quantitative uniqueness of elliptic equations based on a variant of frequency function we improve the vanishing order of solutions for schr o dinger equations which describes quantitative behavior of strong uniqueness continuation property for the first time we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions furthermore strong unique continuation is established for higher order elliptic equations using this variant of frequency function

follmer schweizer decomposition and mean variance hedging for general claims let x be an r d valued special semimartingale on a probability space f f t 0 t t p with decomposition x x 0 m a and the space of all predictable x integrable processes such that dx is in the space s 2 of semimartingales if h is a random variable in l 2 we prove under additional assumptions on the process x that h can be written as the sum of an f 0 measurable random variable h 0 a stochastic integral of x and a martingale part orthogonal to m moreover this decomposition is unique and the function mapping h with its decomposition is continuous with respect to the l 2 norm finally we deduce from this continuity that the subspace of l 2 generated by dx where is closed in l 2 and we give some applications of this result to financial mathematics

gaussian bounds for degenerate parabolic equations abstract let a be a real symmetric degenerate elliptic matrix whose degeneracy is controlled by a weight w in the a 2 or qc class we show that there is a heat kernel w t x y associated to the parabolic equation w u t div a u and w t satisfies classic gaussian bounds w t x y c 1 t n 2 exp c 2 x y 2 t we then use this bound to derive a number of other properties of the kernel

some overdetermined problems related to the anisotropic capacity abstract we characterize the wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the finsler p capacity of a convex set r n with 1 p n in particular we show that if the finsler p capacitary potential u associated to has two homothetic level sets then is wulff shape moreover we show that the concavity exponent of u is q p 1 n p if and only if is wulff shape

variance optimal hedging in discrete time we solve the problem of approximating in 2 a given random variable h by stochastic integrals gt of a given discrete time process x we interpret h as a contingent claim to be paid out at time t x as the price evolution of some risky asset in a financial market and g as the cumulative gains from trade using the hedging strategy as an application we determine the variance optimal strategy which minimizes the variance of the net loss h gt over all strategies

decay estimates for time fractional and other non local in time subdiffusion equations in mathbb r d we prove optimal estimates for the decay in time of solutions to a rather general class of non local in time subdiffusion equations in mathbb r d an important special case is the time fractional diffusion equation which has seen much interest during the last years mostly due to its applications in the modeling of anomalous diffusion processes we follow three different approaches and techniques to study this particular case a estimates based on the fundamental solution and young s inequality b fourier multiplier methods and c the energy method it turns out that the decay behaviour is markedly different from the heat equation case in particular there occurs a critical dimension phenomenon the general subdiffusion case is treated by method b and relies on a careful estimation of the underlying relaxation function several examples of kernels including the ultraslow diffusion case illustrate our results

partial regularity for singular solutions to the monge ampere equation we prove that solutions to the monge ampere inequality in n are strictly convex away from a singular set of hausdorff n 1 dimensional measure zero furthermore we show this is optimal by constructing solutions to det d2u 1 with singular set of hausdorff dimension as close as we like to n 1 as a consequence we obtain w2 1 regularity for the monge ampere equation with bounded right hand side and unique continuation for the monge ampere equation with sufficiently regular right hand side 2015 wiley periodicals inc

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

infinite time blow up for the fractional heat equation with critical exponent we consider positive solutions for the fractional heat equation with critical exponent begin aligned left begin array ll u t delta s u u frac n 2s n 2s text in omega times 0 infty u 0 text on mathbb r n setminus omega times 0 infty u cdot 0 u 0 text in mathbb r n end array right end aligned where omega is a smooth bounded domain in mathbb r n n 4s s in 0 1 u mathbb r n times 0 infty rightarrow mathbb r and u 0 is a positive smooth initial datum with u 0 mathbb r n setminus omega 0 we prove the existence of u 0 such that the solution blows up precisely at prescribed distinct points q 1 ldots q k in omega as t rightarrow infty the main ingredient of the proofs is a new inner outer gluing scheme for the fractional parabolic problems

