analysis of a zig zag array of circular inclusions in a solid under uniaxial tension abstract this paper is concerned with a theoretical analysis of an infinite zig zag array of circular inclusions in an infinite solid under uniaxial tension we take properly defined unit regions assume the complex stress potentials to he in the form of laurent and taylor series expansions and determine the unknown coefficients from the boundary conditions for the unit regions numerical results for the interface stresses and the effect of inclusions on the tensile stiffness of the solid are given for various combinations of the mechanical and geometric parameters the results are fitted to reliable polynomial formulac for the convenience of engineering applications

true scale invariant random spatial networks some aspects of real world road networks seem to have an approximate scale invariance property motivating study of mathematical models of random networks whose distributions are exactly invariant under euclidean scaling this requires working in the continuum plane so making a precise definition is not trivial we introduce an axiomatization of a class of processes we call scale invariant random spatial networks whose primitives are routes between each pair of points in the plane one concrete model based on minimum time routes in a binary hierarchy of roads with different speed limits has been shown to satisfy the axioms and two other constructions based on poisson line processes and on dynamic proximity graphs are expected also to do so we initiate study of structure theory and summary statistics for general processes in the class many questions arise in this setting via analogies with diverse existing topics from geodesics in first passage percolation to transit node based route finding algorithms

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

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

a new method for stokes problems with circular boundaries using degenerate kernel and fourier series summary this study is concerned with the stokes flow of an incompressible fluid of constant density and viscosity with circular boundaries to fully capture the circular boundary the boundary densities in the direct and indirect boundary integral equations bies are expanded in terms of fourier series the kernel functions in either the direct bie or the indirect bie are expanded to degenerate kernels by using the separation of field and source points consequently the improper integrals are transformed to series sum and are easily calculated the linear algebraic system can be established by matching the boundary conditions at the collocation points then the unknown fourier coefficients can be easily determined finally several examples including circular and eccentric domains are presented to demonstrate the validity of the present method five gains were obtained 1 meshless approach 2 free of boundary layer effect 3 singularity free 4 exponential convergence and 5 well posed model copyright q 2007 john wiley sons ltd

estimates for p harmonic functions vanishing on a flat we study p harmonic functions in a domain crn near an m dimensional plane an m flat m where 0 m n 1 in particular let u be a positive p harmonic function with n p vanishing on a porti

analysis of a zig zag array of circular holes in an infinite solid under uniaxial tension abstract this paper is concerned with a theoretical analysis of an infinite zig zag array of circular holes in an infinite solid under uniaxial tension we use properly defined unit regions assume complex stress potentials in the form of laurent series expansions and determine the unknown coefficients from the boundary conditions for the unit regions used numerical results for the maximum stress the stress concentration factor and the effect of perforation on the tensile stiffness of the solid are given for various hole sizes and hole spacings in both directions the results are fitted to reliable polynomial formulae for convenience in engineering applications

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

uniqueness for the signature of a path of bounded variation and the reduced path group we introduce the notions of tree like path and tree like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length we show that the equivalence classes form a group with some similarity to a free group and that in each class there is a unique path that is tree reduced the set of these paths is the reduced path group it is a continuous analogue of the group of reduced words the signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path we identify the paths with trivial signature as the tree like paths and prove that two paths are in tree like equivalence if and only if they have the same signature in this way we extend chen s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parameterisation in the general setting it is suggestive to think of this result as a non commutative analogue of the result that integrable functions on the circle are determined up to lebesgue null sets by their fourier coefficients as a second theme we give quantitative versions of chen s theorem in the case of lattice paths and paths with continuous derivative and as a corollary derive results on the triviality of exponential products in the tensor algebra

from random lines to metric spaces consider an improper poisson line process marked by positive speeds so as to satisfy a scale invariance property actually scale equivariance the line process can be characterized by its intensity measure which belongs to a one parameter family if scale and euclidean invariance are required this paper investigates a proposal by aldous namely that the line process could be used to produce a scale invariant random spatial network sirsn by means of connecting up points using paths which follow segments from the line process at the stipulated speeds it is shown that this does indeed produce a scale invariant network under suitable conditions on the parameter in fact it then produces a parameter dependent random geodesic metric for dd dimensional space d 2d 2 where geodesics are given by minimum time paths moreover in the planar case it is shown that the resulting geodesic metric space has an almost everywhere unique geodesic property that geodesics are locally of finite mean length and that if an independent poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region it is an open question whether the result is a sirsn in aldous sense so placing finite mean length in each compact region but it may be called a pre sirsn

