non element method of solving 2d boundary problems defined on polygonal domains modeled by navier equation the paper presents a non element method of solving boundary problems defined on polygonal domains modeled by corner points to solve these problems a parametric integral equation system pies is used the system is characterized by a separation of the approximation of boundary geometry from the approximation of boundary functions this feature makes it possible to effectively investigate the convergence of the obtained solutions with no need of performing the approximation of boundary geometry the testing examples included confirm high accuracy of the solutions

the neumann problem for the laplacian and the monge kantorovich mass transfer problem abstract we consider the natural neumann boundary condition for the laplacian we study the limit as p of solutions of p u p 0 in a domain with d u p p 2 u p g on we obtain a natural minimization problem that is verified by a limit point of u p and a limit problem that is satisfied in the viscosity sense it turns out that the limit variational problem is related to the monge kantorovich mass transfer problems when the measures are supported on

a hermite interpolation algorithm for hypersingular boundary integrals this paper presents a conforming c1 boundary integral algorithm based on hermite interpolation this work is motivated by the requirement that the surface function multiplying a hypersingular kernel be differentiable at the collocation nodes the unknown surface derivatives utilized by the hermite approximation are determined consistent with other boundary values by writing a tangential hypersingular equation hypersingular equations are primarily invoked for solving crack problems and the focus herein is on developing a suitable approximation for this geometry test calculations for the laplace equation in two dimensions indicate that the algorithm is a promising technique for three dimensional problems

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

symmetric galerkin boundary integral formulation for interface and multi zone problems domains containing an internal boundary such as a bi material interface arise in many applications e g composite materials and geophysical simulations this paper presents a symmetric galerkin boundary integral method for this important class of problems in this situation the physical quantities are known to satisfy continuity conditions across the interface but no boundary conditions are specied the algorithm described herein achieves a symmetric matrix of reduced size moreover the symmetry can also be invoked to lessen the numerical work involved in constructing the system of equations and thus the method is computationally very ecient a prototype numerical example with several variations in the boundary conditions and material properties is employed to validate the formulation and corresponding numerical procedure the boundary element results are compared with analytical solutions and with numerical results obtained with the nite element method 1997 by john wiley sons ltd

boundary integration and interpolation procedures for plate bending in the direct boundary element method dbem for thin plate bending analysis the integration of the kernel shape function products can lead to complications due to a combination of factors the first is the existence of a singularity at the source point leading to the singular kernel which increases in strength as the integral equation is differentiated once for rotation again for moment and yet again for shear the second factor arises from the use of piecewise interpolation polynomials in the first part of the dbem analysis where the unspecified boundary values are computed the numerical procedures are fairly straightforward it is found in this paper that special care must be taken in calculating internal actions at domain points the problems which can be encountered are demonstrated and a modified boundary interpolation procedure is described examples are given which show the superiority of the recommended approach

infinity laplacian equation with strong absorptions abstract we study regularity properties of solutions to reaction diffusion equations ruled by the infinity laplacian operator we focus our analysis in models presenting plateaus i e regions where a non negative solution vanishes identically we obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets in particular we show that the n hausdorff measure of the plateaus boundary is finite for a universal number 0

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

iterative solution of hermite boundary integral equations an efficient iterative method for the solution of the linear equations arising from a hermite boundary integral approximation has been developed along with equations for the boundary unknowns the hermite system incorporates equations for the first order surface derivatives gradient of the potential and is therefore substantially larger than the matrix for a corresponding linear approximation however by exploiting the structure of the hermite matrix a two level iterative algorithm has been shown to provide a very efficient solution algorithm in this approach the boundary function unknowns are treated separately from the gradient taking advantage of the sparsity and near positive definiteness of the gradient equations in test problems the new algorithm significantly reduced computation time compared to iterative solution applied to the full matrix this approach should prove to be even more effective for the larger systems encountered in three dimensional analysis and increased efficiency should come from pre conditioning of the non sparse matrix component

numerical solution of bernoulli type free boundary value problems by variable domain method the variable domain method for a free surface problem is considered as a weighted residual method this considerably simplifies discretization of the problem and at the same time allows implementation of any standard discretization technique discretization of the problem results in a non linear system of equations for discrete values of a field variable and co ordinates of the nodes on the free boundary variables corresponding to a field variable are eliminated and the resulting non linear system is of considerable smaller size some new update strategies of the jacobian together with discussion of the cost effectiveness and covergence rate are introduced and compared to broyden s method

nonexistence results for a class of fractional elliptic boundary value problems abstract in this paper we study a class of fractional elliptic problems of the form s u f x u in u 0 in r n where s 0 1 we prove nonexistence of positive solutions when is star shaped and f is supercritical we also derive a nonexistence result for subcritical f in some unbounded domains the argument relies on the method of moving spheres applied to a reformulated problem using the caffarelli silvestre extension caffarelli and silvestre 2007 11 of a solution of the above problem