doubling inequality and nodal sets for solutions of bi laplace equations we investigate the doubling inequality and nodal sets for the solutions of bi laplace equations a polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for laplace eigenfunctions by logunov in addition we derive an implicit upper bound for the nodal sets of solutions we show two types of doubling inequalities for the solutions of bi laplace equations as a consequence the rate of vanishing is given for the solutions

sharp vanishing order of solutions to stationary schrodinger equations on carnot groups of arbitrary step abstract based on a variant of the frequency function approach of almgren 1 under appropriate assumptions we establish an optimal upper bound on the vanishing order of solutions to stationary schrodinger equations associated to sub laplacian on carnot groups of arbitrary step such a bound provides a quantitative form of strong unique continuation and can be thought of as a subelliptic analogue of the recent results obtained by bakri 3 and zhu 27 for the standard laplacian

scr e martingales and their applications in mathematical finance after introducing a new concept the notion of e martingale we extend the well known doob inequality for 1 p and the burkholder davis gundy inequalities for p 2 to e martingales by means of these inequalities we give sufficient conditions for the closedness of a space of stochastic integrals with respect to a fixed r d valued semimartingale a question which arises naturally in the applications to financial mathematics we also provide a necessary and sufficient condition for the existence and uniqueness of the follmer schweizer decomposition

volume estimates on the critical sets of solutions to elliptic pdes in this paper we study solutions to elliptic linear equations l u i aij x ju bi x iu c x u 0 either on n or a riemannian manifold under the assumption that the coefficient functions aij are lipschitz bounded we focus our attention on the critical set c u x u 0 and the singular set c u x u 0 and more importantly on effective versions of these currently with just the lipschitz regularity of the coefficients the strongest results in the literature say that the singular set is n 2 dimensional however at this point it has not even been shown that hn 2 c unless the coefficients are smooth fundamentally this is due to the need of an regularity theorem that requires higher smoothness of the coefficients as the frequency increases we introduce new techniques for estimating the critical and singular set which avoids the need of any such regularity consequently we prove that if the frequency of u is bounded by then we have the estimates hn 2 c u c 2 and hn 2 s u c 2 depending on whether the equation is critical or not more importantly we prove corresponding estimates for the effective critical and singular sets even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature we also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem 2017 wiley periodicals inc

on doubling inequalities for elliptic systems abstract we prove doubling inequalities for solutions of elliptic systems with an iterated laplacian as diagonal principal part and for solutions of the lame system of isotropic linearized elasticity these inequalities depend on global properties of the solutions

behavior near the boundary of positive solutions of second order parabolic equations ii a boundary backward harnack inequality is proved for positive solutions of second order parabolic equations in non divergence form in a bounded cylinder q q x 0 t which vanish on o q oq x 0 t where q is a bounded lipschitz domain in r this inequality is applied to the proof of the holder continuity of the quotient of two positive solutions vanishirlg on a portion of o q

asymptotic behavior of the solutions of linear and quasilinear elliptic equations on ℝ ℕ we investigate the relationship between the decay at infinity of the right hand side f and solutions u of an equation lu f when l is a second order elliptic operator on r n it is shown that when l is fredholm u inherits the type of decay of f for instance exponential or power like in particular the generalized eigenfunctions associated with all the fredholm eigenvalues of l isolated or not decay exponentially no use is made of spectral theory the result is next extended when l is replaced by a fredholm quasilinear operator various generalizations to other unbounded domains higher order operators or elliptic systems are possible and briefly alluded to but not discussed in detail

approximating random variables by stochastic integrals let x be a semimartingale and the space of all predictable x integrable processes such that dx is in the space 2 of semimartingales we consider the problem of approximating a given random variable h l 2 by a stochastic integral 0 t s dx s with respect to the l 2 norm if x is special and has the form x x 0 m d we construct a solution in feedback form under the assumptions that 2 d is deterministic and that h admits a strong f s decomposition into a constant a stochastic integral of x and a martingale part orthogonal to m we provide sufficient conditions for the existence of such a decomposition and we give several applications to quadratic optimization problems arising in financial mathematics

