long time regularity of solutions of the hele shaw problem abstract in this paper we study the long time behavior of solutions of the one phase hele shaw problem without surface tension we show that after a finite time solutions of the hele shaw problem become starshaped and lipschitz continuous in space based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed

invertibility of a linear combination of two matrices and partial orderings abstract in this paper we present some results relating different matrix partial orderings and the invertibility of a linear combination of ep matrices

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

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

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

fredholmness of a linear combination of operators abstract although it is well known that fredholmness of the linear combination p q c 0 0 does not depend on the choice of scalars if p q b h are idempotents no necessary and sufficient conditions for fredholmness of this linear combination are yet known except for the special case when p and q are orthogonal projectors in this paper using a completely different approach and some results on completion problems of operator matrices we give necessary and sufficient conditions for fredholmness of a sum of two idempotents also we will discuss the more general question when the sum of two operators is a fredholm operator and consider some special cases when fredholmness of a linear combination of two operators is independent of the choice of the scalars on the other hand some special classes of operators for which a linear combination of two operators depends of the choice of the scalars are listed a new proof of a well known result is given

the hele shaw asymptotics for mechanical models of tumor growth models of tumor growth now commonly used present several levels of complexity both in terms of the biomedical ingredients and the mathematical description our first goal here is to formulate a free boundary model of hele shaw type a variant including growth terms starting from the description at the cell level and passing to the stiff limit in the pressure law of state in contrast with the classical hele shaw problem here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics a complete description requires the equation on the cell number density we then go on to consider a more complex model including the supply of nutrients through vasculature and we study the stiff limit for the involved coupled system

applications of cs decomposition in linear combinations of two orthogonal projectors abstract we study the spectrum and the rank of a linear combination of two orthogonal projectors we characterize when this linear combination is ep diagonalizable idempotent tripotent involutive nilpotent generalized projector and hypergeneralized projector also we derive the moore penrose inverse of a linear combination of two orthogonal projectors in a particular case the main tool used here is the cs decomposition

orthogonality conditions and asymptotic stability in the stefan problem with surface tension we prove nonlinear asymptotic stability of steady spheres in the two phase stefan problem with surface tension our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high order energy method

characterizations of the commutators and the anticommutator involving idempotents in this paper we characterize the idempotency the invertibility or generalized invertibility the positivity and the range relations of the commutator c pq qp and the anticommutator d pq qp involving idempotents particular attention is paid to the idempotent operators p and q and some further characteristics when p and q are positive are derived as well

remarks on scaling a model of witten sander type we try to prove rigorously that the perimeter of the large witten sander cluster does not scale as the square root of its area by making a forced comparison with the ill posed hele shaw problem of fluid dynamics the attempt is not completely successful nevertheless some interesting consequences of the comparison are derived

hele shaw type problems with dynamical boundary conditions abstract in this paper we study the nonlinear evolution equation of hele shaw type with dynamical boundary conditions that is the equation u t w f where u h w and h is the heaviside function with boundary condition x w t w k w g where denotes the outward normal vector of the fixed boundary of the domain we prove existence uniqueness and some qualitative properties of the solution

interfaces with a corner point in one dimensional porous medium flow we prove that for a large class of initial distributions the solutions of the initial value problem for the one dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time this happens for example if the initial pressure is o x 2 as x 0 we also give sufficient conditions for the smoothness of the interface which improve previous results

cusp development in hele shaw flow with a free surface free surface flow in a hele shaw cell is considered it has been known for some time that when the fluid region is contracting a finite time blow up can occur in which a cusp forms in the free surface the solution does not exist beyond the time of blow up examples are presented of this blow up and also of a new process in which a cusp of a different type forms in the free surface but after which the solution continues to exist after a transformation of the dependent variable the pressure the new cusps are related to permissible singularities in the free boundary of the obstacle problem

hydrodynamic limits for one dimensional particle systems with moving boundaries we analyze a new class of one dimensional interacting particle systems featuring random boundaries with a random motion that is coupled to the local particle density we show that the hydrodynamic limiting behavior in these systems corresponds to the solution of an appropriate

two phase problems for a class of fully nonlinear elliptic operators lipschitz free boundaries are c 1 γ we prove c 1 regularity of lipschitz free boundaries of two phase problems for a class of homogeneous fully nonlinear elliptic operators f d 2 u x x with holder dependence on x containing convex concave operators

functions of orthogonal projectors involving the moore penrose inverse several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors the results were established in various settings including finite and infinite dimensional vector spaces over either real or complex numbers but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors in the present paper these results are unified and reestablished in a generalized form in a complex euclidean vector space with the generalization obtained by relaxing the nonsingularity assumption and use of the moore penrose inverse instead of the ordinary inverse additionally several new formulae of the type are provided

