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

gradient estimate for solutions to poisson equations in metric measure spaces abstract let x d be a complete pathwise connected metric measure space with a locally ahlfors q regular measure where q 1 suppose that x d supports a local 1 2 poincare inequality and a suitable curvature lower bound for the poisson equation u f on x d moser trudinger and sobolev inequalities are established for the gradient of u the local holder continuity with optimal exponent of solutions is obtained

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

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

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

free boundary regularity for a problem arising in superconductivity this paper concerns regularity properties of the mean field theory of superconductivity the problem is reminiscent of the one studied earlier by l a caffarelli l karp and h shahgholian in connection with potential theory the difficulty introduced in this paper is the existence of several patches where on each patch the solution to the problem may have different constant values however using a refined analysis we reduce the problem to the one patch case at least locally near regular free boundary points using a monotonicity formula due to georg s weiss we characterize global solutions of a related equation hence earlier regularity results apply and we conclude the c1 regularity of the free boundary

optimal regularity in the obstacle problem for kolmogorov operators related to american asian options in this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of kolmogorov type we treat smooth obstacles as well as non smooth obstacles all our proofs follow the same line of thought and are based on blow ups compactness barriers and arguments by contradiction the problem considered arises in financial mathematics when considering path dependent derivative contracts with early exercise feature

the regularity theory for the parabolic double obstacle problem in this paper we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator the result in this paper are generalizations of the theory for the elliptic problem in lee et al calc var partial differ equ 58 3 104 2019 and lee and park the regularity theory for the double obstacle problem for fully nonlinear operator 2018 to parabolic case and also the theory for the parabolic single obstacle problem in caffarelli et al j am math soc 17 4 827 869 2004 to double obstacle case new difficulties in the theory which are generated by the characteristic of parabolic pdes and the existence of the upper obstacle are discussed in detail furthermore the thickness assumptions to have the regularity of the free boundary are carefully considered

gradient estimates for variable coefficient parabolic equations and singular perturbation problems in this article we prove via monotonicity formulas interior and boundary gradient esti mates for solutions to second order parabolic equations in divergence form with dini top order coefficients we then prove uniform lipschitz estimates for solutions of singular perturbation prob lems using the previous results and two phase monotonicity formulas introduction the purpose of this paper is to study the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form with bounded lower order terms and dini continuous coefficients and also to establish uniform spatial lipschitz estimates for some singular perturbation problems in the same context in part 1 we obtain interior and boundary estimates for solutions to these linear equations as is well known the weakest assumption on the modulus of continuity of the top order terms to obtain interior continuity of the gradient of solutions is dini continuity see part 1 for the relevant definitions see for instance g w in the elliptic case the usual approach to such results is to use appropriate potentials here we use a monotonicity formula approach theo rem 1 2 9 which we believe is new the approach has the advantage that it allows us to show that dini continuity of the coefficients at a given point yields bound edness of the gradient at the same point using our monotonicity formula the fact that linear functions are almost solutions and a geometric argument from cl we also obtain under uniform dini continuity of the coefficients a uniform modulus of continuity estimate for gradients of solutions theorem 1 3 1 the fact that the monotonicity formula at a point depends only on the dini continuity at that same point enables us to use a simple reflection argument to extend these results up to the boundary theorem 1 4 5 and theorem 1 4 12 we thus obtain a modulus of continuity estimate up to the boundary for gradients of solutions with smooth dirichlet data under uniform dini continuity of the top order coeffi cients once this is established a compactness argument theorem 1 5 10 and a judicious application of harnack s principle allow us to obtain a hopf maximum principle theorem 1 5 10 corollary 1 5 13 and corollary 1 5 16 for our class of equations

regularity of a free boundary for viscosity solutions of nonlinear elliptic equations our objective in this paper is to analyze the regularity of the boundary of a set omega2 that admits a solution u to the following overdetermined problem f d 2 u chi omega in b u delu 0 in b omega here f is a convex uniformly elliptic operator of homogeneity one b is the unit ball in r n and the equation is satisfied in the viscosity sense we also consider the case of concave f in r 2