on the strong unique continuation of electromagnetic schrodinger equations with random coefficients abstract stochastic partial differential equations spdes with random coefficients are useful mathematical models for the quantification of perturbations in lots of complex engineering and physical applications this paper mainly addresses the strong unique continuation property for the electromagnetic schrodinger operator with random coefficients by the method proposed by n garofalo and f h lin 10 11 appropriate multipliers with physical backgrounds have been introduced to prove a priori estimates moreover its application in an exact controllability problem has been shown in which case the boundary value determines the interior value completely

on unique continuation for the generalized schrodinger equations abstract in this paper we study the unique continuation properties of solutions of the generalized schrodinger equations we prove the carleman inequalities as well as the exponential decay weighted estimates as a consequence we obtain sufficient conditions on the behavior of the solution u at two different times t 0 0 and t 1 1 which guarantee that u 0 is the unique solution

inverse inclusion problem a stable method to determine disks in this paper we are interested in inverse inclusion problem in the plane we derive uniqueness and holder stability results of the inclusion recovery problem using a single boundary measurement under the assumption that the inclusion has a circular shape we also propose a simple minimizing scheme for the recovery of a disk our numerical results show that the holder power in the stability estimate is close to one

a partial answer to a conjecture of b simon concerning unique continuation in his survey paper on schrijdinger semigroups s b simon formulated the following conjecture see s p 5193 let ve kfp the local kato class of potentials then the time independent schrddinger operator h d v has the unique continuation property ucp by this it is meant that given any connected open subset 52 c r the only solution u of hu 0 in 52 that can vanish in an open subset 52 c sz is u 0 a measurable function v on r is said to belong to the kato class k if

quantitative uniqueness for elliptic equations at the boundary of c1 dini domains abstract based on a variant of the frequency function approach of almgren we establish an optimal upper bound on the vanishing order of solutions to variable coefficient schrodinger equations at a portion of the boundary of a c 1 dini domain such bound provides a quantitative form of strong unique continuation at the boundary it can be thought of as a boundary analogue of an interior result recently obtained by bakri and zhu for the standard laplacian

discretisation of fbsdes driven by cadlag martingales abstract we study the discretisation of forward backward stochastic differential equations fbsdes driven by cadlag martingales we prove that under certain conditions imposed on the parameters of the fbsde the time discrete scheme we consider converges to the time continuous equation in the l 2 sense moreover we show that the l 2 norm of the error is of the order of the time step

the nodal set of solutions to some elliptic problems sublinear equations and unstable two phase membrane problem abstract we are concerned with the nodal set of solutions to equations of the form u u q 1 u q 1 in b 1 where 0 q 1 2 b 1 b 1 0 is the unit ball in r n n 2 and u max u 0 u max u 0 are the positive and the negative part of u respectively this class includes the unstable two phase membrane problem q 1 as well as sublinear equations for 1 q 2 we prove the following main results a the finiteness of the vanishing order at every point and the complete characterization of the order spectrum b a weak non degeneracy property c regularity of the nodal set of any solution the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having hausdorff dimension at most n 2 locally finite when n 2 d a partial stratification theorem ultimately the main features of the nodal set are strictly related with those of the solutions to linear or superlinear equations with two remarkable differences first of all the admissible vanishing orders can not exceed the critical value 2 2 q at threshold we find a multiplicity of homogeneous solutions yielding the non validity of any estimate of the n 1 dimensional measure of the nodal set of a solution in terms of the vanishing order as a byproduct we also prove the strong unique continuation property for the unstable obstacle problem corresponding to the case 0 the proofs are based on monotonicity formulae for a 2 parameter family of weiss type functionals blow up arguments and the classification of homogeneous solutions

uniqueness of self similar shrinkers with asymptotically conical ends in this paper we show the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order also we construct non rotationally symmetric self shrinking ends asymptotic to generalized cylinders with rate as fast as any given polynomial

