fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

periodic solutions of allen cahn system with the fractional laplacian abstract we consider periodic solutions of the following nonlinear system associated with the fractional laplacian x x s u x f u x 0 in r where u x u x v x the function f r 2 r is a smooth double well potential we prove the existence of periodic solutions with large period t by using variational methods moreover we draw a conclusion that the second component of periodic solution is identical to zero if the origin is a saddle point of f whereas the second component is not identical to zero if the origin is a local maximum point of f a hamiltonian identity for periodic solutions is also established

a multiplicity result for a nonlinear fractional schrodinger equation in rn without the ambrosetti rabinowitz condition abstract in this paper we study the existence and the multiplicity of positive solutions for the following class of fractional schrodinger equations 2 s s u v x u f u in r n where 0 is a parameter s 0 1 n 2 s v r n r is a continuous positive potential and f r r is a c 1 superlinear nonlinearity which does not satisfy the ambrosetti rabinowitz condition the main result is established by using minimax methods and ljusternik schnirelmann theory of critical points

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

constant scalar curvature metrics with isolated singularities we extend the results and methods of 6 to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on s where is a disjoint union of submanifolds of dimensions between 0 and n 2 2 the existence of solutions with isolated singularities occupies the majority of the paper their existence was previously established by schoen 12 but the proof we give here based on the techniques of 6 is more direct and provides more information about their geometry when is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in 7 and 8

deflation of conjugate gradients with applications to boundary value problems a method for improving the convergence of the standard conjugate gradient method is given this method involves the use of auxiliary subspaces it is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented

overlapping schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media overlapping schwarz preconditioners are constructed and numerically studied for gauss lobatto legendre gll spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by gordon hall transfinite mappings the results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping schwarz preconditioners for standard affine gll spectral elements i e their convergence rate is independent of the number of subdomains of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator while in the case of small overlap the convergence rate depends on the inverse of the overlap size

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

the use of pointwise interpolation in domain decomposition methods with nonnested meshes in this paper the author develops a new technique and a corresponding theory for schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from finite element discretization of elliptic partial differential equations the theory provides an optimal convergence of an additive schwarz algorithm that is constructed with a nonnested coarse space and a not necessarily shape regular subdomain partitioning the theory is also applicable to the graph partitioning algorithms recently developed by cai and saad overlapping domain decomposition algorithms for general sparse matrices preprint 93 027 univ of minnesota and farhat and lesoine internat j numer methods engrg 36 1993 pp 745 764 for problems defined on unstructured meshes

numerical comparison of iterative eigensolvers for large sparse symmetric positive definite matrices abstract the jacobi davidson jd method has been recently proposed for the evaluation of the partial eigenspectrum of large sparse matrices in this work we report a set of numerical experiments that compare this method with other previously proposed techniques deflation accelerated conjugate gradient dacg and lanczos arpack on large sparse symmetric matrices the results obtained by jd and dacg are benchmarked against those obtained with arpack in terms of computational time for the evaluation of s s 100 leftmost eigenpairs the comparison is performed on a number of matrices with dimensions up to a few hundred thousands arising from the discretization of the diffusion equation by means of finite element mixed finite element and finite difference techniques the results show that dacg and jd are equally more efficient than arpack for a small number of eigenpairs with dacg more performing for small s and particularly for s 1 while arpack tends to become competitive when 40 or more eigenpairs are sought for the largest and worst conditioned matrix dacg proves the only viable alternative

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

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

parallel preconditioned conjugate gradient optimization of the rayleigh quotient for the solution of sparse eigenproblems a parallel algorithm based on the multidimensional minimization of the rayleigh quotient is proposed to evaluate the leftmost eigenpairs of the generalized symmetric positive definite eigenproblem the minimization is performed via a conjugate gradient like procedure accelerated by a factorized approximate inverse preconditioner fsai and by a number of block preconditioners the resulting code obtains a high level of parallel efficiency and proves to be comparable with the parpack package on a set of large matrices arising from various discretizations of pdes of elliptic parabolic type

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

spectral problems of a class of non self adjoint one dimensional schrodinger operators abstract in this paper we investigate the one dimensional schrodinger operator l q with complex valued periodic potential q when q l 1 0 1 and q n 0 for n 0 1 2 where q n are the fourier coefficients of q with respect to the system e i 2 n x we prove that the bloch eigenvalues are 2 n t 2 for n z t c and find explicit formulas for the bloch functions then we consider the inverse problem for this operator

periodic solutions for a pseudo relativistic schrodinger equation abstract we study the existence and the regularity of non trivial t periodic solutions to the following nonlinear pseudo relativistic schrodinger equation 0 1 x m 2 m u x f x u x in 0 t n where t 0 m is a non negative real number f is a regular function satisfying the ambrosetti rabinowitz condition and a polynomial growth at rate p for some 1 p 2 1 we investigate such problem using critical point theory after transforming it to an elliptic equation in the infinite half cylinder 0 t n 0 with a nonlinear neumann boundary condition by passing to the limit as m 0 in 0 1 we also prove the existence of a non trivial t periodic weak solution to 0 1 with m 0