on the regularity of a free boundary near contact points with a fixed boundary abstract we investigate the regularity of a free boundary near contact points with a fixed boundary with c 1 1 boundary data for an obstacle like free boundary problem we will show that under certain assumptions on the solution and the boundary function the free boundary is uniformly c 1 up to the fixed boundary we will also construct some examples of irregular free boundaries

multiscale temporal integrators for fluctuating hydrodynamics following on our previous work s delong b e griffith e vanden eijnden and a donev phys rev e 87 033302 2013 we develop temporal integrators for solving langevin stochastic differential equations that arise in fluctuating hydrodynamics our simple predictor corrector schemes add fluctuations to standard second order deterministic solvers in a way that maintains second order weak accuracy for linearized fluctuating hydrodynamics we construct a general class of schemes and recommend two specific schemes an explicit midpoint method and an implicit trapezoidal method we also construct predictor corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow time scales we propose using random finite differences to approximate some of the stochastic drift terms that arise because of the kinetic multiplicative noise in the limiting dynamics we illustrate our integrators on two applications involving the development of giant nonequilibrium concentration fluctuations in diffusively mixing fluids we first study the development of giant fluctuations in recent experiments performed in microgravity using an overdamped integrator we then include the effects of gravity and find that we also need to include the effects of fluid inertia which affects the dynamics of the concentration fluctuations greatly at small wave numbers

on a coefficient conjecture of brannan soit 1 xz 1 z 1 n 1 a n x z n ou x 1 et 0 1 alors a 5 x a 5 1 et l egalite n est vraie que si et seulement si x 1

extremal solutions of the two dimensionall problem of moments by using the theory of the principal function of a hyponormal operator with rank one self commutator one classifies some extremal points of the solutions of the truncatedl problem of moments in two real variables these extremal elements called degenerated solutions of thel problem are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane the main result of the paper characterizes these degenerated solutions from three distinct perspectives complex analysis moment problems and operator theory the interplay between the three points of view each with specific techniques constitutes the main body of the paper

numerical integration of the fluctuating hydrodynamic equations an approach to numerically integrate the landau lifshitz fluctuating hydrodynamic equations is outlined the method is applied to one dimensional systems obeying the nonlinear fourier equation and the full hydrodynamic equations for a dilute gas static spatial correlation functions are obtained from computer generated sample trajectories time series they are found to show the emergence of long range behavior whenever a temperature gradient is applied the results are in very good agreement with those obtained from solving the correlation equations directly

singular limit and homogenization for flame propagation in periodic excitable media this paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media the model involves two parameters the period of the medium l and a singular parameter related to the activation energy the existence of pulsating travelling fronts for fixed and l was proved by berestycki hamel in bh in the present paper we investigate the behaviour of such solutions when more precisely we establish that pulsating travelling fronts behave like travelling waves when the period l is small and we also study the convergence of the solution as goes to zero and l is fixed toward a solution of a free boundary problem

stochastic differential equations for non linear hydrodynamics abstract we formulate the stochastic differential equations for non linear hydrodynamic fluctuations the equations incorporate the random forces through a random stres tensor and random heat flux as in the landau and lifshitz theory however the equations are non linear and the random forces are non gaussian we provide explicit expressions for these random quantities in terms of the well defined increments of the wienner process

regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators this paper is devoted to a proof of regularity near the initial state for solutions to the cauchy dirichlet and obstacle problem for a class of second order differential operators of kolmogorov type the approach used here is general enough to allow us to consider smooth obstacles as well as non smooth obstacles

lipschitz continuity of cheeger harmonic functions in metric measure spaces abstract we use the heat equation to establish the lipschitz continuity of cheeger harmonic functions in certain metric spaces the metric spaces under consideration are those that are endowed with a doubling measure supporting a 1 2 poincare inequality and in addition supporting a corresponding sobolev poincare type inequality for the modification of the measure obtained via the heat kernel examples are given to illustrate the necessity of our assumptions on these spaces we also provide an example to show that in the general setting the best possible regularity for the cheeger harmonic functions is lipschitz continuity