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

new monotonicity formulas and the optimal regularity in the signorini problem with variable coefficients abstract we study the interior signorini or lower dimensional obstacle problem for a uniformly elliptic divergence form operator l div a x with lipschitz continuous coefficients our main result states that similarly to what happens when l the variational solution has the optimal interior regularity c loc 1 1 2 m when m is a codimension one flat manifold which supports the obstacle we achieve this by proving some new monotonicity formulas for an appropriate generalization of the celebrated almgren s frequency functional

model risk and discretisation of locally risk minimising strategies we consider two models for the price process a time continuous jump diffusion and a time discretisation of it then we study the robustness of the related locally risk minimising strategy to this model choice where we focus mainly on hedging asian and spread options using the discretisation scheme and the convergence results on backward stochastic differential equations as studied in khedher and vanmaele 2016 we show that the discrete time locally risk minimising strategies converge to the corresponding continuous time strategies in an l 2 sense we present different numerical examples to illustrate our results

the averaging principle and periodic solutions for nonlinear evolution equations at resonance abstract we study the existence of t periodic solutions t 0 for the first order differential equations being at resonance at infinity where the right hand side is the perturbations of a sectorial operator our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball in terms of special geometrical assumptions imposed on the nonlinearity we also prove that the geometrical assumptions are generalizations of well known landesman lazer and strong resonance conditions the obtained index formula is used to derive the criteria determining the existence of t periodic solutions for the heat equation being at resonance at infinity

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

regularity of solutions to the parabolic fractional obstacle problem in recent years there has been an increasing interest in studying constrained variational problems with a fractional diffusion one of the motivations comes from mathematical finance jumpdiffusion processes where incorporated by merton 14 into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices in contrast with the classical lognormal diffusion model of black and scholes 2 these models allow to take into account large price changes and they have become increasingly popular for modeling market fluctuations both for risk management and option pricing purposes let us recall that an american option gives its holder the right to buy a stock at a given price prior but not later than a given time t 0 if v x represents the rational price of an american option with a payoff at time t 0 then v will solve in the viscosity sense the following obstacle problem min lv v 0 v t

modeling the transport of solutes influenced by multiprocess nonequilibrium a model that explicitly accounts for multiple sources of nonequilibrium is presented the multiprocess nonequilibrium mpne model is specifically formulated for cases where nonequilibrium is caused by a combination of transport and sorption related processes sensitivity analyses were performed to delineate the conditions under which the mpne model reduces to the bicontinuum and to the local equilibrium models these conditions are strongly controlled by the magnitude of the dimensionless rate parameters performance of the mpne model was evaluated with several published data sets an empirical regression equation was successfully used to estimate values for sorption rate constants required in the mpne model the mpne model with values for all parameters obtained from independent sources predicted behavior exhibited by the data this provides support for the validity of the conceptualization upon which the model is based

quantitative uniqueness for elliptic equations with singular lower order terms we use a carleman type inequality of koch and tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms first we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure

boundary estimates for solutions to linear degenerate parabolic equations let omega subset mathbb r n be a bounded nta domain and let omega t omega times 0 t for some t 0 we study the boundary behaviour of non negativesolutions to the equation hu parti

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

non symmetric perturbations of symmetric dirichlet forms abstract we provide a path space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local dirichlet form the representation is a combination of feynman kac and girsanov formulas and extends previously known results in the framework of symmetric diffusion processes through the use of the hardy class of smooth measures which contains the kato class of smooth measures

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

pointwise bounds and blow up for systems of nonlinear fractional parabolic inequalities abstract we investigate nonnegative nonlocal solutions u x t and v x t of the nonlinear system of inequalities 0 t u v 0 t v u in r n r n 1 satisfying the initial conditions u v 0 in r n 0 where and are positive constants specifically using the definition of the fractional heat operator t given in taliaferro 2020 we obtain when they exist optimal pointwise upper bounds on r n 0 for nonnegative nonlocal solutions u and v of this initial value problem with particular emphasis on these bounds as t 0 and as t when 1 we compare our results for nonlocal solutions to those of escobedo and herrero 1991 for classical pointwise solutions

