stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with dirichlet boundary conditions we prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate markov diffusion process in particular our article focuses on the heston stochastic volatility process which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half plane the generator of this process with killing called the elliptic heston operator is a second order degenerate elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half plane in mathematical finance solutions to terminal boundary value or obstacle problems for the parabolic heston operator correspond to value functions for american style options on the underlying asset

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

numerical methods for porous medium equation by an energetic variational approach abstract we study numerical methods for porous media equation pme there are two important characteristics the finite speed propagation of the free boundary and the potential waiting time which make the problem difficult to handle based on different dissipative energy laws we develop two numerical schemes by an energetic variational approach firstly based on f log f as the total energy form of the dissipative law we obtain the trajectory equation and then construct a fully discrete scheme it is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy next based on 1 2 f as the total energy form of the dissipation law we construct a linear numerical scheme for the corresponding trajectory equation both schemes preserve the corresponding discrete dissipation law meanwhile under some smoothness assumption both schemes are second order convergent in space and first order convergent in time each scheme yields a good approximation for the solution and the free boundary no oscillation is observed for the numerical solution around the free boundary furthermore the waiting time problem could be naturally treated which has been a well known difficult issue for all the existing methods due to its linear nature the second scheme is more efficient

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

eventual c regularity and concavity for flows in one dimensional porous media we study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation u t eft u m ight xx inq r imes eft 0 nfty ight ueft x 0 ight u 0 eft x ight forx n r

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

well posedness for a class of thin film equations with general mobility in the regime of partial wetting we establish well posedness for the family of thin film equations left begin array ll h t h n h xxx x 0 quad rm in h 0 h 0 h x 1 quad quad rm on partial h 0 end array right 1 with n in 0 frac 14 5 backslash 1 2 the model 1 with n in 0 3 has been used to describe the evolution of a capillary driven thin liquid droplet on a solid substrate in terms of its height profile h geqq 0 the family of thin film equations 1 provides a model problem to investigate contact line propagation in fluid dynamics under relaxed slip conditions the parabolicity of the fourth order parabolic problem degenerates at the free boundary which leads to a loss of regularity at the moving contact point our solutions are regular in terms of the two variables d x and d x 3 n where d x is the distance to the free boundary the main technical difficulty in the analysis of 1 is related to the loss of regularity at the contact points

on the theory of late stage phase separation in off critically quenched binary systems the late stage of phase separation in off critically quenched binary systems is studied from an unifying viewpoint by numerically solving not only the kinetic equation for the single droplet distribution function r t but also the linear equation with the source term for the structure function s k t both of which were recently derived by the present authors to study the dynamics of phase separation the scaling behaviors r t r t 4fl r r t and s k t km t 3 l kkm t are thus recovered where r t is the average droplet radius and km t the peak position of s k t as a function of k the temporal power behavior of r and km are also shown to agree with those expected on the basis of the lifshitz slyozov wagner theory the universal features of the time independent scaling functions fl and l x are thus investigated explicitly including their volume fraction dependence and the x4 dependence of l x for small x the results are in good agreement with experiments for both scaling functions

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

statistical mechanical theory of coarsening of spherical droplets abstract a new statistical mechanical theory of diffusion controlled droplet coarsening is presented with the aid of a scaling expansion method a spatial graining is carried out in a manner consistent with an expansion in droplet volume fraction to obtain kinetic equations for a single distribution function of droplets it is shown that there two characteristic stages of coarsening depending on their space time scales an intermediate stage and a late stage in both stages new kinetic equations are systematically derived to order these equations have two terms at order a collisionless drift term and a collision term the collision term is shown to be different from the conventional encounter integral discussed by lifshitz and slyozov since the former is of order and describes distant soft collisions while the latter is of order and describes close hard collisions it is shown in both stages that the mean droplet radius increases as the cube root of the time t 1 3 a scaling behavior of the distribution function is also found in both stages in particular in the late stage this scaling behavior is shown to coincide with that obtained by lifshitz and slysov in the limit o it is also pointed out that a naive expansion in powers of breaks down due to the long range nature of interactions among droplets through diffusions fluctuations around the kinetic equations are also explicitly explored they are shown to be nonthermal fluctuations generated by the soft collision process and to be small especially in the late stage they are shown to obey a linear gaussian markov process satisfying a fluctuation dissipation relation of the second kind

