equilibrium points of a singular cooperative system with free boundary abstract in this paper we initiate the study of maps minimising the energy d u 2 2 u d x which due to lipschitz character of the integrand gives rise to the singular euler equations u u u u 0 u u 1 u m our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps our results here concern regularity of the solution as well as that of the free boundary they are achieved by using monotonicity formulas and epiperimetric inequalities in combination with geometric analysis

comparison results between linear and quasilinear elliptic boundary value problems on considere des resultats de comparaison du type suivant on suppose que u c 2 c satisfait un probleme aux valeurs limites elliptique quasi lineaire div a x u u u b x u u dans u sur et on suppose que v est la solution classique du probleme lineaire v const dans v sur on etudie l existence de situations ou u x v x x

a modified gpss method for non hermitian positive definite linear systems this paper is concerned with a modified version of the generalization of positive definite and skew hermitian splitting iteration mgpss to solve the non hermitian positive definite linear systems the theoretical analysis shows that the mgpss iteration method will converge unconditionally numerical examples are reported to confirm the efficiency of the proposed methods

the two phase membrane problem an intersection comparison approach to the regularity at branch points for the two phase membrane problem delta u lambda 2 chi u 0 lambda 2 chi u 0 and lambda 0 we prove in two dimensions that the free boundary is in a neighborhood of each branch poin

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

a note on the spectra and pseudospectra of waveform relaxation operators for linear differential algebraic equations in this note we derived the expressions of the spectra and pseudospectra of waveform relaxation operators for general linear differential algebraic equations of index one the expressions are generalizations of those of lumsdaine and wu siam j sci comput 18 1997 pp 286 304 and the derivation here is nontrivial numerical experiments are given to verify the theoretical results

on the non convexity of solutions in free boundary problems arising in plasma physics and fluid dynamics the purpose of this note is to give counterexamples for the conjectured existence of convex solutions in two closely related free boundary problems the first free boundary problem arises in fluid dynamics the second in the work of laurence and stredulinsky on queer differential equations in plasma physics see 5

convergence analysis of two stage waveform relaxation method for the initial value problems we present a class of two stage waveform relaxation methods for solving the initial value problems of ordinary differential equations by making use of the laplace transform we discuss the convergence of these methods and derive sufficient conditions for guaranteeing their convergence when the system matrices are specifically h matrices also we discuss the monotonicity of the convergence sequence further we compare the sequences generated by two different inner splittings of the same initial value problem

solving shifted linear systems with restarted gmres augmented with error approximations abstract in this paper we investigate a variant of the restarted gmres method for solving a series of large sparse linear systems restarting is carried out by augmenting krylov subspaces with some recently generated error approximations from the seed system the method can preserve a nice property that allows solving the seed and the added linear systems at the cost of only one matrix vector multiplication per iteration compared with solving each added linear system separately the advantage of the new scheme is to lower down the overall cost of solving all added linear systems numerical experiments illustrate the efficiency of the acceleration strategy

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

an unstable elliptic free boundary problem arising in solid combustion we prove a regularity result for the unstable elliptic free boundary problem u u 0 0 1 related to traveling waves in a problem arising in solid combustion the maximal solution and every local minimizer of the energy are regular that is u 0 is locally an analytic surface and u u 0 u u 0 are locally analytic functions moreover we prove a partial regularity result for solutions that are non degenerate of second order here u 0 is analytic up to a set of hausdorff dimension n 2 we discuss possible singularities ams classification 35r35 35j60 35b65

on semi convergence of the parameterized generalized mhss method for singular complex linear systems for solving a class of complex symmetric singular linear systems we propose a parameterized generalized mhss pgmhss iteration method and investigate the semi convergence conditions by analyzing the spectrum of the iteration matrix then we analyze the optimal iteration parameters that minimize the upper bound of the semi convergence factor numerical experiments are used to test the feasibility and the effectiveness of the pgmhss iteration method

a new approach to queer differential equations abstract queer differential equations first arose in the work of harold grad on controlled thermonuclear fusion these relate in particular to models for the slow adiabatic evolution and resistive diffusion of a plasma these are queer in that these share aspects of partial ordinary and functional differential equations in this document the authors give a new way of thinking of these equations by relating these to free boundary problems this is the first of a series of papers intending to demonstrate that solutions of such a queer differential equation can be thought of as limits of solutions of free boundary problems with n free boundaries a long term hope is that this work will complement and further refine existing numerical schemes for finding such solutions