uniform regularity close to cross singularities in an unstable free boundary problem we introduce a new method for the analysis of singularities in the unstable problem delta u chi u 0 which arises in solid combustion as well as in the composite membrane problem our study is confined to points of supercharacteristic growth of the solution i e points at which the solution grows faster than the characteristic invariant scaling of the equation would suggest at such points the classical theory is doomed to fail due to incompatibility of the invariant scaling of the equation and the scaling of the solution in the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded the level set u 0 consists of two c 1 curves meeting at right angles it is important that our result is not confined to the minimal solution of the equation but holds for all solutions

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

a parabolic almost monotonicity formula we prove the parabolic counterpart of the almost monotonicity formula of caffarelli jerison and kening for pairs of functions u x s in the strip s 1 r n x 1 0 satisfying u 0

on applications of hypergeometric functions abstract this is a survey article on the author s involvement over the years with hypergeometric functions we discuss our counter example to one of m robertson s conjectures our results on the omitted values problems brannan s conjecture on the coefficients of a certain power series generalizations of ramanujan s asymptotic formulas for complete elliptic integrals and muir s 1883 approximation to the arc length of an ellipse involving an inequality for some 3f2 s

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

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

rellich type identities for eigenvalue problems and application to the pompeiu problem for classical neumann eigenvalue buckling eigenvalue and clamped plate eigenvalue we give the corresponding rellich type identities as an application of these results then we obtain a new necessary and sufficient condition for a domain without the pompeiu property

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

solutions of fully nonlinear elliptic equations with patches of zero gradient existence regularity and convexity of level curves in this paper we first construct viscosity solutions in the crandall lions sense of fully nonlinear elliptic equations of the form f d 2 u x g x u on u 0 in fact viscosity solutions are surprisingly weak since candidates for solutions are just continuous we only require that the test polynomials p those tangent from above or below to the graph of u at a point x 0 satisfy the correct inequality only if p x 0 0 that is we simply disregard those test polynomials for which p x 0 0 nevertheless this is enough by an appropriate use of the alexandroff bakelman technique to prove existence regularity and in two dimensions for f g cu c 0 and constant boundary conditions on a convex domains to prove that there is only one convex patch

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

numerical methods for the stochastic landau lifshitz navier stokes equations the landau lifshitz navier stokes llns equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes this paper examines explicit eulerian discretizations of the full llns equations several computational fluid dynamics approaches are considered including maccormack s two step lax wendroff scheme and the piecewise parabolic method and are found to give good results for the variance of momentum fluctuations however neither of these schemes accurately reproduces the fluctuations in energy or density we introduce a conservative centered scheme with a third order runge kutta temporal integrator that does accurately produce fluctuations in density energy and momentum a variety of numerical tests including the random walk of a standing shock wave are considered and results from the stochastic llns solver are compared with theory when available and with molecular simulations using a direct simulation monte carlo algorithm

on the positivity of certain trigonometric sums and their applications in this paper we find conditions on the coefficients b k k 1 n such that the corresponding trigonometric cosine and sine sums given respectively by k 1nb email protected 0 and k 1nb email protected 0 for all email protected n are positive using these results we find that the functions f that are in the class of analytic functions a are starlike of certain order in the unit disc d by means of conditions on the taylor coefficients of f as an application we also find conditions such that the cesaro means of order b of f z are close to convex and starlike in d

statistical mechanics of nonlinear hydrodynamic fluctuations abstract the paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics the generalized fokker planck equation for the distribution function of coarse grained densities of conserved quantities is derived from the liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix we have obtained the functional differential fokker planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse grained fluctuating fields an outline of the derivation of fokker planck equations containing the burnett terms is also given the explicit coordinate representation for the hydrodynamic fokker planck equation is discussed in the case of one component simple fluid the general scheme of a change of coarse grained functional variables is developed for hydrodynamic fokker planck equations the corresponding transformation rules are found for drift terms diffusion coefficients and thermodynamic forces the dynamical equations and stationary conditions for averages of functions functionals of hydrodynamic fields are discussed by using the fokker planck operators acting on such functions the explicit form of these operators are found for various sets of fluctuating fields as an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state

