uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

study of coating flow by the finite element method abstract silliman s analysis of slot coating is extended to accommodate film flows with highly bent menisci as in slide and curtain coating by combining polar and cartesian coordinate parametrizations of meniscus shape the nonlinear algebraic equations from the subdomain galerkin weighted residual method and finite element basis functions are solved by newton s method which is perfected for the free boundary problem convergence behavior is examined slot coating results reveal slow flow zones and show that at high capillary number and low metering rate the downstream meniscus can no longer attach to the slot comparison of coyne and elrod s approximate solutions with the finite element solutions shows that their assumption of a parabolic velocity profile leads to reasonable approximations of meniscus shapes

radial symmetry of solution for fractional p laplacian system abstract in this paper we investigate the method of moving planes for fractional p laplacian system we firstly discuss the key ingredients for the method of moving planes such as maximum principle for anti symmetric functions decay at infinity and boundary estimate then we apply the method of moving planes to establish the radial symmetry and the monotonicity of the positive solutions for fractional p laplacian system in a unit ball or in the whole space

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

symmetry of positive solutions for choquard equations with fractional p laplacian abstract this paper deals with the following choquard equations with fractional p laplacian p s u c n t x 2 t n u q u q 1 in r n u 0 on r n where 0 s t 1 2 p p 1 q and n 2 by constructing a decay at infinity theorem and a narrow region principle we establish the radial symmetry of positive solutions based on the generalized direct method of moving planes

uniqueness of positive solutions to some nonlinear neumann problems abstract using the moving plane method we obtain a liouville type theorem for nonnegative solutions of the neumann problem div y a u x y 0 x r n y 0 lim y 0 y a u y x y f u x 0 x r n under general nonlinearity assumptions on the function f r r for any constant a 1 1

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

mixed velocity stress current and potential boundary conditions for stationary mhd flow this paper is concerned with a boundary value problem describing the stationaryflow of a viscous incompressible electrically conducting fluid confined to a bounded region of space under mixed boundary conditions the flow is governed by the navier stokes equations ohm s law and the biot savart law the boundary conditions involve the velocity field stress tensor electric current density and electric potential we derive a mixed variational formulation of the problem which lends itself naturally to finite element discretizations and prove the existence and uniqueness of a small solution under the assumption of sufficiently small data

the sliding method for the nonlocal monge ampere operator abstract we develop a sliding method for the nonlocal monge ampere operator we first establish a narrow region principle in bounded domains which plays an important role in the sliding method then we consider the equation with the nonlocal monge ampere operator and derive the monotonicity of solutions in both bounded domains and the whole space we use a new idea estimating the singular integrals defining the nonlocal monge ampere operator along a sequence of approximate maximum points

a direct method of moving spheres on fractional order equations abstract in this paper we introduce a direct method of moving spheres for the fractional laplacian 2 with 0 2 in which a key ingredient is the narrow region maximum principle as immediate applications we classify non negative solutions for semilinear equations involving the fractional laplacian in r n we prove a non existence result for the prescribing q curvature equation on s n then by combining the direct method of moving planes and moving spheres we establish a liouville type theorem on a half euclidean space we expect to see more applications of this method to many other nonlinear equations involving non local operators

weak imposition of boundary conditions for the navier stokes equations we prove the convergence of a finite element method for the navier stokes equations in which the no slip condition email protected i 0 on c for i 1 2 is imposed by a penalty method and the no penetration condition u n 0 on c is imposed by lagrange multipliers this approach has been studied for the stokes problem in weak imposition of boundary conditions in the stokes problem ph d thesis university of pittsburgh pa 1999 in most flows the reynolds number is not negligible so the email protected u inertial effects are important thus the extension beyond the stokes problem to the navier stokes equations is critical we show existence and uniqueness of the approximate solution and optimal order of convergence can be achieved if the computational mesh follows the real boundary our results for the nonlinear navier stokes equations improve known results for this approach for the stokes problem