a unique continuation property on the boundary for solutions of elliptic equations we prove the following conclusion if is a harmonic function on a smooth domain fl in r n 3 or a solution of a general second order linear elliptic equation on a domain q in r2 and if there are x i 6 d l and constants a b 0 suchthat u x aexp 6 l ol for x e 2 x ol small then u 0 in i2 the decay rate in our results is best possible by the example that u real part of exp l zq 0 a 1 is harmonic but not identically zero in the right complex half plane

connecting orbits for nonlinear differential equations at resonance abstract we study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity where the right hand side is the perturbations of a sectorial operator our aim is to prove an index formula expressing the conley index of associated semiflow with respect to appropriately large ball in terms of special geometrical assumptions imposed on the nonlinearity we also prove that the geometrical assumptions are generalization of the well known in literature landesman lazer and strong resonance conditions obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity

a combined laplace transform and streamline upwind approach for nonideal transport of solutes in porous media a new numerical method for nonideal transport of solutes in porous media is developed by first using the laplace transform to eliminate the time dependency and then the streamline upwind scheme to solve the spatial equations the transient solution is ultimately recovered by an efficient quotient difference inversion algorithm by introducing complex valued artificial dispersion in the weighting functions characteristics of the transient solutions have been successfully addressed the optimum of the complex valued artificial dispersion has been derived for one dimensional problems in multidimensional cases the streamline upwind scheme is modified by adding complex valued artificial dispersion along the streamline within this numerical scheme the grid orientation problems have been successfully treated the limitations on the cell peclet number and on the courant number were greatly relaxed both one dimensional and two dimensional numerical examples are used to illustrate applications of this technique

a remark on a vanishing property for viscosity solutions of fully nonlinear parabolic equations abstract we make the observation that under some natural conditions on f stated in a c in the main text if a viscosity solution of the fully nonlinear parabolic equation f d 2 u d u u u t 0 vanishes to infinite order at x 0 t 0 then there is a small spatial neighborhood b r 0 x 0 t 0 of x 0 t 0 in which u vanishes identically the proof is inspired in an essential way by the ideas employed in the recent paper armstrong and silvestre 2011 2 where the unique continuation property for fully nonlinear elliptic equations was established

prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature let m n g be a complete noncompact riemannian manifold with the curvature bounded between two negative constants given a function k on m n in terms of the behaviors of k at infinite we give a fairly complete answer to when the k can be the scalar curvature function of a complete metric g 1 which is conformal to g

frequency domain approach for the polynomial stability of a system of partially damped wave equations in this paper we study the stability of a system of wave equations which are weakly coupled and partially damped using a frequency domain approach based on the growth of the resolvent on the imaginary axis we establish the polynomial energy decay rate for smooth initial data we show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations

singularities of steady axisymmetric free surface flows with gravity we consider a steady axisymmetric solution of the euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface we analyze stagnation points as well as points on the axis of symmetry at points on the axis of symmetry that are not stagnation points constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation this suggests the presence of downward pointing cusps at those points at stagnation points on the axis of symmetry the unique blowup profile consistent with the invariant scaling of the equation is the garabedian pointed bubble solution with water above air thus at stagnation points on the axis of symmetry with no water above the stagnation point the invariant scaling of the equation cannot be the right scaling a finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve the velocity field scales almost like and is asymptotically given by with a nonzero constant c0 the last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric euler equations by delort while the concentration compactness result alone does not lead to strong convergence in general we prove the convergence to be strong in our application 2014 wiley periodicals inc