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

a parameterized shss iteration method for a class of complex symmetric system of linear equations in this paper we present a parameterized variant of the single step hermitian and skew hermitian shss iteration method for solving a class of complex symmetric system of linear equations we study the convergence properties of the parameterized shss p shss iteration method the choices of the parameters and the quasi optimal parameters numerical experiments are given to verify the effectiveness of the p shss iteration method

discrete time accelerated block successive overrelaxation methods for time dependent stokes equations to further study the application of waveform relaxation methods in fluid dynamics in actual computation this paper provides a general theoretical analysis of discrete time waveform relaxation methods for solving linear daes a class of discrete time waveform relaxation methods named discrete time accelerated block successive overrelaxation dabsor methods is proposed for solving linear daes derived from discretizing time dependent stokes equations in space by using method of lines the analysis of convergence property and optimality of the dabsor method are presented in detail the theoretical results and the efficiency of the dabsor method are verified by numerical experiments

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

boundary behaviour for a singular perturbation problem abstract in this paper we study the boundary behaviour of the family of solutions u e to singular perturbation problem u e e u e u e 1 in b 1 x n 0 x 1 where a smooth boundary data f is prescribed on the flat portion of b 1 here e 1 e e c 0 0 1 0 0 1 t m 0 is an approximation of identity if f z 0 whenever f z 0 then the level sets u e 0 approach the fixed boundary in tangential fashion with uniform speed the methods we employ here use delicate analysis of local solutions along with elaborated version of the so called monotonicity formulas and classification of global profiles

monotonicity theorems for laplace beltrami operator on riemannian manifolds abstract for free boundary problems on euclidean spaces the monotonicity formulas of alt caffarelli friedman and caffarelli jerison kenig are cornerstones for the regularity theory as well as the existence theory in this article we establish the analogs of these results for the laplace beltrami operator on riemannian manifolds as an application we show that our monotonicity theorems can be employed to prove the lipschitz continuity for the solutions of a general class of two phase free boundary problems on riemannian manifolds

efficient single step preconditioned hss iteration methods for complex symmetric linear systems abstract we propose a single step preconditioned variant of hss sphss and an efficient parameterized sphss psphss iteration method for solving a class of complex symmetric linear systems under suitable conditions we analyze the convergence properties of the sphss and psphss iteration methods theoretical analysis shows that the minimal upper bounds for the spectral radius of the sphss and psphss iteration matrices are less than those of the shss and pshss iteration matrices when using the optimal parameters respectively numerical results show that the psphss iteration method has comparable advantage over several other iteration methods whether the experimental optimal parameters are used or not

alternating splitting waveform relaxation method and its successive overrelaxation acceleration for the large sparse implicit linear initial value problem we present a block successive overrelaxation scheme for the alternating direction implicit waveform relaxation method to further accelerate its convergence speed and discuss the convergence property of the resulting iteration method in detail numerical implementations about several non hermitian implicit linear initial value problems show that the alternating direction implicit waveform relaxation method is very effective and the block successive overrelaxation technique really accelerates its convergence speed

multiconstrained variational problems in magnetohydrodynamics equilibrium and slow evolution a computational method is proposed for solving magnetohydrodynamical equilibrium problems with prescribed flux and mass within the magnetic surfaces that foliate the plasma such problems arise in tokamak modeling for instance where they determine either equilibria with given adiabatic profiles or slowly evolving quasi equilibria governed by the grad hogan equations the classical variational principles of kruskal and kulsrud and woltjer which express these problems in terms of energy minimization subject to infinite families of nonlinear nonlocal constraints are taken as the basis for a direct method of solution a natural discretization of the classical constraint families is devised and an iterative algorithm is developed to solve the resulting optimization problems a convergence theory for the algorithm is established and an effective numerical implementation of the method is presented for flux conserving tokamak equilibria some computed examples involving plasma heating and adiabatic compression are described