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

parabolic equations in time dependent reifenberg domains in this paper we define time dependent parabolic reifenberg domains and study lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains the basic assumption is that the principal coefficients are of parabolic bmo space with small parabolic bmo seminorms it is shown that lp estimates hold for time dependent parabolic d reifenberg domains

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

homogenization of the hele shaw problem in periodic spatiotemporal media we consider the homogenization of the hele shaw problem in periodic media that are inhomogeneous both in space and time after extending the theory of viscosity solutions into this context we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous hele shaw type problem with a general possibly nonlinear dependence of the free boundary velocity on the gradient moreover the free boundaries converge locally uniformly in hausdorff distance

on a disjoint idempotent decomposition for linear combinations produced from n commutative tripotent matrices abstract it has been established a 3 n term disjoint idempotent decomposition did for the linear combinations produced from n 2 commutative tripotent matrices their products and their products of power 2 at most the results obtained in this way generalize those in y tian a disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications linear multilinear algebra 59 2011 1237 1246 moreover an algorithm to get a did has been provided finally a numerical example has been given to exemplify the results

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

a dynamical mother body in a hele shaw problem it is well known that in the framework of the internal hele shaw problem the fluid cannot be fully removed through a single point sink because of a cusp formation occurring prior to the moving boundary reaching the sink unless the initial domain is a circle with a sink located in its center we give a definition of a dynamical mother body and use it for developing an algorithm of complete removal of a fluid droplet having an algebraic boundary to illustrate our theory we consider examples where fluid can be completely removed through the sinks distributed along arcs of curves and or points

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

properties of some balayage operators with applications to quadrature domains and moving boundary problems in this paper we define and derive some properties of a fairly wide class of balayage operators we also apply our results to obtain geometric information about quadrature domains and solutions of certain moving boundary problems e g for hele shaw flows actually it was our interest in these applications that was the starting point for the present investigations and the balayage operators were developed rather as a tool our balayage operators are denoted f fp fp r and they depend on an open set r c i and a measure p in r actually we always assume that p has a density also denoted p in l r what f does is replace a given measure p by the nearest one in the energy norm v f p which satisfies v 5 p in r this presupposes that fi has finite energy if this is not the case there is another definition which is the one we will actually use among all distributions u in irn satisfying u i 17 the newtonian potential of p in irn and au i p in r there is a largest one u vp then f p ai by definition the operator of interest in the above mentioned applications is f with r irn p 1 the definition of f y when u has finite energy can be formulated as an elliptic variational inequality or linear complementarity problem and this is one of the standard tools when dealing with free boundary problems of the above kind see l 4 however we think that the balayage point of view is quite natural in our context the reason for considering the balayage operators for more general r and p than r rn p 1 is firstly that we need them in the proof of one of our theorems theorem 4 1 secondly and perhaps more important is that we think that the more general operators have quite a lot of intrinsic interest in particular as they turn out to contain classical balayage of positive measures as a special case namely with r bounded and p 0 recently we also learnt that discrete or numerical versions of these operators have been developed and applied in geophysics during the last three decades by dimiter zidarov who calls the balayage process partial gravi equivalent mass scattering see 5 and also for example 6 the paper is organized as follows in section 2 we define and establish the basic properties of our operators at the end we also explain their relation to quadrature domains and certain moving boundary problems section 3 is devoted to one single theorem theorem 3 1 and its corollaries these results concern the f with r lrn p 1 and give rather good geometric

two phase free boundary problems for parabolic operators smoothness of the front we continue to develop the regularity theory of general two phase free boundary problems for parabolic operators in a 2010 paper we establish the optimal lipschitz regularity of a viscosity solution under the assumptions that the free boundary is locally a flat lipschitz graph and a nondegeneracy condition holds here on one side we improve this result by removing the nondegeneracy assumption on the other side we prove the smoothness of the front the proof relies in a crucial way on a local stability result stating that for a certain class of operators under small perturbations of the coefficients flat free boundaries remain close and flat 2013 wiley periodicals inc

nonsingularity of the difference and the sum of two idempotent matrices abstract gro s and trenkler 1 pointed out that if a difference of idempotent matrices p and q is nonsingular then so is their sum and koliha et al 2 expressed explicitly a condition which combined with the nonsingularity of p q ensures the nonsingularity of p q in the present paper these results are strengthened by showing that the nonsingularity of p q is in fact equivalent to the nonsingularity of any combination ap bq cpq where a 0 b 0 a b c and the nonsingularity of p q is equivalent to the nonsingularity of any combination ap bq cpq where a 0 b 0 a b c