the laplace transform galerkin technique a time continuous finite element theory and application to mass transport in groundwater a time continuous numerical technique referred to as the laplace transform galerkin ltg method is developed and applied to the problem of solute transport in porous media after application of galerkin s procedure and subdivision of the domain into finite elements the method involves a simple application of the laplace transformation to eliminate the temporal derivatives appearing in the space discretized set of ordinary differential equations then by solving the resulting transformed system of algebraic equations in laplace p space numerical inversion of the laplace transformed nodal concentration is performed using the robust and accurate crump 1976 algorithm the crump algorithm permits the concentration to be evaluated from a range of time values from a single set of laplace p space solutions because each of the needed p space solutions are independent the algorithm is well suited for execution on multiprocessor parallel computers it is demonstrated by means of a series of examples that the ltg scheme is capable of providing highly accurate solutions essentially devoid of numerical dispersion for grid peclet numbers in excess of 30 examination of the complex valued laplace domain concentration profiles reveal that they are generally smooth well behaved oscillatory functions compared to the profiles in the time domain thus permitting the use of a coarse finite element grid because of the nature of the laplace transformation the ltg method is particularly well suited to the problem of transient groundwater flow and solute transport in fractured porous media or multiple aquifer aquitard systems based on the dual porosity integrodifferential equation approach

nonlocal and localized analyses of conditional mean transient flow in bounded randomly heterogeneous porous media 1 we consider the numerical prediction of transient flow in bounded randomly heterogeneous porous media driven by random sources initial heads and boundary conditions without resorting to monte carlo simulation after applying the laplace transform to the governing stochastic flow equations we derive exact nonlocal integro differential equations for the mean and variance covariance of transformed head and flux conditioned on measured values of log conductivity y ln k approximating these conditional moment equations recursively to second order in the standard deviation y of y we solve them by finite elements for superimposed mean uniform and convergent flows in a two dimensional domain an alternative conditional mean solution is obtained through localization of the exact moment expressions the nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain a comparison with monte carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with y2 as large as 4 the nonlocal solution is only slightly more accurate than the much simpler localized solution but the latter does not yield information about predictive uncertainty the accuracy of each solution improves markedly with conditioning a preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for monte carlo statistics to stabilize when the same direct matrix solver is used for all three we do not presently know how using iterative solvers would have affected this conclusion this is true whether the laplace inversion and monte carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size the underlying exact and recursive moment equations as well as the proposed computational algorithm are valid in both two and three dimensions only the numerical implementation of our algorithm is two dimensional

monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations abstract we study the strong unique continuation property backwards in time for the nonlocal equation in r n 1 0 1 t s u v x t u s 0 1 our main result theorem 1 2 can be thought of as the nonlocal counterpart of the result obtained in 30 for the case when s 1 in order to prove theorem 1 2 we develop the regularity theory of the extension problem for the equation 0 1 with such theory in hands we establish i a basic monotonicity result for an adjusted frequency function which plays a central role in this paper see theorem 1 3 below ii an extensive blowup analysis of the so called almgren rescalings associated with the extension problem we feel that our work will also be of interest e g in the study of certain basic open questions in free boundary problems as well as in nonlocal segregation problems

a curvilinear finite element model for simulating two well tracer tests and transport in stratified aquifers the problem of solute transport in steady nonuniform flow created by a recharging and discharging well pair is investigated numerical difficulties encountered with the standard galerkin formulation in cartesian coordinates are illustrated an improved finite element solution strategy is presented to overcome the numerical problem the strategy is based on a formulation performed in three dimensional curvilinear coordinates this formulation is developed especially for the simulation of two well injection withdrawal tracer tests conducted in homogeneous or stratified aquifers the formulation can also be applied to other types of problems involving transport in homogeneous and stratified aquifers a three dimensional finite element approximation is derived by two successive applications of upstream weighted residual and subdomain collocation procedures first the weighted residual procedure is formulated in the horizontal flow plane using curvilinear coordinates along and normal to streamlines upstream weighting is included but automatically controlled and limited to one dimension along the main flow direction second the subdomain collocation procedure is formulated in the vertical dimension spatial discretization of the three dimensional flow region is performed using a curvilinear finite element mesh that is point centered in the horizontal plane and block centered in the vertical direction general algorithms are developed for mesh generation time stepping and matrix reduction the solution algorithm is designed to accommodate several thousand nodal unknowns with minimal core storage and cpu time requirement two simulation examples are presented to verify and validate the numerical model in two dimensions using an analytical solution and experimental laboratory and field data a comparative study shows that the proposed formulation is superior to the standard galerkin formulation in cartesian coordinates both in terms of accuracy and computational efficiency the field simulation example indicates the need to conduct a more elaborate three dimensional simulation study allowing for vertical variation in hydraulic conductivity such a study has been completed and is reported in the follow up paper of this series

comment on a combined laplace transform and streamline upwind approach for nonideal transport of solutes in porous media by linlin xu and mark l brusseau