regularity of the free boundary for two phase problems governed by divergence form equations and applications abstract we study a class of two phase inhomogeneous free boundary problems governed by elliptic equations in divergence form in particular we prove that lipschitz or flat free boundaries are c 1 our results apply to the classical prandtl bachelor model in fluiddynamics

optimized schwarz waveform relaxation for advection reaction diffusion equations in two dimensions optimized schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems they are based on a decomposition in space and an iteration where only subproblems in space time need to be solved each subproblem can be simulated using an adapted numerical method for example with local time stepping or one can even use a different model in different subdomains which makes these methods very suitable also from a modeling point of view for rapid convergence however it is important to use effective transmission conditions between the space time subdomains and for best performance these transmission conditions need to take the physics of the underlying evolution problem into account the optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions we study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions we prove comprehensively best approximation results for transmission conditions of robin and ventcel higher order type which can also be used in the various limits for example for the heat equation since we include in our analysis a positive low frequency limiter both in space and time we give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms and which guarantee asymptotically best performance of the iterative methods we finally show extensive numerical experiments including cases not covered by our analysis for example decompositions with cross points we use q1 finite element discretizations in space and forward and backward euler discretizations in time other discretization could also have been considered since all our analysis is at the continuous level and in all cases we measure performance corresponding to our analysis

optimization problems for an energy functional with mass constraint revisited abstract in this paper we report on a variational problem under a constraint on the mass which is motivated by the torsional rigidity and torsional creep following a device by alt caffarelli and friedman we treat instead a problem without constraint but with a penalty term we will complete some of the results of c bandle a wagner optimization problems for weighted sobolev constants calc var partial differential equations 29 2007 481 507 where the existence of a lipschitz continuous minimizer has been established in particular we prove qualitative properties of the optimal shape

some existence results for fully nonlinear elliptic equations of monge ampere type we study the existence and uniqueness of solutions of monge ampere type equations this type of equations has been studied extensively by caffarelli nirenberg spruck and many others see 5 through 8 and the references therein we present some existence and uniqueness results for this type of equations on compact riemannian manifolds with non negative sectional curvature we have also generalized some results in 7

lipschitz regularity of the eigenfunctions on optimal domains we study the optimal sets omega ast subseteq mathbb r d for spectral functionals of the form f big lambda 1 omega ldots lambda p omega big which are bi lipschitz with respect to each of the eigenvalues lambda 1 omega lambda 2 omega ldots lambda p omega of the dirichlet laplacian on omega a prototype being the problem min big lambda 1 omega cdots lambda p omega omega subseteq mathbb r d omega 1 big we prove the lipschitz regularity of the eigenfunctions u 1 ldots u p of the dirichlet laplacian on the optimal set omega ast and as a corollary we deduce that omega ast is open for functionals depending only on a generic subset of the spectrum as for example lambda k omega our result proves only the existence of a lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved

dynamic iteration methods applied to linear dae systems abstract we continue the study of the convergence of dynamic iteration methods by applying them to linear dae systems we show that convergence rate can be studied by similar means as for ode s and that it is critical for convergence to preserve the structure of the dae system when it is split for the iteration

variational formulas on lipschitz domains a rigorous treatment is given of variational formulas for solutions of certain dirichlet problems for the laplace operator on lipschitz domains under interior variations in particular we extend well known variational formulas for the torsional rigidity and for capacity from the class of c1 domains to the class of lipschitz domains a motivation for this work comes from the use of variational methods in the study of prandtl batchelor flows in fluid mechanics

a class of preconditioned generalized local pss iteration methods for non hermitian saddle point problems in this paper a class of new methods based on the positive definite and skew hermitian splitting scheme called preconditioned generalized local positive definite and skew hermitian splitting pglpss methods are considered to solve non hermitian saddle point problems the convergence properties of the proposed methods are analyzed which prove that the pglpss methods are convergent if the iteration parameters and parameter matrices satisfy appropriate conditions some numerical experiments are provided to verify the efficiency of the proposed method showing the competitiveness and efficiency of this novel method over other testing methods whether it served as a preconditioned iteration method or as a preconditioner to the krylov subspace method