elementary derivation of kinetic equations for ostwald ripening a direct and elementary derivation is presented for the kinetic equation of the single droplet distribution function and the variance equation that describe ostwald ripening the derivation is basically algebraical and the screening effects of diffusional interactions among droplets is taken into account automatically by a projection technique the results correct up to the square root of the volume fraction confirm the kinetic equations derived previously by tokuyama and kawasaki except for a correction to the variance equation

local continuity and asymptotic behaviour of degenerate parabolic systems abstract we study the local holder continuity and the asymptotic behaviour of solution u u 1 u k of the degenerate system u t i m u m 1 u i for m 1 and i 1 k which describes the population densities of k species whose diffusions are determined by their total population density u u 1 u k for the local holder continuity we adopt the intrinsic scaling and iteration arguments of degiorgi moser and dibenedetto under some regularity conditions we also prove that the population density of i th species with the population m i converges in c s to the function m i m b m x t as t where b m is the barenblatt profile of the standard porous medium equation with l 1 mass m m 1 m k as a consequence of asymptotic behaviour it is shown that each density becomes a concave function after a finite time

on the martingale problem for degenerate parabolic partial differential operators with unbounded coefficients and a mimicking theorem for ito processes using results from our companion article 26 on a schauder approach to existence of solutions to a degenerate parabolic partial differential equation we solve three intertwined problems motivated by probability theory and mathematical finance concerning degenerate diffusion processes we show that the martingale problem associated with a degenerate elliptic differential operator with unbounded locally holder continuous coefficients on a half space is wellposed in the sense of stroock and varadhan second we prove existence uniqueness and the strong markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable holder continuity properties third for an ito process with degenerate diffusion and unbounded but appropriately regular coefficients we prove existence of a strong markov process unique in the sense of probability law whose onedimensional marginal probability distributions match those of the given ito process

theory of morphological evolution in ostwald ripening abstract the multiple expansion theory for ostwald ripening in a three dimensional system is presented analogously to the case in two dimensional coarsening t imaeda and k kawasaki physica a 164 1990 335 the migration and the shape deformation of droplets caused by the diffusional interaction are taken into account in our kinetic equations of motion the derived equations can be solved numerically within a fairly resonable computation time the role of the mullins sekerka instability in the coarsening kinetics is examined

dynamics of fluctuations in ostwald ripening a new equation of motion for the structure function abstract a new method of finding an equation of motion for the structure function s k t from the kinetic equations recently derived for the late stage of phase separation is presented s k t is shown to be scaled by two kinds of characteristics lengths the average droplet radius r t and the screening length l t as s k t r t 3 s x q q 1 2 with x k r t and q kl t where q is the volume fraction of the minority phase and r t and l t satisfy the power laws t q 1 2 r t and r t t 1 3 the scaled structure function s x q q 1 2 is found to obey a first order differential equation with two source terms which originate from two kinds of fluctuations initial thermal fluctuations and nonthermal fluctuations generated by soft collisions among droplets the soft collision process is shown to produce a new type of narrowing of s x q q 1 2 which is different from the usual one that occurs as q increases the results are compared with other theories

lagrangian approach to the study of level sets application to a free boundary problem in climatology we study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation u t delta u in a mathbb h u mu quad rm in quad q it omega times 0 t where omega subset mathbb r n is a ring shaped domain a and are given positive constants mathbb h cdot is the heaviside maximal monotone graph mathbb h s 1 if s 0 mathbb h 0 0 1 mathbb h s 0 if s 0 such equations arise in climatology the so called budyko energy balance model as well as in other contexts such as combustion we show that under certain conditions on the initial data the level sets it gamma mu mathbf x t u mathbf x t mu are n dimensional hypersurfaces in the x t space and show that the dynamics of is governed by a differential equation which generalizes the classical darcy law in filtration theory this differential equation expresses the velocity of advancement of the level surface through spatial derivatives of the solution u our approach is based on the introduction of a local set of lagrangian coordinates the equation is formally considered as the mass balance law in the motion of a fluid and the passage to lagrangian coordinates allows us to watch the trajectory of each of the fluid particles