a note on the estimation of variances in state space models using the maximum a posteriori procedure this note examines an estimation procedure for the unknown parameters in a state space model proposed by tsang glover and bach the method is based on the maximum a posteriori map principle contrary to the claims of tsang et al it is shown that the algorithm performs very poorly compared to maximum likelihood some insight into the shortcomings of the map procedure is obtained by comparing it to the computation of maximum likelihood estimators by the em algorithm

some free boundary problems recast as nonlocal parabolic equations abstract in this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold the canonical examples would be one phase hele shaw flow as well as its two phase analog we also treat nonlinear versions of both one and two phase problems in the special class of free boundaries that are graphs over r d we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal fractional nonlinear parabolic equation for functions on r d our main observation is that the free boundary condition defines a nonlocal operator having what we call the global comparison property a consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases one obtains in a uniform treatment for all of the problems one and two phase a propagation of modulus of continuity for viscosity solutions of the free boundary flow

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

nonsingularity of linear combinationsof idempotent matrices abstract gros and trenkler siam j matrix anal appl 21 1999 390 pointed out that if a difference of idempotent matrices p 1 and p 2 is nonsingular then so is their sum and koliha et al linear algebra appl in press expressed explicitly a condition which combined with the nonsingularity of p 1 p 2 ensures the nonsingularity of p 1 p 2 in the present note these results are strengthened by showing that the nonsingularity of p 1 p 2 is in fact equivalent to the nonsingularity of any linear combination c 1 p 1 c 2 p 2 wherein c1 c2 0 some other nonsingularity type relationships referring to linear combinations of p 1 and p 2 are also established

regularity of the free boundary for a bernoulli type parabolic problem with variable coefficients abstract in this work the regularity of the free boundary for a class of parabolic free boundary problems with variable coefficients is studied under the main hypothesis that the free boundary is lipschitz the main result is that the free boundary is in fact c 1 surfaces thereby implying that the viscosity solution is in fact classical in this case

two phase problems for linear elliptic operators with variable coefficients lipschitz free boundaries are c 1 γ we prove c 1 regularity of lipschitz free boundaries of two phase problems for linear elliptic operators with holder continuous coefficients

on an inverse type problem for the heat equation in parabolic regular graph domains in this paper we prove some results concerning inverse free boundary type problems below the continuous threshold for the heat equation in the setting of parabolic regular graph domains

an embedding method for the numerical solution of the cathode design problem in electrochemical machining an embedding method is considered for the numerical determination of the cathode shape for a given anode shape in a two dimensional electrolytic cell reference is made to an analytic solution to illustrate the difficulties which have to be overcome by any numerical method applied to this ill posed problem the embedding method searches amongst the direct solutions i e anodes for given cathodes since these resulting from a well posed problem are less troublesome to compute for anodes of the form g x x the transverse direction the embedding method gives good results for less than 30 60 percent of the for which the cathode first contains cusps or becomes triple valued it is shown that the embedding method performs better than some other numerical methods in overcoming the error growth difficulties associated with the illposedness of the problem

on the uniqueness of a solution of a two phase free boundary problem in this paper we study the uniqueness problem of a two phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data we show that in general the comparison principle between the sub and super solutions does not hold and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter examples in various cases in any dimension in one dimension a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed in fact the critical case signifies the change from uniqueness to non uniqueness of a solution of the free boundary problem non uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes such as melting of ice or solidification of water even with the same prescribed boundary data however we prove that a uniqueness theorem is true for the initial boundary value problem of an e evolutionary problem which is the smoothed two phase parabolic free boundary problem

canonical angles of unitary spaces and perturbations of direct complements abstract let n plane1d 4b2 be a given direct sum decomposition we determine the largest number r i such that all subspaces for which the gap between and plane1d 4b2 satisfies have the property dim plane1d 4b2 the problem involves angles between subspaces i e singular values of products of projections

on simultaneous signal estimation and parameter identification using a generalized likelihood approach a common approach to blind deconvolution of bernoulli gaussian processes consists of performing both signal restoration and hyperparameter identification through maximization of a single generalized likelihood criterion it is shown on a simple example that the resulting hyperparameter estimates may not converge toward any meaningful value therefore other more reliable approaches should be adopted whenever possible

characterizations and representations of the group inverse involving idempotents this paper is to present some results on the group invertibility of products and differences of idempotents in addition some formulae for the group inverse of sums differences and products of idempotents are established by using some given idempotents

global stability and decay for the classical stefan problem the classical one phase stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water this is accom plished by solving the heat equation on a time dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary we establish a global in time stability result for nearly spherical geometries and small tempera tures using a novel hybrid methodology which combines energy estimates decay estimates and hopf type inequalities 1 1 the problem formulation we consider the problem of global existence and asymptotic stability of classical solutions to the classical stefan problem describing the evolving free boundary between the liquid and solid phases the temperature of the liquid p t x and the a priori unknown moving phase boundary t must satisfy the following system of equations