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

on the non tangential touch between the free and the fixed boundaries for the two phase obstacle like problem in this paper we consider the following two phase obstacle problem like equation in the unit half ball delta u lambda chi u 0 lambda chi u 0 we classify the angles of touch between the free boundary and the fixed one if the boundary data f satisfies certain conditions these results are connected to the results from andersson et al ark mat 44 1 1 15 2006

on convergence conditions of waveform relaxation methods for linear differential algebraic equations for linear constant coefficient differential algebraic equations we study the waveform relaxation methods without demanding the boundedness of the solutions based on infinite time interval in particular we derive explicit expression and obtain asymptotic convergence rate of this class of iteration schemes under weaker assumptions which may have wider and more useful application extent numerical simulations demonstrate the validity of the theory

a parabolic free boundary problem with double pinning abstract we study the cauchy problem for the equation tue ue e ue in 0 rn as e 0 where the nonlinearity e is assumed to converge to a measure concentrated at 0 in this paper we allow for sign changes of e and ue the solutions are uniformly lipschitz continuous in space and holder continuous in time we show that each limit of ue is a solution of the free boundary problem tu u 0 in u 0 0 rn u 2 u 2 g on u 0 u e the function g in the condition on the free boundary need not be a constant

the uzawa pps iteration methods for nonsingular and singular non hermitian saddle point problems abstract in this paper based on the positive definite and positive semidefinite splitting pps iteration scheme we establish a class of uzawa pps iteration methods for solving nonsingular and singular non hermitian saddle point problems with the 1 1 part of the coefficient matrix being non hermitian positive definite theoretical analyses show that the convergence and semi convergence properties of the proposed methods can be guaranteed under suitable conditions furthermore we consider acceleration of the uzawa pps methods by krylov subspace like gmres methods and discuss the spectral properties of the corresponding preconditioned matrix numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners

on hermitian and skew hermitian splitting iteration methods for the linear matrix equation axb c in this paper a hermitian and skew hermitian splitting hss iteration method for solving the linear matrix equation axb c is proposed this method is formed by extending the corresponding hss iteration method for solving ax b the analysis shows that the hss iteration method will converge under certain assumptions moreover the optimal parameter of this iteration method is presented numerical results show that this new method is more efficient and robust than existing methods

on the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions we prove the existence of convex classical solutions for a general multidimensional multilayer free boundary problem the geometric context of this problem is a nested family of closed convex surfaces except for the innermost and outermost surfaces which are given these surfaces are interpreted as unknown layer interfaces where the layers are the bounded annular domains between them each unknown interface is characterized by a quite general nonlinear equation called a joining condition which relates the first derivatives along the interface of the capacitary potentials in the two adjoining layers as well as the spatial variables a well known special case of this problem involves several stationary immiscible two dimensional flows of ideal fluid related along their interfaces by bernoulli s law

on positive definite and skew hermitian splitting iteration methods for continuous sylvester equation ax xb c in this paper we present a positive definite and skew hermitian splitting pss iteration method for continuous sylvester equations ax xb c with positive definite semi definite matrices the theoretical analysis shows that the pss iteration method will converge unconditionally and the optimal parameter of the new method is presented moreover to reduce the computing cost an inexact variant of the pss iteration method ipss and the analysis of its convergence property in detail have been established numerical results show that this new method and its inexact invariant are efficient and robust solvers for this class of continuous sylvester equations

sor waveform relaxation methods for stochastic differential equations waveform relaxation for solving large scale systems of ordinary differential equations has extended to stochastic differential equations sdes this paper deals with the acceleration of the gauss sediel waveform relaxation method for sdes by successive overrelaxation sor techniques a sufficient condition of the linear convergence is obtained for the continuous time sor waveform relaxation method the discrete time sor waveform relaxation method used in an actual implementation was also studied we first prove that the general numerical schemes are convergent and then the particular method based on euler scheme linearly converge to the euler approximate solution of sdes at last the theory is applied to a model problem and checked against results obtained by numerical experiments