symmetry theorems for the overdetermined eigenvalue problems abstract the well known schiffer conjecture saying that for a smooth bounded domain r n if there exists a positive neumann eigenvalue such that the corresponding neumann eigenfunction u is constant on the boundary of then is a ball in this paper we shall prove that the schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding neumann eigenfunction is constant on the boundary we also prove a similar result to the berenstein conjecture for the overdetermined dirichlet eigenvalue problem

obstacle problem for a non convex fully nonlinear operator abstract in this paper we study a priori c 2 estimate up to the boundary for f d 2 u 0 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator the conditions are variations of conditions for the zero set of the operator in 7

asymptotic estimates for the spatial segregation of competitive systems abstract for a class of population models of competitive type we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity we show that the limiting problem is a remarkable system of differential inequalities which defines the functional class s in 2 by exploiting the regularity theory recently developed in conti et al indiana univ math j to appear for the elements of functional classes of the form s we provide some qualitative and regularity property of the limiting configurations besides for the case of two competing species we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence finally we prove some new liouville type results which allow to have uniform regularity estimates of the solutions

temporal integrators for fluctuating hydrodynamics including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics these langevin equations possess a special fluctuation dissipation structure that needs to be preserved by spatio temporal discretizations in order for the computed solution to reproduce the correct long time behavior in particular numerical solutions should approximate the gibbs boltzmann equilibrium distribution and ideally this will hold even for large time step sizes we describe finite volume spatial discretizations for the fluctuating burgers and fluctuating incompressible navier stokes equations that obey a discrete fluctuation dissipation balance principle just like the continuum equations we develop implicit explicit predictor corrector temporal integrators for the resulting stochastic method of lines discretization these stochastic runge kutta schemes treat diffusion implicitly and advection explicitly are weakly second order accurate for additive noise for small time steps and give a good approximation to the equilibrium distribution even for very strong fluctuations numerical results demonstrate that a midpoint predictor corrector scheme is very robust over a broad range of time step sizes

on the order of starlikeness of the shifted gauss hypergeometric function abstract we determine for several ranges of real parameters the order of starlikeness of the shifted gauss hypergeometric function and we give some consequences of our results in particular some mapping properties of the carlson shaffer convolution operator

fluctuation dissipation theorems in the generic formalism the viscoelastic fluids abstract fluctuation dissipation relations fdr for a maxwell fluid are computed via the generic formalism this formalism is determined by four building blocks two potentials total energy and entropy and two matrices which determine the dynamics the fdr exhibits specific features arising from the type of entropy used in the formalism but the understanding of fluctuations in a given non equilibrium system is reduced to determining a single friction matrix we show that consistent results with fundamental laws of irreversible thermodynamics for maxwell fluid are obtained if it is used the corresponding entropy in lit if structural properties of the fluid are used as thermodynamic variables giving rise to extended irreversible thermodynamics the resulting fdr also shows the desired properties we discuss these results under the scope of the relation among levels of description and statistical properties of fluctuating thermodynamic fields

geometric approach to nonvariational singular elliptic equations in this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems the magnitude of the singularity is measured by a negative parameter 1 for 0 0 in particular we show sharp hausdorff estimates which imply local finiteness of the perimeter of the region u 0 and the mathcal h n 1 almost everywhere weak differentiability property of mathfrak f

on the singular set of the parabolic obstacle problem this paper is devoted to regularity results and geometric properties of the singular set of the parabolic obstacle problem with variable right hand side making use of a monotonicity formula for singular points we prove the uniqueness of blow up limits at singular points these results apply to parabolic obstacle problem with variable coefficients