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

alternative algorithms for the estimation of dynamic factor mimic and varying coefficient regression models this paper provides a general approach to the formulation and estimation of dynamic unobserved component models after introducing the general model two methods for estimating the unknown parameters are presented both are algorithms for maximizing the likelihood function the first is based on the method of scoring the second is the em algorithm a derivative free method each iteration of em requires a kalman filter and smoother followed by straightforward regression calculations the paper suggests using the em methods to quickly locate a neighborhood of the maximum scoring can then be used to pinpoint the maximum and calculate the information matrix

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

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

caloric measure in parabolic flat domains in this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant the logarithm of the density of caloric measure with respect to a certain projective lebesgue measure is of vanishing mean oscillation we also obtain a partial converse to this result our work generalizes to the heat equation recent work of kenig and toro for the laplacian

corner formation for the undercooled stefan problem in this paper a mechanism of corner formation for the one phase undercooled stefan problem in two and three dimensions is described by means of asymptotic methods

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

renormalized solutions for degenerate elliptic parabolic problems with nonlinear dynamical boundary conditions and l 1 data abstract we consider a degenerate elliptic parabolic problem with nonlinear dynamical boundary conditions assuming l 1 data we prove existence and uniqueness in the framework of renormalized solutions particular instances of this problem appear in various phenomena with changes of phase like multiphase stefan problems and in the weak formulation of the mathematical model of the so called hele shaw problem also the problem with non homogeneous neumann boundary condition is included

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

regularity of the free boundary for the one phase hele shaw problem abstract in this paper we prove that in a local neighborhood lipschitz continuous free boundary of a solution of the one phase hele shaw problem is indeed smooth if the solution is lipschitz continuous and non degenerate in the neighborhood

properties of the combinations of commutative idempotents this paper deals with idempotent operators i e p2 p and n potent operators i e pn p we characterize the invertibility the group invertibility and idempotency of the linear combinations of idempotent operators or n potent operators under certain commutativity property imposed on them

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

identifiability of unknown noise covariance matrices for some special cases of a linear time invariant discrete time dynamic system this paper investigates the identification of unknown noise covariance matrices q and r of an lti discrete time dynamic system two algorithms based on maximum a posteriori map and maximum likelihood ml cost functions are evaluated it is demonstrated that the cost functions exhibit local minima versus those elements of q and r which dominate the steady state output covariance matrix the following three special cases are considered 1 single input single output siso system 2 multiinput single output miso systems and 3 single input multioutput simo systems with a diagonal r for these special cases specific identifiability criteria are presented and verified by examples the improvement of the map algorithm over the ml algorithm is also demonstrated

application of variational inequalities to the moving boundary problem in a fluid model for biofilm growth vi betraktar ett problem med rorlig rand motsvarande ett gransfall i en fluid modell for biofilmstillvaxt som foreslagits av j dockery och i klapper siam j appl math 62 3 2001

velocity effects in unstable solidification the supercooled stefan problem with surface tension is considered as a model of unstable solidification with a general anisotropic curvature and velocity dependent boundary condition on the moving interface the problem is reformulated in terms of a history dependent singular integral equation for the velocity of the boundary using this equation a new linear stability analysis of a flat interface is carried out and the smoothing role of velocity dependence and the destabilizing effect of anisotropy are demonstrated the results disagree with previous analyses because transient effects due to the initial temperature field are included and numerical results are presented that confirm the analysis presented here it is found that velocity effects cannot increase the range of unstable modes beyond that permitted by surface tension

the drazin inverse of the linear combinations of two idempotents in the banach algebra abstract in this paper some drazin inverse representations of the linear combinations of two idempotents in a banach algebra are obtained moreover we present counter examples to and establish the corrected versions of two theorems by cvetkovic ilic and deng

uniqueness and existence results on the hele shaw and the stefan problems in this paper we introduce a notion of viscosity solutions for the one phase hele shaw and stefan problems when there is no surface tension we prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of porous medium equation to those of the hele shaw problem

