola garofalo sandro salsa variational inequalities with lack of ellipticity i optimal interior regularity and non degeneracy of the free boundary indiana univ math j 52 2 2003 361 398 mr1976081 2004c 35424 it was proved that weak solutions to the sub elliptic obstacle problem in a carnot group belong to the folland stein optimal lipschitz class the analogue of the well known interior local regularity for the classical obstacle problem however the regularity of the free boundary remained a challenging open problem in this paper we prove that in carnot groups of step r 2 the free boundary is euclidean c1 a near points satisfying a certain thickness condition this constitutes the sub elliptic counterpart of a celebrated result due to caffarelli luis a caffarelli the regularity of free boundaries in higher dimensions acta math 139 3 4 1977 155 184 mr0454350 56 12601

hermitian curvature flow on unimodular lie groups and static invariant metrics we investigate the hermitian curvature flow hcf of leftinvariant metrics on complex unimodular lie groups we show that in this setting the flow is governed by the ricci flow type equation tgt ric gt the solution gt always exists for all positive times and 1 t gt converges as t in cheeger gromov sense to a nonflat left invariant soliton g g moreover up to homotheties on each of these groups there exists at most one left invariant soliton solution which is a static hermitian metric if and only if the group is semisimple in particular compact quotients of complex semisimple lie groups yield examples of compact non k nahler manifolds with static hermitian metrics we also investigate the existence of static metrics on nilpotent lie groups and we generalize a result of enrietti fino and the third author j symplectic geom 10 2012 no 2 203 223 for the pluriclosed flow in the last part of the paper we study the hcf on lie groups with abelian complex structures

the alekseevskii conjecture in low dimensions the long standing alekseevskii conjecture states that a connected homogeneous einstein space g k of negative scalar curvature must be diffeomorphic to mathbb r n this was known to be true only in dimensions up to 5 and in dimension 6 for non semisimple g in this work we prove that this is also the case in dimensions up to 10 when g is not semisimple for arbitrary g besides 5 possible exceptions we show that the conjecture holds up to dimension 8

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains

construction of nice nilpotent lie groups abstract we illustrate an algorithm to classify nice nilpotent lie algebras of dimension n up to a suitable notion of equivalence applying the algorithm we obtain complete listings for n 9 on every nilpotent lie algebra of dimension 7 we determine the number of inequivalent nice bases which can be 0 1 or 2 we show that any nilpotent lie algebra of dimension n has at most countably many inequivalent nice bases

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified

a characterization of bounded symmetric domains by curvature this paper will prove that a bounded homogeneous domain is symmetric if and only if in the bergman metric all sectional curvatures are nonpositive introduction it is well known that a bounded symmetric domain has nonpositive sectional curvature in the bergman metric this paper is devoted to the converse namely that a bounded homogeneous domain equivalently homogeneous siegel domain which has nonpositive sectional curvature in the bergman metric must be a symmetric domain the proof uses the techniques of normal j algebras 13 as well as results of vinberg 16 17 the first author 1 3 and indirectly dorfmeister 5 6 the body of the paper is in two sections in the first we derive certain relations between the dimensions of the root spaces in an irreducible normal j algebra which in the presence of the additional curvature assumption show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root in the second section we show that the first dimension result implies that the cone of the corresponding siegel domain is self dual while the two dimension results together imply that the domain is quasi symmetric the proof is then finished because 3 a quasi symmetric domain with nonpositive sectional curvature in the bergman metric is known to be symmetric we remark that some of the subsidiary results mentioned above are not essentially new however for consistency of presentation it is necessary to have them in the language of normal jalgebras and for that reason they are included there is also overlap here with work of zelow lundquist 21 1 throughout this paper s j will denote a normal j algebra with admissible form w this means that s is a finite dimensional real split solvable lie algebra with almost complex structure j s s such that x y j jx y j x jy jx jy and w is a linear form on s such that the bilinear form o jx y is symmetric positive definite and j invariant note that the assumption that s is real split was incorrectly omitted in 2 let n s s and let a be the orthogonal complement of n in s by the basic structure theorem of pyatetskii shapiro 13 a is a commutative received by the editors july 13 1981 1980 mathematics subiect classification primary 32mi0 53c30 1983 american mathematical society 0002 9947 82 000 0759 09 00

semismooth newton and augmented lagrangian methods for a simplified friction problem in this paper a simplified friction problem and iterative second order algorithms for its solution are analyzed in infinite dimensional function spaces motivated from the dual formulation a primal dual active set strategy and a semismooth newton method for a regularized problem as well as an augmented lagrangian method for the original problem are presented and their close relation is analyzed local as well as global convergence results are given by means of numerical tests we discuss among others convergence properties the dependence on the mesh and the role of the regularization and illustrate the efficiency of the proposed methodologies