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

c1 1 regularity for degenerate elliptic obstacle problems abstract the heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half plane the generator of this process with killing called the elliptic heston operator is a second order degenerate elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half plane in mathematical finance solutions to the obstacle problem for the elliptic heston operator correspond to value functions for perpetual american style options on the underlying asset with the aid of weighted sobolev spaces and weighted holder spaces we establish the optimal c 1 1 regularity up to the boundary of the half plane for solutions to obstacle problems for the elliptic heston operator when the obstacle functions are sufficiently smooth

large scale simulations of the two dimensional cahn hilliard model abstract we review some recent numerical studies of the cahn hilliard model in two dimensions for phase separation with different values of the minority area fraction we find that dynamical scaling is satisfied at sufficiently late times for the pair correlation function the structure factor and the droplet distribution function we study how the shape of these scaling functions change with the area fractions and compare these results with available theoretical predictions the time dependence of the characteristics length is consistent with an asymptotic growth law exponent 1 3 for all area fractions

numerical study of the porous medium equation with absorption variable exponents of nonlinearity and free boundary abstract in this paper we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2d domains with moving boundaries the boundary s movement is governed by an equation prompted by the darcy law and the spatial discretization is defined by a triangulation of the domain at each finite element the solution is approximated by piecewise polynomial functions of degree r 1 using lagrange interpolating polynomials in area coordinates the vertices of the triangles move according to a system of differential equations which is added to the equations of the problem the resulting system is converted into a system of ordinary differential equations in time variable which is solved using a suitable integrator the integrals that arise in the system of ordinary differential equations are calculated using the gaussian quadrature finally we present some numerical results of application of this technique

fluctuation effects in coagulation processes abstract we study diffusion controlled coagulation processes in the system with a finite density of droplets using the fock space formalism we find that the effective reaction rate has a correction to smoluchowski s classical rate constant for three dimensional systems we also predict the scaling form of the scattering function and find that our results qualitatively agree with experimental results

worn stones with flat sides all time regularity of the interface we study the all time regularity of the free boundary problem associated to the deformation of a compact weakly convex surface in 3 with a flat side by its gaussian curvature we show that under certain necessary regularity and non degeneracy initial conditions the interface separating the flat from the strictly convex side remains smooth on 0 t tc up to the vanishing time tc of the flat side

evaluating whittaker functions and maass forms for 3 ℤ we present and compare several algorithms for evaluating jacquet s whittaker functions for sl 3 7 the most suitable algorithm is then applied to the problem of evaluating a maass form for sl 3 7 with known eigenvalues and fourier coefficients

encounter effects on the dynamics of fluctuations in phase separations of off critically quenched binary systems abstract the encounter effect on the dynamics of fluctuations in phase separations of off critically quenched binary systems is studied by performing three dimensional computer simulations of the droplet growth model proposed recently by the authors the dynamical behavior of the scattering structure function including the dynamical scaling the temporal power laws and the crossovers have been investigated at higher volume fractions the encounter effect is shown to strongly influence not only the dynamics of the single droplet size distribution function leading to flattening and broadening of the distribution function but also the dynamics of fluctuations in phase separating systems leading to sharpening and narrowing the structure function

theory of phase separation dynamics in quenched binary mixtures a systematic theory of the dynamics of phase separation of quenched binary systems into metastable states is presented not only the kinetic equation for the single droplet size distribution function f r t but also the linear equation for the structure function s k t which are valid over the entire time region after the nucleation stage are derived from a unified point of view the functions f r t and s k t are shown to satisfy the dynamical scaling relations f r t n t r t f r r t and s k t mathit k mathit m t mathrm ensuremath mathit d ensuremath phi t mathrm ensuremath delta ensuremath psi k mathit k mathit m t with mathit k mathit m mathrm ensuremath 1 t ensuremath propto r mathrm ensuremath phi 1 mathit d where n is the number density of the minority phase r the average droplet radius ensuremath phi the volume fraction of the minority phase mathit k mathit m the peak position of s k t and d 3 here three characteristic stages are then shown to exist after the nucleation stage the growth stage with ensuremath delta 2 where the power laws are given by r ensuremath propto mathit t 1 2 n ensuremath propto mathit t 0 mathit k mathit m ensuremath propto mathit t 0 and ensuremath phi ensuremath propto mathit t 3 2 the intermediate stage with ensuremath delta 1 d where r ensuremath propto mathit t 1 4 n ensuremath propto mathit t mathrm ensuremath 2 3 mathit k mathit m ensuremath propto mathit t mathrm ensuremath 2 9 and ensuremath phi ensuremath propto mathit t 1 12 and the coarsening stage with ensuremath delta 1 d where r ensuremath propto mathit t 1 3 n ensuremath propto mathit t mathrm ensuremath 1 mathit k mathit m ensuremath propto mathit t mathrm ensuremath 1 3 and ensuremath phi ensuremath propto mathit t 0 two crossovers are then observed in the time exponents the time evolution of the scaling functions f ensuremath rho t and ensuremath psi x t are thus studied explicitly including their asymptotic behavior and their dependence on ensuremath phi

multipole expansion in the ostwald ripening theory abstract a multipole expansion theory for ostwald ripening is presented the theory takes into account morphological deformation as well as migration of coarsening droplets numerical computation are performed for the two dimensional ripening process even in a few mode truncation our results reproduce well the direct numerical solution of the laplace equation at a small spatial separation between droplets the importance of these degrees of freedom in the coarsening kinetics is discussed

motion by curvature of planar curves with end points moving freely on a line this paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line we first prove the existence and uniqueness of self similar shrinking solution then we show that the curve shrinks to a point in a self similar manner if initially the curve is a graph

numerical artifacts in the generalized porous medium equation why harmonic averaging itself is not to blame abstract the degenerate parabolic generalized porous medium equation gpme poses numerical challenges due to self sharpening and its sharp corner solutions for these problems we show results for two subclasses of the gpme with differentiable k p with respect to p namely the porous medium equation pme and the superslow diffusion equation spurious temporal oscillations and nonphysical locking and lagging have been reported in the literature these issues have been attributed to harmonic averaging of the coefficient k p for small p and arithmetic averaging has been suggested as an alternative we show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues here we investigate the causes of these numerical artifacts using modified equation analysis the modified equation framework can be used for any type of discretization we show results for the second order finite volume method the observed problems with harmonic averaging can be traced to two leading error terms in its modified equation this is also illustrated numerically through a modified harmonic method mhm that can locally modify the critical terms to remove the aforementioned numerical artifacts

anisotropic mesh adaptation for finite element solution of anisotropic porous medium equation abstract anisotropic porous medium equation apme is developed as an extension of the porous medium equation pme for anisotropic porous media a special analytical solution is derived for apme for time independent diffusion anisotropic mesh adaptation for linear finite element solution of apme is discussed and numerical results for two dimensional examples are presented the solution errors using anisotropic adaptive meshes show second order convergence

many body effects in two dimensional ostwald ripening abstract a theory for ostwald ripening in two dimensions is presented using a many body approach we show the existence of a screening length which had been previously assumed by marqusee j chem phys 81 1984 976 in two dimensional diffusion controlled growth processes we also confirm that the growth law of the critical droplet radius in our model is given by r c t t 1 3

lagrangian approach to the study of level sets ii a quasilinear equation in climatology we study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation u t delta p u f is an element of ah u mu in q omega x 0 t p is an element of 1 infinity where omega subset of r n is a ring shaped domain delta p u is the p laplace operator a and mu are given positive constants and h is the heaviside maximal monotone graph h s 1 if s 0 h 0 0 1 h s 0 if s 0 the mathematical models of this type arise in climatology the case p 3 was proposed and justified by p stone in 1972 we establish the conditions on the initial data which guarantee that the level sets gamma mu t x u x t mu are hypersurfaces study the regularity of gamma mu t and derive the differential equation that governs the dynamics of gamma mu t the analysis is based on the introduction of a system of lagrangian coordinates that transforms the moving surface gamma mu t into a stationary one

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

interfaces in multidimensional diffusion equations with absorption terms we study the properties of interfaces in solutions of the cauchy problem for the nonlinear degenerate parabolic equation ut um aup in rn 0 t n 1 2 3 with the parameters m 1 p 0 a 0 satisfying the condition m p 2 we show that the velocity of the interface t supp u x t is given by the formula v m m 1 um 1 t where is the solution of the degenerate elliptic equation div u aup 0 0 on t the first term expresses the classical darcy law while the second one models the presence of an external force that makes the interface to move in the inward direction we give explicit formulas which represent the interface t as a bijection from 0 it is proved that the solution u and its interface t are analytic functions of time t and that they preserve the initial regularity in the spatial variables we also show that the regularity of the interface velocity v with respect to the spatial variables is better than it was at the initial instant the analysis is based on a special coordinate transformation a local version of lagrangian coordinates that renders the free boundary stationary

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

parallel kinetic monte carlo simulations of two dimensional island coarsening the results of parallel kinetic monte carlo kmc simulations of island coarsening based on a bond counting model are presented our simulations were carried out both as a test of and as an application of the recently developed semirigorous synchronous sublattice sl algorithm by carrying out simulations over long times and for large system sizes the asymptotic coarsening behavior and scaled island size distribution isd were determined our results indicate that while cluster diffusion and coalescence play a role at early and intermediate times at late times the coarsening proceeds via ostwald ripening in addition we find that the asymptotic scaled isd is significantly narrower and more sharply peaked than the mean field theory prediction the dependence of the scaled isd on coverage is also studied our results demonstrate that parallel kmc simulations can be used to effectively extend the time scale over which realistic coarsening simulations can be carried out in particular for simulations of the late stages of coarsening with system size l 1600 and eight processors a parallel efficiency larger than 80 was obtained these results suggest that the sl algorithm is likely to be useful in the future in parallel kmc simulations of more complicated models of coarsening

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

regularity in sobolev spaces for the fast diffusion and the porous medium equation abstract the degenerate parabolic equation u t u m 1 u m 0 is considered in a cylinder 0 t under homogeneous dirichlet boundary values the regularity of weak solutions for the fast diffusion equation m 1 and the porous medium equation m 1 are investigated regularity of u and u m 1 u in weighted sobolev spaces and in fractional order nikolskii spaces are proved

evaluating jacquet s gl n whittaker function algorithms for the explicit symbolic and numeric evaluation of jacquet s whittaker function for the gl n k based generalized upper half plane for n 2 and an implementation for symbolic evaluation in the mathematica package gl n pack are described this requires a comparison of the different definitions of whittaker function which have appeared in the literature

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

on first order corrections to the lsw theory ii finite systems we consider first order corrections to the classical theory by lifshitz slyozov and wagner lsw for systems with a finite number of particles numerical simulations in v e fradkov etal phys rev e 53 3925 3932 1996 show a cross over in the scaling of the correction term from 1 3 to 1 2 is the volume fraction of particles when the system size becomes larger than the screening length we rigorously derive this cross over for the rate of change of the energy starting from the monopole approximation the proof exploits the fact that the rate of change of energy has a variational characterization

schauder a priori estimates and regularity of solutions to boundary degenerate elliptic linear second order partial differential equations abstract we establish schauder a priori estimates and regularity for solutions to a class of boundary degenerate elliptic linear second order partial differential equations furthermore given a c smooth source function we prove c regularity of solutions up to the portion of the boundary where the operator is degenerate boundary degenerate elliptic operators of the kind described in our article appear in a diverse range of applications including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance 10 25 generators of diffusion processes arising in mathematical biology 3 11 and the study of porous media 7 8

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

regularity of interfaces in diffusion processes under the influence of strong absorption we present a method of analysis which allows us to establish the interface equation and to prove lipschitz continuity of interfaces and solutions which appear in a large class of nonlinear parabolic equations and conservation laws posed in one space dimension its main feature is intersection comparison with travelling waves the method is explained on the following study case we consider the cauchy problem for the diffusion absorption model u t left u m right xx u p quad u ge 0 in the range of parameters m 1 0 1 where the support of the solution expands with time and the motion is governed by darcy s law in the strong absorption range there might appear shrinking interfaces and the interface evolution obeys a different mechanism previous methods have failed to provide an adequate analysis of the interface motion and regularity in such a situation

a classical perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate elliptic operators via holomorphic maps abstract we prove existence of solutions to boundary value problems and obstacle problems for degenerate elliptic linear second order partial differential operators with partial dirichlet boundary conditions using a new version of the perron method the elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by daskalopoulos and hamilton 9 in their study of the porous medium equation or the degeneracy of the heston operator 21 in mathematical finance existence of a solution to the partial dirichlet problem on a half ball where the operator becomes degenerate on the flat boundary and a dirichlet condition is only imposed on the spherical boundary provides the key additional ingredient required for our perron method surprisingly proving existence of a solution to this partial dirichlet problem with mixed boundary conditions on a half ball is more challenging than one might expect due to the difficulty in developing a global schauder estimate and due to compatibility conditions arising where the degenerate and non degenerate boundaries touch one cannot directly apply the continuity or approximate solution methods however in dimension two there is a holomorphic map from the half disk onto the infinite strip in the complex plane and one can extend this definition to higher dimensions to give a diffeomorphism from the half ball onto the infinite slab the solution to the partial dirichlet problem on the half ball can thus be converted to a partial dirichlet problem on the slab albeit for an operator which now has exponentially growing coefficients the required schauder regularity theory and existence of a solution to the partial dirichlet problem on the slab can nevertheless be obtained using previous work of the author and c pop 16 our perron method relies on weak and strong maximum principles for degenerate elliptic operators concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate elliptic operators and maximum and comparison principle estimates previously developed by the author 13