singularities in hele shaw flows we study analytically and numerically singularities in an interface flow driven by surface tension and a sink in a two dimensional hele shaw cell our analysis proves that singularity formation is inevitable in generic situations our numerical simulations suggest that the hele shaw solution develops singularities via the interface reaching the sink before all the fluid is sucked out a corner is formed right at the tip of the interface that touches the sink

the nullity and rank of linear combinations of idempotent matrices abstract baksalary and baksalary j k baksalary o m baksalary nonsingularity of linear combinations of idempotent matrices linear algebra appl 388 2004 25 29 proved that the nonsingularity of p 1 p 2 where p 1 and p 2 are idempotent matrices is equivalent to the nonsingularity of any linear combinations c 1 p 1 c 2 p 2 where c 1 c 2 0 and c 1 c 2 0 in the present note this result is strengthened by showing that the nullity and rank of c 1 p 1 c 2 p 2 are constant furthermore a simple proof of the rank formula of gros and trenkler j gros g trenkler nonsingularity of the difference of two oblique projectors siam j matrix anal appl 21 1999 390 395 is obtained

the difference and sum of projectors abstract the aim of this paper is to present new results on the invertibility of the sum of projectors and new relations between the nonsingularity of the difference and the sum of projectors we also give simple proofs without referring to rank theory of some results on the nonsingularity of the difference of projectors obtained by gros and trenkler siam j matrix anal appl 21 1999 390

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

on invertibility of combinations of k potent operators abstract some properties of combinations c 1 p 1 c 2 p 2 c 3 p 1 s p 2 k 1 s where p 1 and p 2 are two different nonzero k potents c 1 c 2 and c 3 are three nonzero complex numbers and positive integers k 2 and s k 1 are obtained furthermore the invertibility the group invertibility and the k potency of the linear combinations of k potents are investigated under certain commutativity properties imposed on them

regularity for the one phase hele shaw problem from a lipschitz initial surface in this paper we show that if the lipschitz constant of the initial free boundary is small then for small positive time the solution is smooth and satisfies the hele shaw equation in the classical sense a key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data

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

well posedness for the classical stefan problem and the zero surface tension limit we develop a framework for a unified treatment of well posedness for the stefan problem with or without surface tension in the absence of surface tension we establish well posedness in sobolev spaces for the classical stefan problem we introduce a new velocity variable which extends the velocity of the moving free boundary into the interior domain the equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature solutions to the classical stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation in which the moving free boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem with our methodology we simultaneously find the required stability condition for well posedness and obtain new estimates for the regularity of the moving free boundary finally we prove that solutions of the stefan problem with positive surface tension sigma converge to solutions of the classical stefan problem as sigma to 0

local and global existence results for anisotropic hele shaw flows in this paper we study a moving boundary problem for an anisotropic two phase hele shaw flow using a regularization technique we prove existence of a local solution under suitable conditions on the initial free boundary we obtain a global solution and study its asymptotic behaviour

on the projectors ff and f f abstract a particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors namely ff and f f determined by a complex square matrix f and its moore penrose inverse f various functions of the projectors are considered from the point of view of their nonsingularity idempotency nilpotency or their relation to the known classes of matrices such as ep bi ep gp dr or sr this part of the paper was inspired by benitez and rakocevic j benitez v rakocevic matrices a such that aa a a are nonsingular appl math comput 217 2010 3493 3503 further characteristics of ff and f f with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues are derived as well besides establishing selected exemplary results dealing with ff and f f the paper develops a general approach whose applicability extends far beyond the characteristics provided therein

on the perturbation of the group generalized inverse for a class of bounded operators in banach spaces abstract given a bounded operator a on a banach space x with drazin inverse a d and index r we study the class of group invertible bounded operators b such that i a d b a is invertible and r b n a r 0 we show that they can be written with respect to the decomposition x r a r n a r as a matrix operator b b 1 b 12 b 21 b 21 b 1 1 b 12 where b 1 and b 1 2 b 12 b 21 are invertible several characterizations of the perturbed operators are established extending matrix results we analyze the perturbation of the drazin inverse and we provide explicit upper bounds of b a d and b b a d a we obtain a result on the continuity of the group inverse for operators on banach spaces

obstacle problem for a class of parabolic equations of generalized p laplacian type abstract we study nonlinear parabolic pdes with orlicz type growth conditions the main result gives the existence of a unique solution to the obstacle problem related to these equations to achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution moreover we prove that if the obstacle is continuous so is the solution

regularity of solutions to a parabolic free boundary problem with variable coefficients abstract in this work the optimal lipschitz regularity of viscosity solutions for a type of parabolic free boundary problem with variable coefficients is proved under the main assumption that the free boundary is lipschitz