optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

optimal results for the fractional heat equation involving the hardy potential abstract in this paper we study the influence of the hardy potential in the fractional heat equation in particular we consider the problem p u t s u u x 2 s u p c f in 0 t u x t 0 in 0 t u x t 0 in r n 0 t u x 0 u 0 x if x where n 2 s 0 s 1 s is the fractional laplacian of order 2 s p 1 c 0 0 1 and u 0 f 0 are in a suitable class of functions the main results in the article are 1 optimal results about existence and instantaneous and complete blow up in the linear problem p 0 where the best constant in the fractional hardy inequality n s provides the threshold between existence and nonexistence to obtain local sharp estimates of the solutions it is required to prove a weak harnack inequality for a weighted operator that appears in a natural way 2 the existence of a critical power p s in the semilinear problem p 1 be such that a if p p s the problem has no weak positive supersolutions and a phenomenon of complete and instantaneous blow up happens b if p p s there exists a positive solution for a suitable class of nonnegative data

on overdetermined problems for a general class of nonlocal operators abstract we study the overdetermined problem for a large family of non local operators given by generators of subordinate brownian motions in particular this family includes the fractional laplacian relativistic stable operators etc we consider these problems in bounded domains exterior domains and in annular domains and we show that under suitable conditions the domains and solutions are both radially symmetric our method uses both analytic and probabilistic tools

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

a non local regularization of first order hamilton jacobi equations abstract in this paper we investigate the regularizing effect of a non local operator on first order hamilton jacobi equations we prove that there exists a unique solution that is c 2 in space and c 1 in time in order to do so we combine viscosity solution techniques and green s function techniques viscosity solution theory provides the existence of a w 1 solution as well as uniqueness and stability results a duhamel s integral representation of the equation involving the green s function permits to prove further regularity we also state the existence of c solutions in space and time under suitable assumptions on the hamiltonian we finally give an error estimate in l norm between the viscosity solution of the pure hamilton jacobi equation and the solution of the integro differential equation with a vanishing non local part

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

optimal existence and uniqueness theory for the fractional heat equation abstract we construct a theory of existence uniqueness and regularity of solutions for the fractional heat equation t u s u 0 0 s 1 posed in the whole space r n with data in a class of locally bounded radon measures that are allowed to grow at infinity with an optimal growth rate we consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions which is given in one direction by the representation formula in the other one by the initial trace we review many of the typical properties of the solutions in particular we prove optimal pointwise estimates and new harnack inequalities

aleksandrov bakelman pucci maximum principles for a class of uniformly elliptic and parabolic integro pde abstract we prove generalized aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes of hamilton jacobi bellman isaacs types whose pde parts are either uniformly elliptic or uniformly parabolic the proofs of these results are based on the classical aleksandrov bakelman pucci maximum principles for the elliptic and parabolic pdes and an iteration procedure using solutions of pucci extremal equations we also provide proofs of nonlocal versions of the classical aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes

integro partial differential equations with singular terminal condition in this paper we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro partial differential equation both with a singular terminal condition singularity means that at the final time the value of the solution can be equal to infinity different types of regularity of this viscosity solution are investigated sobolev holder or strong regularity

principal eigenvalues of a class of nonlinear integro differential operators abstract we consider a class of nonlinear integro differential operators and prove existence of two principal half eigenvalues in bounded smooth domains with exterior dirichlet condition we then establish simplicity of the principal eigenfunctions in viscosity sense maximum principles continuity property of the principal eigenvalues with respect to domains etc we also prove an anti maximum principle and study existence result for some nonlinear problem via rabinowitz bifurcation type results

a study of pattern forming systems with a fully nonlocal interaction kernel of possibly changing sign abstract a nonlocal model with a possibly sign changing kernel is proposed to study pattern formation problems in physical and biological systems with self organization properties one defines pre solutions of a related linear problem and gives sufficient conditions for pre solutions to be jump discontinuous solutions of the nonlocal model among a multitude of solutions a selection criterion is used to single out more significant critical solutions these solutions are further classified into minimizing solutions maximizing solutions and saddle solutions minimizing solutions are the most relevant for patterned states and they exist if the kernel changes sign a non negative kernel on the other hand behaves in some ways like the gradient term in the standard local model

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

regularity for solutions of nonlocal parabolic equations ii we prove boundary regularity and a compactness result for parabolic nonlocal equations of the form ut iu f where the operator i is not necessarily translation invariant as a consequence of this and the regularity results for the translation invariant case we obtain c1 interior estimates in space for nontranslation invariant operators under some hypothesis on the time regularity of the boundary data

on the differentiability of the solution to the hamilton jacobi equation with critical fractional diffusion abstract we prove that the hamilton jacobi equation for an arbitrary hamiltonian h locally lipschitz but not necessarily convex and fractional diffusion of order one critical has classical c 1 solutions the proof is achieved using a new holder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free

heat kernels and analyticity of non symmetric jump diffusion semigroups let d geqslant 1 and alpha in 0 2 consider the following non local and non symmetric levy type operator on mathbb r d begin aligned fancyscript l kappa alpha f x hbox p v int mathbb r d f x z f x frac kappa x z z d alpha mathord mathrm d z end aligned where 0 kappa 0 leqslant kappa x z leqslant kappa 1 kappa x z kappa x z and kappa x z kappa y z leqslant kappa 2 x y beta for some beta in 0 1 using levi s method we construct the fundamental solution also called heat kernel p kappa alpha t x y of fancyscript l kappa alpha and establish its sharp two sided estimates as well as its fractional derivative and gradient estimates we also show that p kappa alpha t x y is jointly holder continuous in t x the lower bound heat kernel estimate is obtained by using a probabilistic argument the fundamental solution of fancyscript l kappa alpha gives rise a feller process x mathbb p x x in mathbb r d on mathbb r d we determine the levy system of x and show that mathbb p x solves the martingale problem for fancyscript l kappa alpha c 2 b mathbb r d furthermore we show that the c 0 semigroup associated with fancyscript l kappa alpha is analytic in l p mathbb r d for every p in 1 infty a maximum principle for solutions of the parabolic equation partial t u fancyscript l kappa alpha u is also established as an application of the main result of this paper sharp two sided estimates for the transition density of the solution of mathord mathrm d x t a x t mathord mathrm d y t is derived where y is a rotationally symmetric stable process on mathbb r d and a x is a holder continuous d times d matrix valued function on mathbb r d that is uniformly elliptic and bounded

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

min max formulas for nonlocal elliptic operators on euclidean space abstract an operator satisfies the global comparison property if anytime a function touches another from above at some point then the operator preserves the ordering at the point of contact this is characteristic of degenerate elliptic operators including nonlocal and nonlinear ones in previous work the authors considered such operators in riemannian manifolds and proved they can be represented by a min max formula in terms of levy operators in this note we revisit this theory in the context of euclidean space with the intricacies of the general riemannian setting gone the ideas behind the original proof of the min max representation become clearer moreover we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

boundary regularity for fully nonlinear integro differential equations we study fine boundary regularity properties of solutions to fully nonlinear elliptic integro differential equations of order 2s with s 0 1 we consider the class of nonlocal operators l l 0 which consists of all the infinitesimal generators of stable levy processes belonging to the class l0 of caffarelli silvestre for fully nonlinear operators i elliptic with respec tt ol w e prove that solutions to iu f in u 0 in r n satisfy u d s c s for all 0 where d is the distance to a ndf l we expect the holder exponent s to be optimal or almost optimal for general right hand sides f l moreover we also expect the class l to be the largest scale invariant subclass of l0 for which this result is true in this direction we show that the class l0 is too large for all solutions to behave like d s the constants in all the estimates in this paper remain bounded as th eo rder of the equation approaches 2

the evans krylov theorem for nonlocal parabolic fully nonlinear equations abstract in this paper we prove the evans krylov theorem for nonlocal parabolic fully nonlinear equations

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the common randomness capacity of a network of discrete memoryless channels we generalize our previous results on generating common randomness at two terminals to a situation where any finite number of agents interconnected by an arbitrary network of independent point to point discrete memoryless channels wish to generate common randomness by interactive communication over the network our main result is an exact characterization of the common randomness capacity of such a network i e the maximum number of bits of randomness that all the agents can agree on per step of communication as a by product we also obtain a purely combinatorial result viz a characterization of the incidence vectors of the spanning arborescences rooted at a specified vertex in a digraph and having exactly one edge exiting the root as precisely the extreme points of a certain unbounded convex polyhedron described by a system of linear inequalities

spectral approximation of a variable coefficient fractional diffusion equation in one space dimension abstract in this article we consider the approximation of a variable coefficient two sided fractional diffusion equation fde having unknown u by introducing an intermediate unknown q the variable coefficient fde is rewritten as a lower order constant coefficient fde a spectral approximation scheme using jacobi polynomials is presented for the approximation of q q n the approximate solution to u u n is obtained by post processing q n an a priori error analysis is given for q q n and u u n two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates

on fractional elliptic equations in lipschitz sets and epigraphs regularity monotonicity and rigidity results we consider a nonlocal equation set in an unbounded domain with the epigraph property we prove symmetry monotonicity and rigidity results in particular we deal with halfspaces coercive epigraphs and epigraphs that are flat at infinity these results can be seen as the nonlocal counterpart of the celebrated article berestycki et al commun pure appl math 50 11 1089 1111 1997

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

parameter estimation for fractional transport a particle tracking approach 1 space fractional advection dispersion models provide attractive alternatives to the classical advection dispersion equation for model applications that exhibit early arrivals and plume skewness this paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data a particle tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares a simple model of concentration variance based on the particle tracking approach identifies the optimal weights

some free boundary problems recast as nonlocal parabolic equations abstract in this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold the canonical examples would be one phase hele shaw flow as well as its two phase analog we also treat nonlinear versions of both one and two phase problems in the special class of free boundaries that are graphs over r d we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal fractional nonlinear parabolic equation for functions on r d our main observation is that the free boundary condition defines a nonlocal operator having what we call the global comparison property a consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases one obtains in a uniform treatment for all of the problems one and two phase a propagation of modulus of continuity for viscosity solutions of the free boundary flow

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

interior regularity for nonlocal fully nonlinear equations with dini continuous terms abstract this paper is concerned with interior regularity of viscosity solutions of non translation invariant nonlocal fully nonlinear equations with dini continuous terms we obtain c regularity estimates for the nonlocal equations by perturbative methods and a version of a recursive evans krylov theorem

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

cordes nirenberg type estimates for nonlocal parabolic equations in this paper we obtain cordes nirenberg type estimates for nonlocal parabolic equations on the more flexible solution space lt l 1 than the classical solution space b rtn consisting of all bounded functions on rtn

robust dpg method for convection dominated diffusion problems we propose and analyze a discontinuous petrov galerkin dpg method for convection dominated diffusion problems that provides robust l 2 error estimates for the field variables which are quasi optimal in the energy norm a key feature of the method is to construct test functions defined by a variational formulation with bilinear form test norm specifically designed for the goal of robustness a main theoretical ingredient is a stability analysis of the adjoint problem numerical experiments underline our theoretical results and in particular confirm robustness of the dpg method for well chosen test norms

further time regularity for nonlocal fully nonlinear parabolic equations we establish holder estimates for the time derivative of solutions of nonlocal parabolic equations under mild assumptions for the boundary data as a consequence we are able to extend the evans krylov estimate for rough kernels to parabolic equations 2016 wiley periodicals inc

a petrov galerkin spectral element method for fractional elliptic problems abstract we develop a new c 0 continuous petrov galerkin spectral element method for one dimensional fractional elliptic problems of the form 0 d x u x u x f x 1 2 subject to homogeneous boundary conditions we employ the standard modal spectral element basis and the jacobi poly fractonomials as the test functions zayernouri and karniadakis 2013 we formulate a new procedure for assembling the global linear system from elemental local mass and stiffness matrices the petrov galerkin formulation requires performing elemental local construction of mass and stiffness matrices in the standard domain only once moreover we efficiently obtain the non local history stiffness matrices in which the non locality is presented analytically for uniform grids we also investigate two distinct choices of basis test functions i local basis test functions and ii local basis with global test functions we show that the former choice leads to a better conditioned system and accuracy while the latter results in ill conditioned linear systems and therefore that is not an efficient and a proper choice of test function we consider smooth and singular solutions where the singularity can occur at boundary points as well as in the interior domain we also construct two non uniform grids over the whole computational domain in order to capture singular solutions finally we perform a systematic numerical study of non local effects via full and partial history fading in order to further enhance the efficiency of the scheme

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

on the loss of continuity for super critical drift diffusion equations we show that there exist solutions of drift diffusion equations in two dimensions with divergence free super critical drifts that become discontinuous in finite time we consider classical as well as fractional diffusion however in the case of classical diffusion and time independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local l 1 norm of the drift which is a super critical quantity

regularity results on a class of doubly nonlocal problems abstract the purpose of this article is twofold first an issue of regularity of weak solution to the problem p see below is addressed secondly we investigate the question of h s versus c 0 weighted minimizers of the functional associated to problem p and then give applications to existence and multiplicity results

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

the evans krylov theorem for nonlocal fully nonlinear equations we prove a regularity result for solutions of a purely integro differential bellman equation this regularity is enough for the solutions to be understood in the classical sense if we let the order of the equation approach two we recover the theorem of evans and krylov about the regularity of solutions to concave uniformly elliptic partial differential equations

a dirichlet problem for nonlocal degenerate elliptic operators with internal nonlinearity abstract we study a dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities with very mild assumptions on the boundary datum we prove existence and uniqueness of the solution in the viscosity sense if we further assume uniform ellipticity then the solution is shown to be classical and even smooth if both the operator and the boundary datum are smooth

the fractional schrodinger equation with general nonnegative potentials the weighted space approach abstract we study the dirichlet problem for the stationary schrodinger fractional laplacian equation s u v u f posed in bounded domain r n with zero outside conditions we consider general nonnegative potentials v l l o c 1 and prove well posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f important properties of the solutions such as its boundary behaviour are derived the case of super singular potentials that blow up near the boundary is given special consideration since it leads to so called flat solutions we comment on related literature

regularization by 12 laplacian and vanishing viscosity approximation of hjb equations abstract we investigate the regularizing effect of adding small fractional laplacian with critical fractional exponent 1 2 to a general first order hjb equation our results include some regularity estimates for the viscosity solutions of such perturbations making the solutions classically well defined most importantly we use these regularity estimates to study the vanishing viscosity approximation to first order hjb equations by 1 2 laplacian and derive an explicit rate convergence for the vanishing viscosity limit

identification via channels the authors main finding is that any object among doubly exponentially many objects can be identified in blocklength n with arbitrarily small error probability via a discrete memoryless channel dmc if randomization can be used for the encoding procedure a novel doubly exponential coding theorem is presented which determines the optimal r that is the identification capacity of the dmc as a function of its transmission probability matrix this identification capacity is a well known quantity namely shannon s transmission capacity for the dmc

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

holder estimates for non local parabolic equations with critical drift abstract in this paper we extend previous results on the regularity of solutions of integro differential parabolic equations the kernels are non necessarily symmetric which could be interpreted as a non local drift with the same order as the diffusion we provide a growth lemma and a harnack inequality which can be used to prove higher regularity estimates

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

identification in the presence of feedback a discovery of new capacity formulas a study is made of the identification problem in the presence of a noiseless feedback channel and the second order capacity c sub f resp c sub f for deterministic resp randomized encoding strategies is determined several important phenomena are encountered 1 although feedback does not increase the transmission capacity of a discrete memoryless channel dmc it does increase the second order identification capacity 2 noise increases c sub f 3 the structure of the new capacity formulas is simpler than c e shannon s 1948 familiar formula this has the effect that proofs of converses become easier than in the authors previous work

lipschitz regularity for censored subdiffusive integro differential equations with superfractional gradient terms in this paper we are interested in integro differential elliptic and parabolic equations involving nonlocal operators with order less than one and a gradient term whose coercivity growth makes it the leading term in the equation we obtain lipschitz regularity results for the associated stationary dirichlet problem in the case when the nonlocality of the operator is confined to the domain feature which is known in the literature as censored nonlocality as an application of this result we obtain strong comparison principles which allow us to prove the well posedness of both the stationary and evolution problems and steady ergodic large time behavior for the associated evolution problem

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

regularity properties of nonlocal minimal surfaces via limiting arguments abstract we prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s 1 as a consequence we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1

a weak galerkin finite element method for high dimensional time fractional diffusion equation abstract this paper is concerned with the application of a weak galerkin finite element method wg fem to the time fractional diffusion equation the wg fem with l1 formula and the fast evaluation scheme based on wg fem are designed the optimal convergence rates for both semi discrete and fully discrete wg fem schemes are obtained and the stability analysis for the semi discrete wg fem is derived numerical experiments are implemented to verify the theoretical results

infinitely many solutions for fractional kirchhoff schrodinger poisson systems in this paper we study the existence of infinitely many solutions for a fractional kirchhoff schrodinger poisson system based on variational methods especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by kajikiya for the critical case we obtain infinitely many solutions for the system under certain assumptions the novelties of this article lie in the appearance of the possibly degenerate kirchhoff function and weak assumptions on the nonlinear term which are quite mild in this paper we study the existence of infinitely many solutions for a fractional kirchhoff schrodinger poisson system based on variational methods especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by kajikiya for the critical case we obtain infinitely many solutions for the system under certain assumptions the novelties of this article lie in the appearance of the possibly degenerate kirchhoff function and weak assumptions on the nonlinear term which are quite mild

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

identification plus transmission over channels with perfect feedback we determine the region of all identification and transmission rate pairs achievable over a discrete memoryless channel dmc with perfect and instantaneous feedback for both randomized and deterministic encoding as a by product we also have a new proof of kemperman s 1973 strong converse to shannon s coding theorem for dmc s with feedback

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

fractal hamilton jacobi kpz equations nonlinear and nonlocal evolution equations of the form u t lu u q where c is a pseudodifferential operator representing the infinitesimal generator of a levy stochastic process have been derived as models for growing interfaces in the case when the continuous brownian diffusion surface transport is augmented by a random hopping mechanism the goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the diffusive linear and hyperbolic nonlinear terms posed in the whole space r n and supplemented with nonnegative bounded and sufficiently regular initial conditions

obstacle problems for integro differential operators regularity of solutions and free boundaries we study the obstacle problem for integro differential operators of order 2s with s in 0 1 our main result establish that the free boundary is c 1 gamma and u in c 1 s near all regular points namely we prove the following dichotomy at all free boundary points x 0 in partial u varphi i either u x varphi x c d 1 s x o x x 0 1 s alpha for some c 0 ii or u x varphi x o x x 0 1 s alpha where d is the distance to the contact set u varphi moreover we show that the set of free boundary points x 0 satisfying i is open and that the free boundary is c 1 gamma and u in c 1 s near those points these results were only known for the fractional laplacian 2 and are completely new for more general integro differential operators the methods we develop here are purely nonlocal and do not rely on any monotonicity type formula for the operator thanks to this our techniques can be applied in the much more general context of fully nonlinear integro differential operators we establish similar regularity results for obstacle problems with convex operators

inhomogeneous dirichlet boundary value problems of space fractional diffusion equations and their finite element approximations we prove the wellposedness of the galerkin weak formulation and petrov galerkin weak formulation for inhomogeneous dirichlet boundary value problems of constant or variable coefficient conservative caputo space fractional diffusion equations we also show that the weak solutions to their riemann liouville analogues do not exist in general in addition we develop an indirect finite element method for the dirichlet boundary value problems of caputo fractional differential equations which reduces the computational work for the numerical solution of variable coefficient fractional diffusion equations from o n 3 to o n and the memory requirement from o n 2 to o n on any quasiuniform space partition we further prove a nearly sharp error estimate for the method which is expressed in terms of the smoothness of the prescribed data of the problem only we carry out numerical experiments to investigate the performance of the method in comparison with the galerkin finite element method

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

a new regularization mechanism for the boltzmann equation without cut off we apply recent results on regularity for general integro differential equations to derive a priori estimates in holder spaces for the space homogeneous boltzmann equation in the non cut off case we also show an a priori estimate in l infty which applies in the space inhomogeneous case as well provided that the macroscopic quantities remain bounded

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

nonlinear fractional elliptic systems with boundary measure data existence and a priori estimates abstract we are concerned with positive solutions of the fractional elliptic system e s u f v in s v g u in where is an arbitrary domain in r n n 2 s s 1 2 1 and f g c l o c r for some 0 1 we establish universal a priori estimate for positive solutions of e then for c 2 bounded domain we prove the existence of positive solutions of e with prescribed boundary value u and v where are positive bounded measure on and discuss regularity property of the solutions

dirichlet heat kernel estimates for rotationally symmetric levy processes in this paper we consider a large class of purely discontinuous rotationally symmetric levy processes we establish sharp two sided estimates for the transition densities of such processes killed upon leaving an open set d w hend is a fat open set the sharp two sided estimates are given in terms of surviving probabilities and the global transition density of the levy process when d is a c 1 1 open set and the levy exponent of the process is given by 2 with being a complete bernstein function satisfying a mild growth condition at infinity our two sided estimates are explicit in terms of the distance function to the boundary of d and the levy density of x this gives an affirmative answer to the conjecture posted in chen kim and song global heat kernel estimates for relativistic stable processes in half space like open sets potential anal 36 2012 235 261 our results are the first sharp two sided dirichlet heat kernel estimates for a large class of symmetric levy processes with general levy exponents we also derive an explicit lower bound estimate for symmetric levy processes on r d in terms of their levy exponents

regularity theory for the fractional harmonic oscillator abstract in this paper we develop the theory of schauder estimates for the fractional harmonic oscillator h x 2 0 1 more precisely a new class of smooth functions c h k is defined in which we study the action of h in fact these spaces are those adapted to the operator h hence the suited ones for this type of regularity estimates in order to prove our results an analysis of the interaction of the hermite riesz transforms with the holder spaces c h k is needed that we believe of independent interest

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

an analytic approach to purely nonlocal bellman equations arising in models of stochastic control abstract given a bounded domain r d and two integro differential operators l 1 l 2 of the form l j u x p v u x u y k j x y x y d y we study the fully nonlinear bellman equation 0 1 max j 1 2 l j u x a j x u x f j x 0 in with dirichlet boundary conditions here a j f j r are non negative functions we prove the existence of a non negative function u r which satisfies 0 1 almost everywhere the main difficulty arises through the nonlocality of l j and the absence of regularity near the boundary

finite difference finite element method for two dimensional time space fractional bloch torrey equations with variable coefficients on irregular convex domains abstract in magnetic resonance imaging of the human brain the diffusion process of tissue water is considered in the complex tissue environment of cells membranes and connective tissue models based on fractional order bloch torrey equations are known to provide insights into tissue structures and the microenvironment in this paper we consider new two dimensional multi term time and space fractional bloch torrey equations with variable coefficients on irregular convex domains which involve the caputo time fractional derivative and the riemann liouville space fractional derivative an unstructured mesh galerkin finite element method is used to discretize the spatial fractional derivative while for each time fractional derivative we use the l 1 scheme on a temporal graded mesh the stability and convergence of the fully discrete scheme are proved numerical examples are given to verify the efficiency of our method

regularity results for nonlocal equations by approximation we obtain c1 regularity estimates for nonlocal elliptic equations that are not necessarily translation invariant using compactness and perturbative methods and our previous regularity results for the translation invariant case

regularity theory for general stable operators parabolic equations abstract we establish sharp interior and boundary regularity estimates for solutions to t u l u f t x in i with i r and r n the operators l we consider are infinitesimal generators of stable levy processes these are linear nonlocal operators with kernels that may be very singular on the one hand we establish interior estimates obtaining that u is c 2 s in x and c 1 2 s in t whenever f is c in x and c 2 s in t in the case f l we prove that u is c 2 s in x and c 1 in t for any 0 on the other hand we study the boundary regularity of solutions in c 1 1 domains we prove that for solutions u to the dirichlet problem the quotient u d s is holder continuous in space and time up to the boundary where d is the distance to this is new even when l is the fractional laplacian