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

a local boundary integral equation method for potential problems boundary integral equation boundary element methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed further benefits would be obtained if attention could be confined to information at one small part of the boundary the particular region of interest in a given problem a local boundary integral equation method based on a taylor series expansion of the unknown function is developed to do this for two dimensional potential problems governed by laplace s equation very accurate local values of the function and its derivatives can be obtained the method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques

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

on the treatment of corners in the boundary element method abstract a new technique for treating surface discontinuities within boundary element calculations is proposed multiple nodes are used to represent the geometry and the necessary additional equations are obtained by differentiating the usual boundary element integral equation in deriving expressions for the resulting singular integrals it is found that constraints must be placed on the functional approximation at the discontinuity an algorithm for incorporating these constraints is developed and numerical tests for an exactly solvable three dimensional laplace equation problem are presented

an optimization problem with volume constraint for a degenerate quasilinear operator abstract we consider the optimization problem of minimizing u p d x with a constraint on the volume of u 0 we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

on schrodinger operators with multipolar inverse square potentials abstract positivity essential self adjointness and spectral properties of a class of schrodinger operators with multipolar inverse square potentials are discussed in particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established

bezier curves in the modification of boundary integral equations bie for potential boundary values problems the paper presents a modification of the classical boundary integral equation method biem for two dimensional potential boundary values problems the proposed modification consists in describing the boundary geometry by means of bezier curves as a result of this analytical modification of the biem a new parametric integral equation system pies was obtained the kernels of these equations include the geometry of the boundary this new pies is no longer defined on the boundary as in the case of the biem but on the straight line for any given domain the solution of the new pies does not require a boundary discretization since it can be reduced merely to an approximation of boundary functions to solve this pies a pseudospectral method has been proposed and the results obtained were compared with exact solutions

modification of the boundary integral equation for the navier lame equations in 2d elastoplastic problems and its numerical solving summary the paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2d elastoplastic problems the modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements this was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation for approximation of plastic strains the lagrange polynomials with various number and arrangement of interpolation nodes were used reliability of the modification was verified on examples with analytical solutions copyright 2016 john wiley sons ltd

existence of classical solutions to a free boundary problem for the p laplace operator i the exterior convex case in this paper we prove under convexity assumptions for the data the existence of classical solutions for a bernoulli type free boundary problem with the p laplacian as the governing operator the method employed here originates from a pioneering work of a beurling where he proves the existence for the harmonic case in the plane though with no geometrical restrictions

a minimum problem with free boundary in orlicz spaces abstract we consider the optimization problem of minimizing g u u 0 d x in the class of functions w 1 g with u 0 w 0 1 g for a given 0 0 and bounded w 1 g is the class of weakly differentiable functions with g u d x the conditions on the function g allow for a different behavior at 0 and at we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface also we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the lipschitz regularity of the weak solutions and the c 1 regularity of their free boundaries near flat free boundary points

hardy lieb thirring inequalities for fractional schrodinger operators we show that the lieb thirring inequalities on moments of negative eigenvalues of schrodinger like operators remain true with possibly different constants when the critical hardy weight c x 2 is subtracted from the laplace operator we do so by first establishing a sobolev inequality for such operators similar results are true for fractional powers of the laplacian and the hardy weight and in particular for relativistic schrodinger operators we also allow for the inclusion of magnetic vector potentials as an application we extend for the first time the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge z 2 for less than some critical value

on fractional multi singular schrodinger operators positivity and localization of binding abstract in this work we investigate positivity properties of nonlocal schrodinger type operators driven by the fractional laplacian with multipolar critical and locally homogeneous potentials on one hand we develop a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions while on the other hand we study the localization of binding for this kind of potentials combining these two tools and performing an inductive procedure on the number of poles we establish necessary and sufficient conditions for the existence of a configuration of poles that ensures the positivity of the corresponding schrodinger operator

source field superposition analysis of two dimensional potential problems this method for problems governed by laplace s equation involves representing the potential field in a given solution domain as a superposition of fields generated by a number of sousces

