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

pointwise estimates and stability for dispersive diffusive shock waves we study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion along with the usual lax shocks such equations are known to admit undercompressive shocks we unify the treatment of these two cases by introducing a new wave tracking method based on instantaneous projection giving improved estimates even in the lax case another important feature connected with the introduction of dispersion is the treatment of a non sectorial operator an immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all l p norms p 1 for the lax case and p 1 for the undercompressive case our approach extends immediately to the case of certain scalar equations of higher order and would also appear suitable for extension to systems

asymptotic stability of traveling wave solutions for perturbations with algebraic decay abstract for a class of scalar partial differential equations that incorporate convection diffusion and possibly dispersion in one space and one time dimension the stability of traveling wave solutions is investigated if the initial perturbation of the traveling wave profile decays at an algebraic rate then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate and optimal intermediate results that combine temporal and spatial decay are obtained the proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations

parabolic minkowski convolutions of solutions to parabolic boundary value problems abstract we introduce a new kind of convolution which is a sort of parabolic version of the classical supremal convolution of convex analysis this operation allows us to compare solutions of different parabolic problems in different domains as examples of applications of our main result we study the parabolic concavity of solutions to parabolic boundary value problems analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term we also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time proving some optimality of the ball

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

lp decay rates for perturbations of degenerate scalar viscous shock waves abstract this contribution shows how to obtain l p decay rates with 2 p for zero mass perturbations of degenerate scalar viscous shock waves using energy methods the proof is based upon previous work by matsumura and nishihara comm math phys 165 1 1994 83 96 by extending their weighted energy estimates to l p norms and by obtaining sharp decay rates for the antiderivative of the perturbation with the aid of basic interpolation inequalities the analysis applies to shocks of all orders of degeneracy and it is elementary

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

distortion product otoacoustic emission input output characteristics in normal hearing and hearing impaired human ears dpoae input output i o functions were measured at 7f2 frequencies 1 to 8 khz f2 f1 1 22 over a range of levels 5 to 95 db spl in normal hearing and hearing impaired human ears l1 l2 was level dependent in order to produce the largest 2f1 f2 responses in normal ears system distortion was determined by collecting dp data in six different acoustic cavities these data were used to derive a multiple linear regression model to predict system distortion levels the model was tested on cochlear implant users and used to estimate system distortion in all other ears at most but not all f2 s measurements in cochlear implant ears were consistent with model predictions at all f2 frequencies the ears with normal auditory thresholds produced i o functions characterized by compressive nonlinear regions at moderate levels with more rapid growth at low and high stimulus levels as auditory threshold increased dpoae threshold increased accompanied by dpoae amplitude reductions notably over the range of levels where normal ears showed compression the slope of the i o function was steeper in impaired ears the data from normal hearing ears resembled direct measurements of basilar membrane displacement in lower animals data from ears with hearing loss showed that the compressive region was affected by cochlear damage however responses at high levels of stimulation resembled those observed in normal ears

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

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

asymptotic stability and bifurcation of time periodic solutions for the viscous burgers equation abstract we consider the dirichlet boundary value problem for the viscous burgers equation with a time periodic force on a one dimensional finite interval under the boundedness assumption on the external force we prove the existence of the time periodic solution by using the galerkin method and schaefer s fixed point theorem furthermore we show that this time periodic solution is unique and time asymptotically stable in the h 1 sense under an additional smallness condition on the external force it is naturally expected that when the amplitude of the external force increases and crosses a certain critical value the time periodic solution is no longer asymptotically stable in the last part of the article to support our theory numerical experiments are carried out to investigate the exchange of stabilities of the time periodic solutions when the amplitude of the force crosses the first critical value we numerically find this critical value at which the stable solutions turn into the unstable ones

power concavity in weakly coupled elliptic and parabolic systems abstract in this paper we start the investigation of concavity properties of solutions to systems of pde s in convex domains in particular we prove that suitable powers of solutions to some weakly coupled elliptic and parabolic systems are concave

on uniqueness of weak solutions for the thin film equation abstract in any number of space variables we study the cauchy problem related to the fourth order degenerate diffusion equation s h h h 0 this equation derived from a lubrication approximation also models the surface tension dominated flow of a thin viscous film in the hele shaw cell our focus is on uniqueness of weak solutions in the complete wetting regime when a zero contact angle between liquid and solid is imposed in this case we transform the problem by zooming into the free boundary and look at small lipschitz perturbations of a quadratic stationary solution in the perturbational setting the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide optimal conditions on the initial data under which a unique solution exists in fact this solution can be used to define a class of functions in which the original initial value problem has a unique weak solution moreover we show that the interface between empty and wetted regions is an analytic hypersurface in time and space

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

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

local tracking and stability for degenerate viscous shock waves abstract we study the pointwise behavior of perturbed degenerate sonic shock waves for scalar conservation laws with nonconstant diffusion building on the pointwise green s function approach of zumbrun and howard indiana u math j 47 3 1998 741 we extend the linear analysis to an equation with nonintegrable coefficients in lieu of working with the integrated equation we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems indiana u math j 47 3 1998 741

a schauder approach to degenerate parabolic partial differential equations with unbounded coefficients abstract motivated by applications to probability and mathematical finance we consider a parabolic partial differential equation on a half space whose coefficients are suitably holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half space we establish existence and uniqueness of solutions in weighted holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients in our companion article feehan and pop 12 we apply the main result of this article to show that the martingale problem associated with a degenerate elliptic partial differential operator is well posed in the sense of stroock and varadhan

distortion product otoacoustic emissions cochlear source contributions and clinical test performance it has been proposed that the clinical accuracy of distortion product otoacoustic emissions dpoaes is affected by the interaction of distortion and reflection sources contributing to the response this study evaluated changes in dichotomous decision test performance and threshold prediction accuracy when dpoae source contribution was controlled data were obtained from 205 normal and impaired ears with l2 ranging from 0 to 80db spl and f2 2 and 4khz data were collected for control conditions no suppressor f3 and with f3 presented at three levels that previously had been shown to reduce the reflection source contribution the results indicated that controlling source contribution with a suppressor did not improve diagnostic accuracy as reflected by relative operating characteristic curve area and frequently resulted in poorer test performance compared to control conditions likewise correlations between dpoae and behavioral thresholds were not strengthened when using the suppressors to control sou

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

asymptotic l p stability for transition fronts in cahn hilliard systems abstract we consider the asymptotic behavior of perturbations of transition front solutions arising in cahn hilliard systems on r such equations arise naturally in the study of phase separation processes and systems describe cases in which three or more phases are possible when a cahn hilliard system is linearized about a transition front solution the linearized operator has an eigenvalue at 0 due to shift invariance which is not separated from essential spectrum in cases such as this nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate our main result asserts that if initial perturbations are small in l 1 l then spectral stability a necessary condition for stability defined in terms of an appropriate evans function implies asymptotic nonlinear stability in l p for all 1 p

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

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

well posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge we investigate regularity and well posedness for a fluid evolution model in the presence of a three phase contact point we consider a fluid evolution governed by darcy s law after linearization we obtain a nonlocal third order operator which contains the dirichlet neumann operator on the wedge with opening angle epsilon 0 we show well posedness and regularity for this linear evolution equation in the limit of vanishing opening angle we show the convergence of solutions to a fourth order degenerate parabolic operator related to the thin film equation in the course of the analysis we introduce and characterize a new type of sum of weighted sobolev spaces which are suitable to capture the singular limit as epsilon to 0 in particular the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem

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

holder regularity of solutions of degenerate elliptic and parabolic equations abstract we establish the alexandroff bakelman pucci estimate the harnack inequality and the holder continuity of solutions to degenerate parabolic equations of the non divergence form lu xa 11 u xx 2 x a 12 u xy a 22 u yy b 1 u x b 2 u y u t g on x 0 with bounded measurable coefficients we also establish similar regularity results in the corresponding elliptic case

pointwise green s function bounds for multidimensional scalar viscous shock fronts we determine detailed pointwise bounds for the green s function of the linearized operator about a multidimensional scalar viscous shock front these extend the pointwise semigroup methods introduced by howard and zumbrun in the one dimensional case to multidimensions sharpening lp estimates obtained by goodman and miller using a weighted norm approach moreover our results apply to shocks of arbitrary strength as previous results did not as described in a companion paper the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear lp asymptotic behavior of the front under small perturbation the analysis of the multidimensional case involves several new features not found in the one dimensional case concerned with the geometry of propagating signals

automated pure tone threshold estimations from extrapolated distortion product otoacoustic emission dpoae input output functions a promising approach to the prediction of pure tone thresholds through the estimation of dpoae thresholds by input output functions was recently published by boege and janssen j acoust soc am 111 1810 1818 2002 on the basis of their results a device that enables automated measurements of these thresholds was recently developed the purpose of the current study was to evaluate the reliability of this instrument for the objective assessment of hearing loss in 101 ears with either normal hearing or with cochlear hearing loss of up to 50 db hl the median difference between pure tone hearing and dpoae thresholds was approximately 2 db for individual subjects however dpoae thresholds differed from pure tone thresholds by up to 40 db we find therefore that the clinical benefits of this method are probably limited

the one phase free boundary problem for the p laplacian with non constant bernoulli boundary condition our objective here is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a bernoulli type boundary gradient condition and with the p laplacian as the governing operator the main theorems of this paper assert that the exterior and the interior free boundary problem with a bernoulli law i e with a prescribed pressure a x on the free streamline of the flow have convex solutions provided the initial domains are convex the continuous function a x is subject to certain convexity properties in our earlier results we have considered the case of constant a x in the lines of the proof of the main results we also prove the semi continuity up to the boundary of the gradient of the p capacitary potentials in convex rings with c 1 boundaries

holder regularity and uniqueness theorem on weak solutions to the degenerate keller segel system abstract in this paper we present local holder estimates for the degenerate keller segel system k s m below in the range of m 1 and q 2 before a blow up of solutions to deal with difficulties caused by the degeneracy of the operator we find uniform estimates depending on sup norm of the density function and modified the energy estimates and intrinsic scales considered in porous medium equation as its application the uniqueness of weak solution to k s m is also showed for the case q max 1 m 2 2 in the class of holder continuous functions by proving l 1 contraction in this class

convexity and all time c regularity of the interface in flame propagation we consider the following one phase free boundary problem find u such that u 0 and with qt n 0 t under the condition that o is convex and log uo is concave we show that the convexity of t and the concavity of log u t are preserved under the flow for 0 t t as long as t and u on t are smooth as a consequence we show the existence of a smooth up to the interface solution on 0 t tc with tc denoting the extinction time of t we also provide a new proof of a short time existence with c2 initial data on the general domain 2002 john wiley sons inc

c0 estimates and smoothness of solutions to the parabolic equation defined by kimura operators abstract kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics their infinitesimal generator is an elliptic differential operator whose second order coefficients matrix degenerates on the boundary of the domain in this article we consider the inhomogeneous initial value problem defined by generators of kimura diffusions and we establish c 0 estimates which allows us to prove that solutions to the inhomogeneous initial value problem are smooth up to the boundary of the domain where the operator degenerates even when the initial data is only assumed to be continuous

the use of distortion product otoacoustic emission suppression as an estimate of response growth distortion product otoacoustic emission dpoae levels in response to primary pairs f2 2 or 4 khz l2 ranging from 20 to 60 db spl l1 0 4l2 39 db were measured with and without suppressor tones f3 which varied from 1 octave below to 12 octave above f2 in normal hearing subjects suppressor level l3 varied from 5 to 85 db spl dpoae levels were converted into decrements by subtracting the level in the presence of the suppressor from the level in the absence of a suppressor dpoae decrement vs l3 functions showed steeper slopes when f3 f2 this pattern is similar to other measurements of response growth such as direct measures of basilar membrane motion single unit rate level functions suppression of basilar membrane motion and discharge rate suppression from lower animals as l2 increased the l3 necessary to maintain 3 db of suppression increased at a rate of about 1 db db when f3 was approximately equal to f2 but increased more slowly when f3 f2 functions r

long time existence of smooth solution for the porous medium equation in a bounded domain in this paper we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 ut umin 0 u x 0 u0 0in u x t 0for x where m 1 is the permeability the proof is based on the short time existence of cs2 smooth solution the global cs1 estimate the holder estimate of divergence type degenerate equation with measurable coefficients and cs1 estimate of mixed type equation with lipschitz coefficients

parabolic power concavity and parabolic boundary value problems this paper is concerned with power concavity properties of the solution to the parabolic boundary value problem begin aligned p quad left begin array l quad l partial t u vardelta u f x t u nabla u text in quad varomega times 0 infty u x t 0 text on quad partial varomega times 0 infty u x 0 0 text in quad varomega end array right end aligned where varomega is a bounded convex domain in mathbf r n and f is a nonnegative continuous function in varomega times 0 infty times mathbf r times mathbf r n we give a sufficient condition for the solution of p to be parabolically power concave in overline varomega times 0 infty

large time geometric properties of solutions of the evolution p laplacian equation abstract we establish the behavior of the solutions of the degenerate parabolic equation u t u p 2 u p 2 posed in the whole space with nonnegative continuous and compactly supported initial data we prove a nonlinear concavity estimate for the pressure away from the maximum point the estimate has important geometric consequences it implies that the support of the solution becomes convex for large times and converges to a ball in dimension one we know also that the pressure itself eventually becomes p concave in several dimensions we prove concavity but for a small neighborhood of the maximum point

heat kernels of second order complex elliptic operators and applications we study the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on rn we obtain gaussian bounds without further assumption ifn 2 and when the principal part has holder continuous coefficients ifn 3 a thorough study of the boundedness properties of the green operator is made in dimension 1 we construct the fundamental solution of these operators in dimension 2 boundedness results of the maximal accretive square roots onlpsobolev spaces and holder spaces are obtained whenn 2 under holder continuity assumption on the coefficients boundedh functional calculi onlp 1 p are also discussed

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

pointwise green s function approach to stability for scalar conservation laws we study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law obtaining an estimate independent of shock strength we find that for a perturbation with initial data decaying algebraically or slower the perturbation decays in time at the rate of decay of the integrated initial data in any l p norm p 1 stability in any l p norm is a direct consequence the approach taken is that of obtaining pointwise estimates on the perturbation through a duhamel s principle argument that employs recently developed pointwise estimates on the green s function for the linearized equation c 1999 john wiley sons inc

distance to the line in the heston model abstract the main object of study in the paper is the distance from a point to a line in the riemannian manifold associated with the heston model we reduce the problem of computing such a distance to certain minimization problems for functions of one variable over finite intervals one of the main ideas in this paper is to use a new system of coordinates in the heston manifold and the level sets associated with this system in the case of a vertical line the formulas for the distance to the line are rather simple for slanted lines the formulas are more complicated and a more subtle analysis of the level sets intersecting the given line is needed we also find simple formulas for the heston distance from a point to a level set as a natural application we use the formulas obtained in the present paper in the study of the small maturity limit of the implied volatility in the heston model

smooth solution for the porous medium equation in a bounded domain abstract in this paper we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 u t u m m 1 with zero boundary condition on the fixed boundary the flux u m is nonzero while the gradient of pressure u m 1 is zero as a consequence the parabolic equation above becomes degenerate the proof is based on a definition of weighted space c s 2 corresponding to the given degeneracy and the schauder estimates in its linearized equation 0 2 w t x n a i j x d i j w in c s 2 for 0 m 1 m 1

multidimensional viscous shocks i degenerate symmetrizers and long time stability received by the editors september 18 2002 2000 mathematics subject classification primary 35l60 secondary 35b35 35b45 35k55 35l65 76l05