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

the helmholtz equation in heterogeneous media a priori bounds well posedness and resonances abstract we consider the exterior dirichlet problem for the heterogeneous helmholtz equation i e the equation a u k 2 n u f where both a and n are functions of position we prove new a priori bounds on the solution under conditions on a n and the domain that ensure nontrapping of rays the novelty is that these bounds are explicit in k a n and geometric parameters of the domain we then show that these a priori bounds hold when a and n are l and satisfy certain monotonicity conditions and thereby obtain new results both about the well posedness of such problems and about the resonances of acoustic transmission problems i e a and n discontinuous where the transmission interfaces are only assumed to be c 0 and star shaped the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for c convex interfaces with strictly positive curvature

a parabolic problem with a fractional time derivative we study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative our main result is a de giorgi nash moser holder regularity theorem for solutions in a divergence form equation we also prove results regarding existence uniqueness and higher regularity in time

extension properties and boundary estimates for a fractional heat operator the square root of the heat operator sqrt partial t delta can be realized as the dirichlet to neumann map of the heat extension of data on mathbb r n 1 to mathbb r n 2 in this n

reply to comment on a combined laplace transform and streamline upwind approach for nonideal transport of solutes in porous media by linlin xu and mark l brusseau

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

some new monotonicity formulas and the singular set in the lower dimensional obstacle problem we construct two new one parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem the first one is a family of weiss type formulas geared for points of any given homogeneity and the second one is a family of monneau type formulas suited for the study of singular points we show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity this allows to prove a structural theorem for the singular set our approach works both for zero and smooth non zero lower dimensional obstacles the study in the latter case is based on a generalization of almgren s frequency formula first established by caffarelli salsa and silvestre

wiener s criterion for parabolic equations with variable coefficients and its consequences dans un ensemble borne de r n 1 on etudie la regularite des points limites pour le probleme de dirichlet pour un operateur parabolique a coefficients lisses on donne une caracterisation geometrique de ces points limites qui sont reguliers

the dirichlet to neumann operator via hidden compactness abstract we show that to each symmetric elliptic operator of the form a k a k l l c on a bounded lipschitz domain r d one can associate a self adjoint dirichlet to neumann operator on l 2 which may be multi valued if 0 is in the dirichlet spectrum of a to overcome the lack of coerciveness in this case we employ a new version of the lax milgram lemma based on an indirect ellipticity property that we call hidden compactness we then establish uniform resolvent convergence of a sequence of dirichlet to neumann operators whenever the underlying coefficients converge uniformly and the second order limit operator in l 2 has the unique continuation property we also consider semigroup convergence

on the regularity of the non dynamic parabolic fractional obstacle problem abstract in the class of the non dynamic fractional obstacle problems of parabolic type it is shown existence uniqueness apriori bounds of the solution and optimal regularity of the space derivatives of the solution furthermore at free boundary points of positive parabolic density it is proven that the time derivative of the solution is holder continuous at such free boundary points space time regularity of the corresponding free boundary is obtained for any fraction s 0 1

positive and sign changing solutions to a nonlinear choquard equation abstract we consider the problem u w x u 1 x u p u p 2 u u h 0 1 where is an exterior domain in r n n 3 0 n p 2 2 n n 2 w c 0 r n inf r n w 0 and w x v 0 as x under symmetry assumptions on and w which allow finite symmetries and some assumptions on the decay of w at infinity we establish the existence of a positive solution and multiple sign changing solutions to this problem having small energy

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100