time structure invariance criteria for closure approximations in many areas of physics time evolution equations for moments of distributions are expressed in terms of higher order moments closure approximations are then introduced in an ad hoc fashion to reduce the higherorder moments to functions of the lower order ones herein the time structure invariance of the poisson bracket as manifested through the jacobi identity is used to derive constraint relationships on these approximations these constraints severely limit the allowable functionality of general closures and help to define the boundaries within which future investigations should concentrate s1063 651x 97 10410 x

a free boundary problem for the heat equation arising in flame propagation we introduce a new free boundary problem for the heat equation of interest in combustion theory it is obtained in the description of laminar flames as an asymptotic limit for high activation energy the problem asks for the determination of a domain in space time q c rn x 0 t and a function u x t 0 defined in q such that ut au in q u takes certain initial conditions u x 0 uo x for x e qo oqfn t 0 and two conditions are satisfied on the free boundary r a2 n t 0 u o and u1 1 where u1 denotes the derivative of u along the spatial exterior normal to r we approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data

examination of phenomenological coefficient matrices within the canonical model of field theory of thermodynamics the field equations of linear irreversible thermodynamics have been deduced from hamilton s principle the hamiltonian formalism has been considered as a theory of conservative systems without dissipative processes in this paper we present the field equations of linear irreversible thermodynamics that are deduced from a hamiltonian principle first we present the canonical mathematical model for purely dissipative transport processes then introducing a lie algebra of the potentials with the help of an algebraic type transformation we examine the physical processes in this algebra we expect that two kinds of descriptions of the same physical situation develop into two such descriptions in time which describe the same physical situations as well since the given transformation is a dynamical transformation it leaves the lagrangian invariant in the sense of the above mentioned expectation we expect that the entropy density function and the entropy production density function which pertains to the same physical situation have to be invariant under that transformation which leaves the lagrangian invariant it is shown that these are satisfied if the phenomenological coefficient matrices are symmetric s1063 651x 97 08305 0

almost monotonicity formulas for elliptic and parabolic operators with variable coefficients in this paper we extend the results of caffarelli jerison and kenig ann of math 2 155 2002 and caffarelli and kenig amer j math 120 1998 by establishing an almost monotonicity estimate for pairs of continuous functions satisfying u 0 lu 1 u u 0 in an infinite strip global version or a finite parabolic cylinder localized version where l is a uniformly parabolic operator lu l a b c u div a x s u b x s u c x s u s u with double dini continuous a and uniformly bounded b and c we also prove the elliptic counterpart of this estimate this closes the gap between the known conditions in the literature both in the elliptic and parabolic case imposed on u in order to obtain an almost monotonicity estimate at the end of the paper we demonstrate how to use this new almost monotonicity formula to prove the optimal c 1 1 regularity in a fairly general class of quasi linear obstacle type free boundary problems

a phenomenological non equilibrium theory for the thermodynamics of kelvin bodies abstract from a generalized gibbs equation and a general form for entropy flux we develop a non linear theory for the thermodynamics of several rheological bodies known as kelvin bodies in accordance with irreversible extended thermodynamics we deduce a non stationary transport equation which contains new cross effect terms in so doing we obtain a complete hyperbolic system which describes the evolution of the above bodies

bracket formulation of diffusion convection equations abstract an abstract diffusion convection equation is introduced it is shown that many well known and experimentally tested time evolution equations of physical systems e g the boltzmann and the enskog kinetic equations and the navier stokes fourier fluid mechanics equations are identical to or well approximated by realizations of the abstract diffusion convection equation the physical content of the abstract diffusion convection equation is compatibility with equilibrium thermodynamics it means that properties of solutions of the abstract diffusion convection equation are mathematical formulations of the experience on which equilibrium thermodynamics is based the requirement that all admissible time evolution equations are realizations of the abstract diffusion convection equation is used to find admissible closures in the bbgky and the maxwell grad hierarchies and admissible constitutive relations in generalized hydrodynamics