stability of rarefaction waves in viscous media we study the time asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one dimensional viscous media with strictly hyperbolic flux functions our main result is to show that solutions of perturbed rarefaction data converge to an approximate burgers rarefaction wave for initial perturbations w 0 with small mass and localized as w 0 x mathcal o x 1 the proof proceeds by iteration of a pointwise ansatz for the error using integral representations of its various components based on green s functions we estimate the green s functions by careful use of the hopf cole transformation combined with a refined parametrix method as a consequence of our method we also obtain rates of decay and detailed pointwise estimates for the error this pointwise method has been used successfully in studying stability of shock and constant state solutions new features in the rarefaction case are time varying coefficients in the linearized equations and error waves of unbounded mass mathcal o log t these diffusion waves have amplitude mathcal o t 1 2logt in linear degenerate transversal fields and mathcal o t 1 2logt in genuinely nonlinear transversal fields a distinction which is critical in the stability proof

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

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

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

the use of cumulative distributions to determine critical values and levels of confidence for clinical distortion product otoacoustic emission measurements distortion product otoacoustic emission dpoae input output functions were measured at nine f2 frequencies ranging from 500 to 8000 hz in 210 normal hearing and hearing impaired subjects in a companion paper stover et al j acoust soc am 100 956 967 1996 l1 l2 was held constant at 10 db and l2 was varied from 65 to 10 db spl in 5 db steps based upon analyses using clinical decision theory it was demonstrated that dpoae amplitudes for 65 55 db spl primaries l1 l2 and dpoae thresholds resulted in the greatest separation between normal and impaired ears in this paper the data for these two conditions were recast as cumulative distributions which not only describe the extent of overlap between normal and impaired distributions but also provide the measured value i e the specific dpoae amplitude or threshold for any combination of hit and false alarm rates from these distributions confidence limits were constructed for both dpoae amplitude and threshold to determine the degree of ce

stability of periodic solutions of conservation laws with viscosity pointwise bounds on the green function we investigate stability of periodic traveling wave solutions of systems of conservation laws with viscosity within the abstract evans function framework established by r a gardner our main result generalizing the work of zumbrun and howard in the traveling front or pulse setting is to establish sharp pointwise bounds on the green function for the linearized evolution equations provided that an appropriate evans function condition applies to the linearized operator about the wave this condition is equivalent to a spectral stability criterion introduced by schneider in the context of periodic reaction diffusion waves an immediate consequence is that strong spectral stability in the sense of schneider implies linearized l1 l p asymptotic stability for all p 1 on the other hand we show that the strict version of schneider s condition generically fails in the conservation law setting leading to complicated new metastable behavior reminiscent of that seen for degenerate neutrally stable families in the traveling front or pulse case our results apply also to the reaction diffusion setting sharpening at the linearized level results obtained by schneider using weighted norm and bloch decomposition methods

pointwise estimates on the green s function for a scalar linear convection diffusion equation abstract pointwise estimates are found on green s functions for the scalar linear convection diffusion equations that arise when a scalar conservation law with non constant diffusion is linearized about a viscous shock profile of arbitrary strength the estimates take the form of gaussian kernels centered around paths determined by the typically different asymptotic states of the convection function the analysis extends the spectral transform method to the non constant coefficient case

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

infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains we study the positivity and regularity of solutions to the fractional porous me dium equations ut sum 0 in 0 for m 1 and s 0 1 with dirichlet boundary data u 0 in 0 n and nonnegative initial condition u 0 u0 0 our first result is a quantitative lower bound for solutions that holds for all positive times t 0 as a consequence we find a global harnack principle stating that for any t 0 solutions are comparable to ds m where d is the distance to this is in sharp contrast with the local case s 1 where the equation has finite speed of propagation after this we study the regularity of solutions we prove that solutions are classical in the interior c in x and c1 in t and establish a sharp cxs m regularity estimate up to the boundary our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form ut f u 0 in both in bounded and unbounded domains 2016 wiley periodicals inc