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

characterization of the domain chaos convection state by the largest lyapunov exponent using numerical integrations of the boussinesq equations in rotating cylindrical domains with realistic boundary conditions we have computed the value of the largest lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths we study in particular the domain chaos state which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects we compare our results with those from egolf et al nature 404 733 2000 who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short lived spatially localized dislocation nucleation events we also show that the quantity lambda1 is not intensive for aspect ratios gamma over the range 20 gamma 40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism

delaunay type singular solutions for the fractional yamabe problem we construct delaunay type solutions for the fractional yamabe problem with an isolated singularity begin aligned delta gamma w c n gamma w frac n 2 gamma n 2 gamma w 0 quad text in quad mathbb r n backslash 0 end aligned we follow a variational approach in which the key is the computation of the fractional laplacian in polar coordinates

ground states solutions for a non linear equation involving a pseudo relativistic schrodinger operator in this paper we prove the existence regularity and symmetry of a ground state for a nonlinear equation in the whole space involving a pseudo relativistic schrodinger operator

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

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

global bifurcation results for a fractional equation in rn abstract in this paper we study the existence of positive solution for the following class of fractional equation p s u g x f u in r n 0 u x 1 x r n lim x u x 0 where n 2 s s 0 1 0 f r r is a continuous function g r n r is a bounded continuous function that changes sign and s u is the fractional laplacian the main tools used are a theorem due to crandall and rabinowitz 14 and the global bifurcation theorem due to rabinowitz 23

quasi optimal schwarz methods for the conforming spectral element discretization the spectral element method is used to discretize self adjoint elliptic equations in three dimensional domains the domain is decomposed into hexahedral elements and in each of the elements the discretization space is the set of polynomials of degree n in each variable a conforming galerkin formulation is used the corresponding integrals are computed approximately with gauss lobatto legendre gll quadrature rules of order n and a lagrange interpolation basis associated with the gll nodes is used fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method the conforming finite element space on the gll mesh consisting of piecewise q1 or p1 functions produces a stiffness matrix kh that is known to be spectrally equivalent to the spectral element stiffness matrix kn kh is replaced by a preconditioner tilde k h which is well adapted to parallel computer architectures the preconditioned operator is then tilde k h 1 k n techniques for nonregular meshes are developed which make it possible to estimate the condition number of tilde k h 1 k n where tilde k h is a standard finite element preconditioner of kh based on the gll mesh two finite element based preconditioners the wirebasket method of smith and the overlapping schwarz algorithm for the spectral element method are given as examples of the use of these tools numerical experiments performed by pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions

domain decomposition methods for eigenvalue problems abstract this paper proposes several domain decomposition methods to compute the smallest eigenvalue of linear self adjoint partial differential operators let us be given a partial differential operator on a domain which consists of two nonoverlapping subdomains suppose a fast eigenvalue solver is available for each of these subdomains but not for the union of them the first proposed method is a scheme which determines an appropriate boundary condition at the interface separating the two regions this boundary condition can be derived from the zero of a nonlinear operator which plays the same role for the eigenvalue problem as the steklov poincare operator for the linear equation an iterative method can be used to solve this nonlinear equation yielding the exact boundary condition at the interface this enables the determination of the eigenpair on the whole domain the same concept can be applied to a schwarz alternating method for the eigenvalue problem in case the subdomains overlap a nonoverlapping scheme in the spirit of schur complement will also be discussed these ideas are also applicable to a domain imbedding algorithm

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

periodic solutions for the one dimensional fractional laplacian abstract in this paper we are concerned with the construction of periodic solutions of the nonlocal problem s u f u in r where s stands for the s laplacian s 0 1 we introduce a suitable framework which allows by means of regularity to link the searching of such solutions into the existence of the ones of a semilinear problem in a suitable hilbert space then by a bifurcation theory from eigenvalues of odd multiplicity and also variational method that avoid the constant solutions we get existence theorems which are lately enlightened with the analysis of some examples in particular multiplicity results for generalized benjamin ono equation are obtained