semilinear integro differential equations i odd solutions with respect to the simons cone abstract this is the first of two papers concerning saddle shaped solutions to the semilinear equation l k u f u in r 2 m where l k is a linear elliptic integro differential operator and f is of allen cahn type saddle shaped solutions are doubly radial odd with respect to the simons cone x x r m r m x x and vanish only on this set by the odd symmetry l k coincides with a new operator l k o which acts on functions defined only on one side of the simons cone x x and that vanish on it this operator l k o which corresponds to reflect a function oddly and then apply l k has a kernel on x x which is different from k in this first paper we characterize the kernels k for which the new kernel is positive and therefore one can develop a theory on the saddle shaped solution the necessary and sufficient condition for this turns out to be that k is radially symmetric and k is a strictly convex function assuming this we prove an energy estimate for doubly radial odd minimizers and the existence of saddle shaped solution in a subsequent article part ii further qualitative properties of saddle shaped solutions will be established such as their asymptotic behavior a maximum principle for the linearized operator and their uniqueness

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

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

coupled generalized nonlinear stokes flow with flow through a porous medium in this article we analyze the flow of a fluid through a coupled stokes darcy domain the fluid in each domain is non newtonian modeled by the generalized nonlinear stokes equation in the free flow region and the generalized nonlinear darcy equation in the porous medium a flow rate is specified along the inflow portion of the free flow boundary we show existence and uniqueness of a variational solution to the problem we propose and analyze an approximation algorithm and establish a priori error estimates for the approximation

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

monotonicity of positive solutions for fractional p systems in unbounded lipschitz domains abstract in this paper without imposing any decay assumptions on the solutions at infinity we first develop an unbounded narrow region principle for the fractional p laplacian systems an essential ingredient to carry on the direct method of moving planes then we combine this direct method with the sliding method to obtain the monotonicity of positive bounded solutions for cooperative systems involving the fractional p laplacian in unbounded lipschitz domains we believe that the new ideas introduced here will become useful tools in studying qualitative properties of solutions for other types of nonlinear nonlocal systems in various unbounded domains

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

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

symmetry and decay of traveling wave solutions to the whitham equation abstract this paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model an exponential decay result for supercritical solitary wave solutions is given moreover it is shown that all such solitary wave solutions are symmetric and monotone on either side of the crest the proof is based on the method of moving planes furthermore a close relation between symmetric and traveling wave solutions is established

a liouville s theorem for a fractional elliptic system with indefinite nonlinearities abstract in this paper we consider a fractional elliptic system with indefinite nonlinearities we prove that every positive bounded solution of the system is monotone increasing along x 1 direction by using the direct method of moving planes based on this monotonicity we obtain the nonexistence of positive solutions and a liouville type results for the fractional elliptic system as an application we get the liouville type results for the fractional elliptic system with cubic nonlinearities

the equations of stationary incompressible magnetohydrodynamics with mixed boundary conditions abstract this paper deals with the questions of existence uniqueness and finite element approximation of solutions to the equations of steady state magnetohydrodynamics with mixed boundary conditions posed on a bounded three dimensional domain the boundary conditions for the velocity equations are of dirichlet neumann and mixed type these boundary conditions are important when considering free boundary value problems problems on artificially truncated domains and control problems which are governed by these equations

symmetric properties of positive solutions for fully nonlinear nonlocal system abstract in this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain bounded or unbounded in r n via using a direct method of moving planes this extends the results in wang and niu 2017 and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order

maximum principles for the fractional p laplacian and symmetry of solutions abstract in this paper we consider nonlinear equations involving the fractional p laplacian p s u x c n s p p v r n u x u y p 2 u x u y x y n s p d y f x u we prove a maximum principle for anti symmetric functions and obtain other key ingredients for carrying on the method of moving planes such as a variant of the hopf lemma a boundary estimate lemma which plays the role of the narrow region principle then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p laplacian in a unit ball and in the whole space we believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators

on the structure of the nodal set and asymptotics of least energy sign changing radial solutions of the fractional brezis nirenberg problem abstract in this paper we study the asymptotic and qualitative properties of least energy radial sign changing solutions of the fractional brezis nirenberg problem ruled by the s laplacian in a ball of r n when s 0 1 and n 6 s as usual is the positive parameter in the linear part in u we prove that for sufficiently small such solutions cannot vanish at the origin we show that they change sign at most twice and their zeros coincide with the sign changes moreover when s is close to 1 such solutions change sign exactly once finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a tower of bubbles as 0 i e the positive and negative parts concentrate at the same point with different concentration speeds