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

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

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

further efforts to predict pure tone thresholds from distortion product otoacoustic emission input output functions recently boege and janssen j acoust soc am 111 1810 1818 2002 fit linear equations to distortion product otoacoustic emission dpoae input output i o functions after the dpoae level in db spl was converted into pressure in pa significant correlations were observed between these dpoae thresholds and audiometric thresholds the present study extends their work by 1 evaluating the effect of frequency 2 determining the behavioral thresholds in those conditions that did not meet inclusion criteria and 3 including a wider range of stimulus levels dpoae i o functions were measured in as many as 278 ears of subjects with normal and impaired hearing nine f2 frequencies 500 to 8000 hz in 12 octave steps were used l2 ranged from 10 to 85 db spl 5 db steps and l1 was set according to the equation l1 0 4l2 39 db kummer et al j acoust soc am 103 3431 3444 1998 for l2 levels up to 65 db spl beyond which l1 l2 for the same conditions as those used by boege and janssen we o

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

asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems abstract we study the fully nonlinear parabolic equation f d 2 u m u t 0 in 0 m 1 with the dirichlet boundary condition and positive initial data in a smooth bounded domain r n provided that the operator f is uniformly elliptic and positively homogeneous of order one we prove that the renormalized limit of parabolic flow u x t as t is the corresponding positive eigenfunction which solves f d 2 p 0 in where 0 p 1 m 1 and 0 is the corresponding eigenvalue we also show that some geometric property of the positive initial data is preserved by the parabolic flow under the additional assumptions that is convex and f is concave as a consequence the positive eigenfunction has such geometric property that is log is concave in the case p 1 and 1 p 2 is concave for 0 p 1

asymptotic stability analysis for transition front solutions in cahn hilliard systems abstract we consider the asymptotic behavior of perturbations of transition front solutions arising in cahn hilliard systems on r such equations arise naturally in the study of phase separation and systems describe cases in which three or more phases are possible when a cahn hilliard system is linearized about a transition front solution the linearized operator has an eigenvalue at 0 due to shift invariance which is not separated from essential spectrum in cases such as this nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate our main result asserts that spectral stability a necessary condition for stability defined in terms of an appropriate evans function implies nonlinear stability

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

periodogram based tests for distortion product otoacoustic emissions distortion product otoacoustic emissions dpoaes are an important nonbehavioral measure of cochlear function which provides a close analogue of the behavioral pure tone audiogram dpoaes are sinusoidal distortion products dps produced by nonlinearities in the healthy cochlea detection of dps is accomplished in the fourier domain with a periodogram based test the test compares the power in the dp periodogram bin to a noise estimate derived from a certain number of the surrounding bins statistical properties of this test to date have only been examined by constructing receiver operator characteristics curves derived from dpoae measurements in normal and hearing impaired individuals in this paper the null distribution of this order statistic based test is explicitly derived and via simulations intended to mimic the nonwhite features of real ear noise measurements the power of the test is demonstrated these simulations demonstrate that a local f test is more powerful than this dpoae test with critical values that are easier to calculate although the power of both tests increase with an increasing number of bins the improvement is negligible at around four bins since the power of both tests decrease at lower dp frequencies it is not recommended to use a large number of bins

distortion product otoacoustic emission test performance when both 2 f1 f2 and 2 f2 f1 are used to predict auditory status the objective of this study was to determine whether distortion product otoacoustic emission dpoae test performance defined as its ability to distinguish normal hearing ears from those with hearing loss can be improved by examining response and noise amplitudes at 2 f1 f2 and 2 f2 f1 simultaneously in addition there was interest in knowing whether measurements at both dps and for several primary frequency pairs can be used in a multivariate analysis to further optimize test performance dpoae and noise amplitudes were measured at 2 f1 f2 and 2 f2 f1 for 12 primary levels l2 from 10 to 65 db spl in 5 db steps and 9 pairs of primary frequencies 0 5 to 8 khz in 1 2 octave steps all data were collected in a sound treated room from 70 subjects with normal hearing and 80 subjects with hearing loss subjects had normal middle ear function at the time of the dpoae test based on standard tympanometric measurements measurement based stopping rules were used such that the test terminated when the noise

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