chebyshev pseudospectral solution of the stokes equations using finite element preconditioning the stokes equations are solved by a chebyshev pseudospectral method on a rectangular domain as the resulting system of algebraic equations is very difficult to factorize a preconditioning is designed using a finite element technique the fem solver constitutes the masterpiece of a richardson iteration process several finite elements are investigated the 9 nodes lagrangian element b q 2 b q 1 the b q 1 b q 0 element and the b q 1 b q 1 element due to brezzi and pitkaranta an eigenvalue analysis is carried out in order to pinpoint the characteristic features of each precondition it is shown that the b q 2 b q 1 element yields the best convergence results the power of this choice is demonstrated on theoretical solutions and on the regularized square cavity problem

overlapping schwarz methods on unstructured meshes using non matching coarse grids we consider two level overlapping schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions standard finite element interpolation from the coarse to the fine grid may be used our theory requires no assumption on the substructures that constitute the whole domain so the substructures can be of arbitrary shape and of different size the global coarse mesh is allowed to be non nested to the fine grid on which the discrete problem is to be solved and neither the coarse mesh nor the fine mesh need be quasi uniform in addition the domains defined by the fine and coarse grid need not be identical the one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which neumann boundary conditions are given in this general setting our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes the condition number of the preconditioned system depends only on the possibly small overlap of the substructures and the size of the coarse grid but is independent of the sizes of the subdomains

a novel domain decomposition framework for the ground state solution of bose einstein condensates abstract in this study we present a new type of domain decomposition algorithm to obtain the ground state solution for bose einstein condensates using our proposed algorithm instead of directly solving the nonlinear eigenvalue problem one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method and subsequently solve a small scale nonlinear eigenvalue problem on an enriched space simultaneously because solving large scale nonlinear eigenvalue problem directly is time intensive our algorithm can obviously improve the efficiency of producing simulation for bose einstein condensates in addition any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework which makes the algorithm considerably flexible two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

a spectral collocation method for the navier stokes equations abstract a fourier chebyshev spectral method for the incompressible navier stokes equations is described it is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time in this fully spectral algorithm a preconditioned iterative technique is used for solving the implicit equations arising from semi implicit treatment of pressure mean advection and vertical diffusion terms the algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity

nonlocal delaunay surfaces abstract we construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic cylindrically symmetric and decreasing these surfaces may be seen as a nonlocal analogue of the classical delaunay surfaces onduloids for small volume most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls in fact we give explicit quantitative bounds on these facts

the spectral expansion for a nonself adjoint hill operator with a locally integrable potential abstract we construct the spectral expansion for the one dimensional schrodinger operator l d 2 dx 2 q x x in l 2 where q x is a 1 periodic lebesgue integrable on 0 1 and complex valued potential we obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator l t for t 0 generated by this operation in l 2 0 1 and the t periodic boundary conditions using it we prove that the eigenfunctions and associated functions of l t form a riesz basis in l 2 0 1 for t 0 then we find the spectral expansion for the operator l

further study on periodic solutions of elliptic equations with a fractional laplacian abstract we obtain some existence theorems for periodic solutions to several linear and nonlinear equations involving fractional laplacian we also prove that the lower bound of all periods for semilinear elliptic equations involving fractional laplacian is not larger than some exact positive constant hamiltonian identity modica type inequalities and an estimate of the energy for periodic solutions are also established

on fractional gjms operators we describe a new interpretation of the fractional gjms operators as generalized dirichlet to neumann operators associated to weighted gjms operators on naturally associated smooth metric measure spaces this gives a geometric interpretation of the caffarelli silvestre extension for when 0 1 and both a geometric interpretation and a curved analogue of the higher order extension found by r yang for when 1 we give three applications of this correspondence first we exhibit some energy identities for the fractional gjms operators in terms of energies in the compactified poincare einstein manifold including an interpretation as a renormalized energy second for 1 2 we show that if the scalar curvature and the fractional q curvature q2 of the boundary are nonnegative then the fractional gjms operator p2 is nonnegative third by assuming additionally that q2 is not identically zero we show that p2 satisfies a strong maximum principle 2014 wiley periodicals inc

regularity for the fractional gelfand problem up to dimension 7 abstract we study the problem s u e u in a bounded domain r n where is a positive parameter more precisely we study the regularity of the extremal solution to this problem our main result yields the boundedness of the extremal solution in dimensions n 7 for all s 0 1 whenever is for every i 1 n convex in the x i direction and symmetric with respect to x i 0 the same holds if n 8 and s 0 28206 or if n 9 and s 0 63237 these results are new even in the unit ball b 1

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

radial fractional laplace operators and hessian inequalities abstract in this paper we deduce a formula for the fractional laplace operator s on radially symmetric functions useful for some applications we give a criterion of subharmonicity associated with s and apply it to a problem related to the hessian inequality of sobolev type r n k k 1 u k 1 d x c r n u f k u d x where f k is the k hessian operator on r n 1 k n 2 under some restrictions on a k convex function u in particular we show that the class of u for which the above inequality was established in ferrari et al 5 contains the extremal functions for the hessian sobolev inequality of x j wang 1994 15 this is proved using logarithmic convexity of the gaussian ratio of hypergeometric functions which might be of independent interest