sharp threshold of blow up and scattering for the fractional hartree equation abstract we consider the fractional hartree equation in the l 2 supercritical case and find a sharp threshold of the scattering versus blow up dichotomy for radial data if m u 0 s s c s c e u 0 m q s s c s c e q and m u 0 s s c s c u 0 h s 2 m q s s c s c q h s 2 then the solution u t is globally well posed and scatters if m u 0 s s c s c e u 0 m q s s c s c e q and m u 0 s s c s c u 0 h s 2 m q s s c s c q h s 2 the solution u t blows up in finite time this condition is sharp in the sense that the solitary wave solution e i t q x is global but not scattering which satisfies the equality in the above conditions here q is the ground state solution for the fractional hartree equation

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

a liouville theorem for a class of fractional systems in r n abstract let 0 2 be any real number in this paper we investigate a class of fractional elliptic systems of the form 2 u x f v x 2 v x g u x x r n u v 0 x r n applying the iteration method and the direct method of moving planes for the fractional laplacian without any decay assumption on the solutions at infinity we prove the liouville theorem of nonnegative solutions under some natural conditions on f and g

the sliding methods for the fractional p laplacian abstract in this paper we develop a sliding method for the fractional p laplacian we first obtain the key ingredient needed in the sliding method in a bounded domain the narrow region principle then using nonlinear equations involving the fractional p laplacian in both bounded domains and in the whole space we illustrate how this new sliding method can be employed to obtain monotonicity of solutions during these processes we introduce a new idea estimating the singular integrals defining the fractional p laplacian along a sequence of approximate maximum points we believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

finite volume methods for convection diffusion problems derivation stability and error analysis in both discrete h 1 and l 2 norms for cell centered finite volume approximations of convection diffusion problems are presented various upwind strategies are investigated the theoretical results are illustrated by numerical examples

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

penalty method with p1 p1 finite element approximation for the stokes equations under the slip boundary condition we consider the p1 p1 or p1b p1 finite element approximations to the stokes equations in a bounded smooth domain subject to the slip boundary condition a penalty method is applied to address the essential boundary condition u cdot n g on partial omega which avoids a variational crime and simultaneously facilitates the numerical implementation we give o h 1 2 epsilon 1 2 h epsilon 1 2 error estimate for velocity and pressure in the energy norm where h and epsilon denote the discretization parameter and the penalty parameter respectively in the two dimensional case it is improved to o h epsilon 1 2 h 2 epsilon 1 2 by applying reduced order numerical integration to the penalty term the theoretical results are confirmed by numerical experiments

efficient parallel algorithms for parabolic problems domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation implicit schemes are used in order to march with large time steps parallelization is realized by approximating interface values using explicit computation various techniques are examined including a multistep second order explicit scheme and a one step high order scheme we show that the resulting schemes are of second order global accuracy in space and stable in the sense of osher or in l infty they are optimized with respect to the parallel efficiency

weak imposition of no slip conditions in finite element methods abstract we begin the analysis of two schemes for imposing weakly the no slip boundary conditions in the navier stokes equations the first is via a lagrange multiplier constraining no tangential velocity on the boundary and the second is via penalization of tangential velocities along boundaries our analysis in this first step is for the stokes problem although the motivation for relaxing this essential boundary condition comes from higher reynolds number flow

explicit implicit domain decomposition method with modified upwind differences for convection diffusion equations based on domain decomposition and multi layer explicit computation technique one new explicit implicit algorithm is presented for convection diffusion equations which has high parallelism and applies modified upwind differences to discretize diffusion term by the analysis of auxiliary difference equations and the maximum principle the resulting procedure is shown to be of second order global accuracy in space numerical experiments illustrating the accuracy efficiency and parallelism are shown

continuation techniques for a penalty approximation of the navier stokes equations continuation techniques are used to solve a penalty finite element approximation of the navier stokes equations sufficient conditions are given for convergence of the euler newton continuation method in a reynolds parameter to an isolated solution of the finite element problem numerical results are presented for the driven cavity problem the approach is extended to arc length continuation to treat problems with singular points on the continuation path

nonexistence of positive solutions for an indefinite fractional p laplacian equation abstract in this paper we consider the following fractional p laplacian equation with indefinite nonlinearities p s u x x 1 u q x x r n where 0 s 1 p q 1 2 we use the direct method of moving planes to prove the solution of above fractional p laplacian equation is monotone increasing along x 1 direction then obtain that the above equation possess no positive bounded solution on the whole space there is no decay condition at infinity on solution for the fractional p laplacian and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p laplacian here we also cannot find a suitable auxiliary function for this fractional p laplacian to circumvent these difficulties we use the following two estimate to study the property of solution i we estimate the singular integrals defining the fractional p laplacian along a sequence of approximate maxima ii we estimate the lower and upper bound of the fractional p laplacian equation