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

a simple example for comparing generic with rational non equilibrium thermodynamics during the last 15 years a bracket formalism of dissipative continuum physics has been developed which resulted in a formulation which is shortly denoted as generic generic has been applied to different problems of continuum thermodynamics often in this way that a well known problem was reformulated in generic formalism to learn some more about the generic procedure we consider a gas which is contained in a cylinder closed by a piston moving with friction we treat this simple discrete system with rational non equilibrium thermodynamics by using liu s procedure and for comparison also with the generic formalism both different procedures yield in the same results especially in the same entropy production differences similarities and fundamental presuppositions of both formalisms are discussed and compared

regularity in the obstacle problem for parabolic non divergence operators of hormander type in this paper we continue the study initiated in 15 concerning the obstacle problem for a class of parabolic non divergence operators structured on a set of vector fields x x1 xq in rn with c coefficients satisfying hormander s finite rank condition i e the rank of lie x1 xq equals n at every point in rn in 15 we proved under appropriate assumptions on the operator and the obstacle the existence and uniqueness of strong solutions to a general obstacle problem the main result of this paper is that we establish further regularity in the interior as well as at the initial state of strong solutions compared to 15 we in this paper assume in addition that there exists a homogeneous lie group g rn ring operator such that x1 xq are left translation invariant on g and such that x1 xq are homogeneous of degree one

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

on a new thermodynamic description of viscoelastic materials abstract a new thermodynamic approach based on extended irreversible thermodynamics is proposed to describe viscoelastic materials the state space is widened by including the stress tensor and the heat flux vector among the set of independent state variables these extra variables are satisfying evolution equations taking the form of first order time differential equations the present formalism encompasses the classical models of maxwell kelvin voigt poynting thomson as well as the thermoelastic body and the newtonian fluid this is achieved without appealing to an artificial decomposition of the stress and strain tensors into elastic and inelastic parts

cheeger harmonic functions in metric measure spaces revisited let x d be a complete metric measure space with a locally doubling measure that supports a local weak l2 poincare inequality by assuming a heat semigroup type curvature condition we prove that cheeger harmonic functions are lipschitz continuous on x d gradient estimates for cheeger harmonic functions and solutions to a class of non linear poisson type equations are presented

optimal regularity for the no sign obstacle problem in this paper we prove the optimal regularity for a general obstacle type problem under the assumption that is where n is the newtonian potential this is the weakest assumption for which one can hope to get regularity as a by product of the regularity we are able to prove that under a standard thickness assumption on the zero set close to a free boundary point the free boundary is locally a graph close to provided f is dini this completely settles the question of the optimal regularity of this problem which has been the focus of much attention during the last two decades 2012 wiley periodicals inc

regularity properties of a free boundary near contact points with the fixed boundary in the upper half of the unit ball b x 0 let u and omega a domain in r n x is an element of r n x 1 0 solve the following overdetermined problem deltau chi omega in b u

on quadrature domains and the schwarz potential abstract let t be a distribution in r n whose support is compact a domain in r n is said to be a quadrature domain with respect to t if supp t and h h t for all functions h which are harmonic and integrable over we denote this by q t hl 1 this is equivalent to saying that there is a solution u called the modified schwarz potential to the following overdetermined cauchy problem u 1 t in and u u 0 on in this paper we investigate different properties of quadrature domains we show that if t is a fixed distribution then bounded domains in q t hl1 are uniformly bounded later we prove that if the modified schwarz potential of a domain is a polynomial then is a half space we also study the uniqueness problem namely we show that if d and are in q t hl1 v and u are the msps of d and respectively and sup d u 0 then meets x v x consequently if v 0 then d 0

divergence form operators in reifenberg flat domains we study the boundary regularity of solutions of elliptic operators in divergence form with c 0 coefficients or operators which are small perturbations of the laplacian in non smooth domains we show that as in the case of the laplacian there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain

irreducible tensor description iii thermodynamics of a low temperature phonon gas let one assume that the interacting phonon gas whose behavior is governed by the boltzmann peierls equation inhabits an insulating crystal at sufficiently low temperature then within the framework of a single acoustic phonon branch and of an isotropic long wavelength approximation to the dispersion relation the simplest acceptable version of extended irreversible thermodynamics based upon the nine moment system of differential equations for the slow and fast gas state variables is carefully investigated it is clearly demonstrated that in virtue of the structure simplifications just mentioned the conceptually different macroscopic kinetic and variational procedures which are discussed in this paper appear complementary to each other finally with the help of a suitable contraction of the nine moment system of field equations for callaway s relaxation model a slightly generalized nonlinear variant of ordinary low temperature phonon hydrodynamics is explicitly derived