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

analysis of a free boundary at contact points with lipschitz data in this paper we consider a minimization problem for the functional j u b 1 u 2 2 u 0 2 u 0 in the upper half ball b 1 n n 2 subject to a lipschitz continuous dirich

convexity of level sets for elliptic problems in convex domains or convex rings two counterexamples this paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings constant boundary conditions are imposed on the single component of the boundary when the domain is convex or on each of the two components of the boundary when the domain is a convex ring a function is called quasiconcave if its superlevel sets defined in a suitable way when the domain is a convex ring are all convex in this paper we prove that the superlevel sets of the solutions do not always inherit the convexity or ring convexity of the domain namely we give two counterexamples to this quasiconcavity property the first one for some two dimensional convex domains and the second one for some convex rings in any dimension

a gradient bound for the grad kruskal kulsrud functional ab stract we announce a gradient bound for minimizers of a variational problem arising in the grad kruskal kulsrud model for the equilibrium of a confined plasma the variational problem involves derivatives of the nondecreasing rearrangement of minimizers in the case of a convex domain our results answer in the negative a question raised by grad concerning the possibly singular behavior of the magnetic field at the point of maximum flux it completes a researchprogram begun in 1983 in whichan approach using free boundaries was initiated to handle the nonlinear nonlocal nature of the problem limiting minimizers of the variational problem are shown to have bounded gradient and satisfy a weak equation which for one model problem takes the form where are respectively the nondecreasing rearrangement and distribution function of plasmas and variational problems with global constraints the grad kruskal kulsrud model hereafter gkk ghs kk for the confinement of a thermonuclear plasma is a realistic one in which the nonlinearities are nonlocal nonlinear functionals of the solution here we apply the variational problem to axisymmetric magnetic bottles known as tokamaks ch these devices are at the forefront of experimental research today tftr at princeton jet at culham associated with the plasma in equilibrium is an energy expressed in terms of the magnetic field b 1991 mathematics subject classification primary 35r35 35d10 34jxx secondary 35q35

free transmission problems we study transmission problems with free interfaces from one random medium to another solutions are required to solve distinct partial differential equations rm l and rm l within their positive and negative sets respectively a corresponding flux balance from one phase to another is also imposed we establish existence and l infty bounds of solutions we also prove that variational solutions are non degenerate and develop the regularity theory for solutions of such free boundary problems

asymmetric hermitian and skew hermitian splitting methods for positive definite linear systems in this paper efficient iterative methods for the large sparse non hermitian positive definite systems of linear equations based on the hermitian and skew hermitian splitting of the coefficient matrix are studied these methods include an asymmetric hermitian skew hermitian ahss iteration and its inexact version the inexact asymmetric hermitian skew hermitian iahss iteration which employs some krylov subspace methods as its inner process we theoretically study the convergence properties of the ahss method and the iahss method moreover the contraction factor of the ahss iteration is derived numerical examples illustrating the effectiveness of both ahss and iahss iterations are presented

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

some new monotonicity formulas and the singular set in the lower dimensional obstacle problem we construct two new one parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem the first one is a family of weiss type formulas geared for points of any given homogeneity and the second one is a family of monneau type formulas suited for the study of singular points we show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity this allows to prove a structural theorem for the singular set our approach works both for zero and smooth non zero lower dimensional obstacles the study in the latter case is based on a generalization of almgren s frequency formula first established by caffarelli salsa and silvestre

efficient preconditioned nhss iteration methods for solving complex symmetric linear systems abstract based on the new hss nhss iteration method introduced by pour and goughery 2015 we propose a preconditioned variant of nhss p nhss and an efficient parameterized p nhss ppnhss iteration methods for solving a class of complex symmetric linear systems the convergence properties of the p nhss and the ppnhss iteration methods show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess when the parameters are properly chosen moreover we discuss the quasi optimal parameters which minimize the upper bounds for the spectral radius of the iteration matrices numerical results show that the ppnhss iteration method is superior to several iteration methods whether the experimental optimal parameters are used or not

a single step iteration method for non hermitian positive definite linear systems abstract an efficient single step iteration method is presented for solving the large sparse non hermitian positive definite linear systems we theoretically prove that this method converges to the unique solution of the system of linear equations under suitable restrictions moreover we derive an upper bound for the spectral radius of the new iteration matrix furthermore we consider acceleration of the new iteration by krylov subspace methods and some special properties of the new preconditioned matrix are proposed numerical experiments on a few model problems are presented to further examine the effectiveness of our new method