reaction diffusion equations for the infinity laplacian abstract we derive sharp regularity for viscosity solutions of an inhomogeneous infinity laplace equation across the free boundary when the right hand side does not change sign and satisfies a certain growth condition we prove geometric regularity estimates for solutions and conclude that once the source term is comparable to a homogeneous function then the free boundary is a porous set and hence has zero lebesgue measure additionally we derive a liouville type theorem when near the origin the right hand side grows not faster than third degree homogeneous function we show that if a non negative viscosity solution vanishes at a point then it has to vanish everywhere

hermitian cubic boundary elements for two dimensional potential problems a hermite interpolation based formulation is presented for the boundary element analysis of two dimensional potential problems two three noded hermitian cubic elements hce are introduced for the modelling of corners or points with non unique tangents on the boundary these elements along with the usual two noded hce are used in numerical examples the results obtained show that faster convergence can be achieved using hce compared with using lagrange interpolation type quadratic elements qe for about the same amount of computing resources

degenerate scale problem when solving laplace s equation by bem and its treatment in this paper laplace problems are solved by using the dual boundary element method bem it is found that a degenerate scale problem occurs if the conventional bem is used in this case the influence matrix is rank deficient and numerical results become unstable both the circular and elliptical bars are studied analytically in the continuous system in the discrete system the fredholm alternative theorem in conjunction with the svd singular value decomposition updating documents is employed to sort out the spurious mode which causes the numerical instability three regularization techniques method of adding a rigid body mode hypersingular formulation and cheef combined helmholtz exterior integral equation formulation concept are employed to deal with the rank deficiency problem the addition of a rigid body term c in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e c the torsion rigidities are obtained and compared with analytical solutions numerical examples including elliptical square and triangular bars were demonstrated to show the failure of conventional bem in case of the degenerate scale after employing the three regularization techniques the accuracy of the proposed approaches is achieved copyright 2004 john wiley sons ltd

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

an optimization problem with volume constraint in orlicz spaces we consider the optimization problem of minimizing g u dx in the class of functions w1 g with a constraint on the volume of u 0 the conditions on the function g allow for a different behavior at 0 and at we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed

estimates for the optimal constants in multipolar hardy inequalities for schrodinger and dirac operators by expanding squares we prove several hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities these inequalities involve the l2 norm such results are generalized to an arbitrary number of singularities and compared with standard results given by the ims method the generalized version of hardy inequalities with several singularities is equivalent to some spectral information on a schrodinger operator involving a potential with several inverse square singularities we also give a generalized hardy inequality for dirac operators in the case of a potential having several singularities of coulomb type which are critical for dirac operators

three dimensional fracture simulation with a single domain direct boundary element formulation an efficient hypersingular boundary integral equation method for three dimensional fracture mechanics was presented in a previous paper the details of the numerical implementation of this method are further discussed herein in particular an algorithm for achieving the required differentiability of the crack surface displacement function is discussed to illustrate the utility of the method computational results for several strongly interacting multiple crack geometries are presented the calculated stress intensity factors are in excellent agreement with those obtained by an approximate analytical method due to kachanov and laures

semilinear elliptic equations for the fractional laplacian with hardy potential abstract in this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the hardy potential

an optimization problem with volume constraint with applications to optimal mass transport in this manuscript we study the following optimization problem with volume constraint min left frac 1 p int omega nabla v pdx int partial omega gv ds colon v in w 1 p left omega right text and v 0 leq alpha right here g is acontinuous function and alpha is a fixed constant such that 0 0 we prove that a minimizer exists and satisfies left begin array ccl delta p u p 0 text in u p 0 cup u p 0 5pt nabla u p p 2 frac partial u p partial nu g text on partial omega cap partial u p 0 cup u p 0 5pt u p 0 alpha end array right next we analyze the limit as p to infty we obtain that any sequence of weak solutions converges up to a subsequence lim limits p j to infty u p j x u infty x uniformly in overline omega and uniform limits u infty are solutions to the maximization problem with volume constraint max left int partial omega gv ds colon v in w 1 infty left omega right nabla v l infty omega leq 1 text and v 0 leq alpha right furthermore we obtain the limit equation that is verified by u infty in the viscosity sense finally it turns out that such a limit variational problem is connected to the monge kantorovich mass transfer problem with the involved measures are supported on partial omega and along the limiting free boundary partial u infty neq 0

an optimization problem with volume constraint we consider the optimization problem of minimizing dirichlet integral int omega nabla u 2 with volume constraint on the set u 0 we use a penalization method which for small values of the penalization parameter leads to a solution of the original problem