on the foundations of extended irreversible thermodynamics a theory of macroscopic systems which takes as independent variables the slow conserved ones plus the fast dissipative fluxes is carefully analyzed at three levels of description macroscopic thermodynamic microscopic projection operators and mesoscopic fluctuation theory such a description is compared with the memory function approach based only on the conserved variables we find that the first theory is richer and wider than the second one and some misunderstandings in this connection are clarified and discussed

regularity of a free boundary in parabolic potential theory we study the regularity of the free boundary in a stefan type problem 1u t u in d rn r u u 0 on d with no sign assumptions on u and the time derivative t u

on the foundations of the stochastic immersed boundary method we explore the theoretical foundations for the inclusion of thermal fluctuations in the immersed boundary method for simulating microscale fluid systems with immersed flexible structures as in cellular and subcellular biology we investigate in particular the physical validity of the thermal forcing scheme with respect to the coupling of fluid and immersed structural degrees of freedom and non equilibrium conditions we discuss also the shortcomings of a natural alternative scheme in which the thermal fluctuations are applied directly to the structural degrees of freedom through langevin type dynamics

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

c1 1 regularity in semilinear elliptic problems in this paper we give an astonishingly simple proof of c1 1 regularity in elliptic theory our technique yields both new simple proofs of old results as well as new optical results the setting we ll consider is the following let u be a solution to where b is the unit ball in n f x t is a bounded lipschitz function in x and ft is bounded from below then we prove that u c1 1 b1 2 our method is a simple corollary to a recent monotonicity argument due to caffarelli jerison and kenig 2002 wiley periodicals inc

geometric and energetic criteria for the free boundary regularity in an obstacle type problem we consider an obstacle type problem u f x in d u u 0 on d where d is a given open set in n and is an unknown open subset of d the problem originates in potential theory in connection with harmonic continuation of potentials the qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign using geometric and energetic criteria in delicate combination we show the c1 1 regularity of the solutions and the regularity of the free boundary below the lipschitz threshold for the right hand side

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

hausdorff measure and stability for the p obstacle problem 2 p in this paper we analyze the free boundary for the inhomogeneous obstacle problem with zero obstacle governed by the degenerate operator div a u f u 0 where f is a positive lipschitz function and a u is of the p laplacian type i e a u u p 2 u 2 p as a result of our analysis we obtain stability and finite n 1 hausdorff measure of the free boundary our technique is a modified version of that of l a caffarelli who obtained similar results for p 2

preservation of thermodynamic structure in model reduction based on the availability of an invariant manifold we develop a model reduction procedure that preserves thermodynamic structure more concretely we construct the poisson and irreversible brackets of the general equation for the nonequilibrium reversible irreversible coupling of nonequilibrium thermodynamics by means of the ideas originally introduced for handling constraints the general ideas are then applied to the kramers problem that is the description of transitions between two potential wells separated by a high barrier this example reveals how a fortuitous cancellation mechanism that allows a logarithmic entropy to generate a linear diffusion equation is inherited by a master equation resulting from model reduction

a parabolic two phase obstacle like equation for the parabolic obstacle problem like equation delta u partial derivative t u lambda chi u 0 lambda chi u 0 boolean or partial derivative u 0 is in a neighborhood of each branc

up to boundary regularity for a singular perturbation problem of p laplacian type abstract in this paper we are interested in establishing up to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular degenerate elliptic operator our main result states if r n is a c 1 domain f c 1 for some 0 1 and u e verifies div a x u e u e e u e in 0 u e 1 in u e f on where e 0 e t 1 e t e and 0 t b 0 t 1 r e t d t m 0 with some positive constants b and m then there exists a constant c 0 independent of e such that u e l c