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

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

overlapping schwarz methods for fekete and gauss lobatto spectral elements the classical overlapping schwarz algorithm is here extended to the triangular tetrahedral spectral element tsem discretization of elliptic problems this discretization based on fekete nodes is a generalization to nontensorial elements of the tensorial gauss lobatto legendre quadrilateral spectral elements qsem the overlapping schwarz preconditioners are based on partitioning the domain of the problem into overlapping subdomains solving local problems on these subdomains and solving an additional coarse problem associated with either the subdomain mesh or the spectral element mesh the overlap size is generous i e one element wide in the tsem case while it is minimal or variable in the qsem case the results of several numerical experiments show that the convergence rate of the proposed preconditioning algorithm is independent of the number of subdomains n and the spectral degree p in case of generous overlap otherwise it depends inversely on the overlap size the proposed preconditioners are also robust with respect to arbitrary jumps of the coefficients of the elliptic operator across subdomains

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity low order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill conditioning when the material approaches the incompressible limit if only the displacement variables are used mixed finite elements using both displacement and pressure variables provide a well known remedy but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging two level overlapping schwarz preconditioners for the almost incompressible system of linear elasticity discretized by mixed finite elements with discontinuous pressures are constructed and analyzed the preconditioned systems are accelerated either by a gmres generalized minimum residual method applied to the resulting discrete saddle point problem or by a pcg preconditioned conjugate gradient method applied to a positive definite although extremely ill conditioned reformulation of the problem obtained by eliminating all pressure variab

projection techniques for iterative solution of ax b with successive right hand sides we present two projection techniques for computing approximate solutions to linear systems of the form ax n b n for a sequence n 1 2 e g such as arises from time discretization of a partial differential equation the inexpensive approximate solutions can be used as initial guesses for iterative solution of the system resulting in significantly reduced computational expense examples of two and three dimensional incompressible navier stokes calculations are presented in which x represents the pressure and a is a discrete poisson operator in flows containing significant dynamic activity these projection techniques lead to as much as a two fold reduction in solution time

overlapping schwarz preconditioned eigensolvers for spectral element discretizations abstract model generalized eigenproblems associated with self adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered their numerical approximation is based on gauss lobatto legendre conforming spectral elements defined by gordon hall transfinite mappings the resulting discrete eigenproblems are solved iteratively with a locally optimal block preconditioned conjugate gradient lobpcg method accelerated by an overlapping schwarz preconditioner several numerical tests show the good convergence properties of the proposed preconditioned eigensolver such as its scalability and quasi optimality in the discretization parameters which are analogous to those obtained for overlapping schwarz preconditioners for linear systems

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom

semilinear elliptic equations with the pseudo relativistic operator on a bounded domain abstract we study the dirichlet problem for the semilinear equations involving the pseudo relativistic operator on a bounded domain m 2 m u u p 1 u in with the dirichlet boundary condition u 0 on here p 1 and the operator m 2 m is defined in terms of spectral decomposition in this paper we investigate existence and nonexistence of a nontrivial solution depending on the choice of p m and precisely we show that i if p is not h 1 subcritical p n 2 n 2 and is star shaped the equation has no nontrivial solution for all m 0 i i if p is not h 1 2 supercritical 1 p n 1 n 1 then there exists a least energy solution for all m 0 and any bounded domain i i i finally in the intermediate range n 1 n 1 p n 2 n 2 the problem has a nontrivial solution provided that m is sufficiently large and the problem 0 1 u u p 1 u in u 0 on admits a non degenerate nontrivial solution for example when is a ball or an annulus

on higher dimensional singularities for the fractional yamabe problem a nonlocal mazzeo pacard program we consider the problem of constructing solutions to the fractional yamabe problem which are singular at a given smooth submanifold for which we establish the classical gluing method of mazzeo and pacard j differential geom 1996 for the scalar curvature in the fractional setting this proof is based on the analysis of the model linearized operator which amounts to the study of a fractional order ordinary differential equation ode thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal odes note however that no traditional phase plane analysis is available here instead we first provide a rigorous construction of radial fast decaying solutions by a blowup argument and a bifurcation method then second we use conformal geometry to rewrite this nonlocal ode giving a hint of what a nonlocal phase plane analysis should be third for the linear theory we use complex analysis and some non euclidean harmonic analysis to examine a fractional schrodinger equation with a hardy type critical potential we construct its green s function deduce fredholm properties and analyze its asymptotics at the singular points in the spirit of frobenius method surprisingly enough a fractional linear ode may still have a 2 dimensional kernel as in the second order case