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

liouville theorem and classification of positive solutions for a fractional choquard type equation abstract let n 2 0 2 and 0 n we prove that the equation 2 u 1 x n u p u p 1 in r n has no positive solution if 1 p n n we also classify all positive solutions to the equation in the critical case p n n

properties of positive solutions to nonlocal problems with negative exponents in unbounded domains abstract we study nonlinear nonlocal problems involving the fractional p laplacian operator and nonlinearities with negative exponents in the whole space or in an unbounded epigraph domain the symmetry and monotonicity results are proved by using a direct method of moving planes

positive solutions to a fractional equation with singular nonlinearity abstract in this paper we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain r n n 2 s p s u k x u f u in u 0 in u 0 in r n here 0 s 1 0 0 and f r r is a positive c 2 function k r is a holder continuous function in which behave as dist x near the boundary with 0 2 s first for any 0 and for small enough we prove the existence of solutions to p next for a suitable range of values of we show the existence of an unbounded connected branch of solutions to p emanating from the trivial solution at 0 for a certain class of nonlinearities f we derive a global multiplicity result that extends results proved in 2 to establish the results we prove new properties which are of independent interest and deal with the behavior and holder regularity of solutions to p

the finite element method with lagrange multipliers for domains with corners we study the convergence of the finite element method with lagrange multipliers for approximately solving the dirichlet problem for a second order elliptic equation in a plane domain with piecewise smooth boundary assuming mesh refinements around the corners we construct families of boundary subspaces that are compatible with triangular lagrange elements in the interior and we carry out the error analysis of the resulting approximations in weighted sobolev spaces

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

symmetry and non existence of positive solutions for fractional p laplacian systems abstract chen and li 2018 developed maximum principles for the fractional p laplacian which carry out the direct method of moving planes to the fractional p laplacian equation in this paper we generalize their method to the system case and obtain a symmetry result for the fractional p laplacian system on the whole space we also consider the system on the upper half space and obtain a non existence result

solutions of fully nonlinear nonlocal systems abstract in this paper we consider the system involving fully nonlinear nonlocal operators f u x c n p v r n g u x u y x y n d y f v x f v x c n p v r n g v x v y x y n d y g u x for carrying on the method of moving planes a narrow region principle and a decay at infinity are established then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space furthermore non existence of positive solutions to the nonlinear system on a half space is derived

indefinite fractional elliptic problem and liouville theorems abstract in this paper we consider an indefinite fractional elliptic problem a corresponding liouville type theorem is established furthermore we obtain a priori estimate for solutions in a bounded domain by blowing up and rescaling we also classify the solutions of some degenerate elliptic equation originated from the fractional laplacian

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

finite difference scheme for parabolic problems on composite grids with refinement in time and space finite difference schemes for transient convection diffusion problems on grids with local refinement in time and space are constructed and studied the construction utilizes a modified upwind approximation and linear interpolation at the slave nodes the proposed schemes are implicit of backward euler type and unconditionally stable error analysis is presented in the maximum norm and convergence estimates are derived for smooth solutions optimal approximation results for ratios between the spatial and time discretization parameters away from the cfl condition are shown finally numerical examples illustrating the theory are given

liouville theorems involving the fractional laplacian on a half space abstract let r n be the upper half euclidean space and let be any real number between 0 and 2 consider the following dirichlet problem involving the fractional laplacian 1 2 u u p x r n u 0 x r n instead of using the conventional extension method of caffarelli and silvestre 8 we employ a new and direct approach by studying an equivalent integral equation 2 u x r n g x y u p y d y applying the method of moving planes in integral forms we prove the non existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions