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

a randomized algorithm for two servers in cross polytope spaces it has been a long standing open problem to determine the exact randomized competitiveness of the 2 server problem that is the minimum competitiveness of any randomized online algorithm for the 2 server problem for deterministic algorithms the best competitive ratio that can be obtained is 2 and no randomized algorithm is known that improves this ratio for general spaces for the line bartal et al 1998 2 give a 15578 competitive algorithm but their algorithm is specific to the geometry of the line we consider here the 2 server problem over cross polytope spaces m 2 4 we obtain an algorithm with competitive ratio of 1912 and show that this ratio is best possible this algorithm gives the second non trivial example of metric spaces with better than2 competitive ratio the algorithm uses a design technique called the knowledge state technique a method not specific to m 2 4

infinite dimensional linear programming approach to singularstochastic control we consider a multidimensional singular stochastic control problem with state dependent diffusion matrix and drift vector and control cost depending on the position and direction of displacement of the controlled process the objective is to minimize the total expected discounted cost we write an equivalent infinite dimensional linear programming problem on a subspace of the space conjugate to cal c bbb r n times cal c bbb r n times b where b is the unit sphere in bbb r n we write a dual linear program and prove absence of duality gap the dual program characterizes the optimal cost function as a maximal solution to the variational inequality with gradient constraints

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

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

regularization by kinetic undercooling of blow up in the ill posed stefan problem it is well known that the one dimensional supercooled stefan problem possesses solutions that blow up in finite time the asymptotics of such solutions have been analyzed by herrero and velazquez european j appl math 7 1996 pp 119 150 here we consider the effect of kinetic undercooling as a regularizing mechanism to prevent the formation of such singularities and study the continuation of the solution through the near blow up regime the asymptotics of solutions and interfaces are described for small values of the kinetic undercooling parameter it is shown that in this limit the interface jumps over an interval determined by the latent heat and by the initial data specifically in dimensionless variables if the temperature profile at blow up is denoted by u x t c where t c is the finite blow up time then the interface jumps over the interval in which u x t c lambda where lambda is the latent heat

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

on the k server conjecture we prove that the work function algorithm for the k server problem has a competitive ratio at most 2 k 1 manasse et al 1988 conjectured that the competitive ratio for the k server problem is exactly k it is trivially at least k previously the best known upper bound was exponential in k our proof involves three crucial ingredients a quasiconvexity property of work functions a duality lemma that uses quasiconvexity to characterize the configuration that achieve maximum increase of the work function and a potential function that exploits the duality lemma

the stefan problem with small surface tension the stefan problem with small surface tension e is considered assuming that the classical stefan problem with e 0 has a smooth free boundary r we denote the temperature of the solution by 00 and consider an approximate solution 00 su for the case where e 7 0 e small we first establish the existence and uniqueness of u and then investigate the effiect of u on the free boundary r it is shown that small surface tension affects the free boundary r radically differently in the two phase problem than in the one phase problem 0 introduction in the classical formulation of the two phase stefan problem the temperatures ow and si of water and ice satisfy the following conditions on the interface tm xn t 0 0 1 vx 0w vx vx ivxo ot 0 2 ow oi the first condition is the cbnservation of energy the functions sw oi further satisfy the heat equation in the water and ice sets respectively as well as initial and boundary conditions on the fixed portions of the boundary for definiteness we shall take in this paper the initial geometry to be as in figure 1 namely the fixed boundary 0d is surrounded by water region g go d and the water is surrounded by ice the fixed boundary does not change in time but the free boundary the water ice interface will of course change with time the one phase stefan problem arises when oi 0 in the ice region or ow 0 in the water region taking for definiteness the case oi 0 the interface conditions are 0 3 vx ow vx4 st received by the editors october 20 1989 1980 mathematics subject classification 1985 revision primary 35r35 secondary 35k20 35k85 this work is partially supported by national science foundation grant dms 86 12880 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

linear programming and average optimality of markov control processes on borel spaces unbounded costs this paper is concerned with the linear programming formulation of markov control processes with borel state and action spaces and the average cost ac criterion the one stage cost function may be unbounded a linear program ep and its dual ep are introduced their values inf ep and sup ep bound the value say inf ac of the ac problem i e sup ep leq inf ac leq inf ep conditions are provided for the existence of no duality gap viz sup ep inf ep and also for strong duality so that both ep and ep are solvable and their optimal values satisfy max ep min ep the latter implies i the existence of an ac optimal control policy and that ii the ac optimality equation holds almost everywhere these results are applied to a general vector valued additive noise system with quadratic costs

linear programming formulation for optimal stopping problems optimal stopping problems for continuous time markov processes are shown to be equivalent to infinite dimensional linear programs over a space of pairs of measures under very general conditions the measures involved represent the joint distribution of the stopping time and stopping location and the occupation measure of the process until it is stopped these measures satisfy an identity for each function in the domain of the generator which is sufficient to characterize the stochastic process finite dimensional linear programs obtained using markov chain approximations are solved in two examples to illustrate the numerical accuracy of the linear programming formulation

existence of optimal policies for semi markov decision processes using duality for infinite linear programming semi markov decision processes on borel spaces with deterministic kernels have many practical applications particularly in inventory theory most of the results from general semi markov decision processes do not carry over to a deterministic kernel since such a kernel does not provide smoothness we develop infinite dimensional linear programming theory for a general stochastic semi markov decision process we give conditions general enough to allow deterministic kernels for solvability and strong duality of the resulting linear programs by using the developed linear programming theory we give conditions for the existence of a stationary deterministic policy for deterministic kernels which is optimal among all possible policies

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

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

an optimal on line algorithm for k servers on trees the k server problem is investigated when the metric space is a tree for this case an on line k competitive algorithm for k servers is presented the competitiveness ratio k is optimal the algorithm is memoryless in the sense that it does not use any information from the past

on the two phase stefan problem with interfacial energy and entropy abstract the classical stefan theory for the melting of a solid or the freezing of a liquid is too simplistic to describe phenomena such as supercooling in which a liquid supports temperatures below its freezing point or superheating the analog for solids or dendritic growth in which simple shapes evolve to complicated tree like structures this paper develops a general theory for two phase phenomena of this type it develops partial differential equations satisfied in the phase regions and free boundary conditions satisfied on the interface between phases and gives arguments which indicate that the resulting boundary value problems predict the formation of dendrites keywords equilibrium liapunoo functions mathematical physics

computation of sharp phase boundaries by spreading the planar and spherically symmetric cases abstract the sharp interface that arises from any of the major transition problems classical or modified stefan etc can be smoothed using the phase field approach as a numerical tool the basic idea is that the thickness of the interface can be regarded as a mathematical free parameter which can be stretched beyond its physical value for computational convenience the computations in one dimensional space and n dimensions with radial symmetry indicate that this is an efficient method for dealing with stiff equations and results in a very accurate interface determination without explicit tracking the question of optimizing the interfacial thickness with respect to grid size is also considered empirically the technique also provides a numerical verification of the concept of an unstable critical radius of solidification

sample path average optimality for markov control processes the authors consider a markov control process with borel state and actions spaces unbounded costs and under the long run sample path average cost criterion they prove that under very weak assumptions on the transition law and a moment assumption for the one step cost there exists a stationary policy with invariant probability distribution v that is sample path average cost optimal for v almost all initial states in addition every expected average cost optimal stationary policy is in fact liminf sample path average cost optimal and strongly expected average cost optimal

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