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

intrinsic geometry and analysis of diffusion processes and l variational problems the aim of this paper is twofold first we obtain a better understanding of the intrinsic distance of diffusion processes precisely a for all n 1 the diffusion matrix a is weak upper semicontinuous on if and only if the intrinsic differential and the local intrinsic distance structures coincide b if n 1 or if n 2 and a is weak upper semicontinuous on the intrinsic distance and differential structures always coincide c if n 2 and a fails to be weak upper semicontinuous on the non coincidence of the intrinsic distance and differential structures depend on the geometry of the non weak upper semicontinuity set of a second for an arbitrary diffusion matrix a we show that the intrinsic distance completely determines the absolute minimizer of the corresponding l variational problem and then obtain the existence and uniqueness for given boundary data we also give an example of a diffusion matrix a for which there is an absolute minimizer that is not of class c1 when a is continuous we also obtain the linear approximation property of the absolute minimizer

new properties of convex functions in the heisenberg group we prove some new properties of the weakly h convex functions recently introduced by danielli garofalo and nhieu as an interesting application of our results we prove a theorem of busemann feller alexandrov type in the heisenberg groups h n 1 2

on the dirichlet and serrin problems for the inhomogeneous infinity laplacian in convex domains regularity and geometric results given an open bounded subset of mathbb r n which is convex and satisfies an interior sphere condition we consider the pde delta infty u 1 in subject to the homogeneous boundary condition u 0 on we prove that the unique solution to this dirichlet problem is power concave precisely 3 4 concave and it is of class c1 we then investigate the overdetermined serrin type problem formerly considered in buttazzo and kawohl int math res not pp 237 247 2011 obtained by adding the extra boundary condition nabla u a on by using a suitable p function we prove that if satisfies the same assumptions as above and in addition contains a ball which touches at two diametral points then the existence of a solution to this serrin type problem implies that necessarily the cut locus and the high ridge of coincide in turn in dimension n 2 this entails that must be a stadium like domain and in particular it must be a ball in case its boundary is of class c2

applied alternating method to analyze the stress concentration around interacting multiple circular holes in an infinite domain abstract this paper presents an efficient alternating method for analyzing the interactions among multiple circular holes in a two dimensional infinite domain an analytical solution is derived for a single circular hole in an infinite domain subjected to the arbitrary tractions across the circle boundary to achieve this purpose this analytical solution correlates with a successive iterative superposition process capable of satisfying the prescribed boundary condition for each circular hole of the problem in addition several perforated plate problems are solved to demonstrate the proposed methods validity the computed results and the available referenced solutions closely corresponds to each other and indicates the methods accuracy and efficiency

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

boundary differentiability of infinity harmonic functions abstract we prove that an infinity harmonic function u x c is differentiable at a boundary point x 0 if both and the boundary data g x are differentiable at x 0 this work improved a former result of changyou wang and yifeng yu 2012 10 they obtained the boundary differentiability of u x with the c 1 assumption on and g x

weak fubini property and infinity harmonic functions in riemannian and sub riemannian manifolds we examine the relationship between infinity harmonic functions absolutely minimizing lipschitz extensions strong absolutely minimizing lipschitz extensions and absolutely gradient minimizing extensions in carnot carath eodory spaces using the weak fubini property we show that absolutely minimizing lipschitz extensions are infinity harmonic in any sub riemannian manifold

c1 regularity for infinity harmonic functions in two dimensions a continuous function open image in new window is said to be infinity harmonic if it satisfies the pde

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

an embedding method for modeling micromechanical behavior and macroscopic properties of composite materials this paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber reinforced composites and perforated materials the material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions the rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated fourier series the analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method the boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation all of the integrals arising in the method can be evaluated analytically as a result the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials

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

boundary asymptotic and uniqueness of solutions to the p laplacian with infinite boundary values we prove boundary asymptotics to solutions of weighted p laplacian equations that take infinite value on the boundary of a bounded domain uniqueness of such solutions would then follow as a consequence our results extend previously known results by allowing weights that are unbounded in the domain

asymptotic estimates of boundary blow up solutions to the infinity laplace equations abstract in this paper we study the asymptotic behavior of boundary blow up solutions to the equation u b x f u in where is the laplacian the nonlinearity f is a positive increasing function in 0 and the weighted function b c is positive in and may vanish on the boundary we first establish the exact boundary blow up estimates with the first expansion when f is regularly varying at infinity with index p 3 and the weighted function b is controlled on the boundary in some manner furthermore for the case of f s s p 1 c g s with the function g normalized regularly varying with index q 0 we obtain the second expansion of solutions near the boundary it is interesting that the second term in the asymptotic expansion of boundary blow up solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the boundary blow up problems involving the classical laplacian

derivation of the aronsson equation for c1 hamiltonians it is proved herein that any absolute minimizer u for a suitable hamiltonian h c r r u is a viscosity solution of the aronsson equation hp du u x h du u x x 0 in u the primary advance is to weaken the assumption that h c used by previous authors to the natural condition that h c

boundary blow up solutions to degenerate elliptic equations with non monotone inhomogeneous terms abstract given a non negative and non trivial continuous real valued function h on 0 such that h x 0 0 for all x we study the boundary value problem bvp u h x u in u on where r n n 2 is a bounded domain and is the laplacian a degenerate elliptic operator in this paper we investigate conditions on the inhomogeneous term h x t under which problem bvp admits a solution or fails to admit a solution in c some notable features of this work are that h x t is not required to have any special structure and no monotonicity condition is imposed on h x t furthermore h x t may be allowed to vanish in either of the variables

the second order expansion of boundary blow up solutions for infinity laplacian equations abstract in this paper by using karamata regular variation theory and a perturbed argument we study the second order expansion of viscosity solutions to boundary blow up elliptic problem u b x f u x u where is a bounded domain with c 2 boundary in r n b c is non negative and non trivial in f c 1 0 is a normalized regularly varying function at infinity with index 3

relaxation of hamilton jacobi equations we study the relaxation of hamilton jacobi equations the relaxation in our terminology is the following phenomenon the pointwise supremum over a certain collection of subsolutions in the almost everywhere sense of a hamilton jacobi equation yields a viscosity solution of the convexified hamilton jacobi equation this phenomenon has recently been observed in 13 in eikonal equations we show in this paper that this relaxation is a common phenomenon for a wide range of hamilton jacobi equations

loosening of elastic inclusions abstract a numerical method is presented for simulating the occurrence of localized slip and separation along the interfaces of multiple randomly distributed circular elastic inclusions in an infinite elastic plane the method is an extension of a direct boundary integral approach previously described elsewhere for solving problems involving perfectly bonded circular inclusions here we allow displacement discontinuities to develop along the inclusion matrix interfaces in accordance with a linear mohr coulomb yield condition combined with a tensile strength cut off the displacements tractions and displacement discontinuities on the inclusion boundaries are all represented by truncated fourier series and an explicit iterative algorithm is adopted to determine zones of slip and separation under the prevailing loading conditions several examples are given to demonstrate the accuracy and generality of the approach

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

a galerkin boundary integral method for multiple circular elastic inclusions with homogeneously imperfect interfaces a galerkin boundary integral method is presented to solve the problem of an infinite isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with homogeneously imperfect interfaces problems of interest might involve thousands of inclusions with no restrictions on their locations except that the inclusions may not overlap sizes and elastic properties the tractions are assumed to be continuous across the interfaces and proportional to the corresponding displacement discontinuities the analysis is based on a numerical solution of a complex hypersingular integral equation with the unknown tractions and displacement discontinuities at each circular boundary approximated by truncated complex fourier series the method allows one to calculate the stress and displacement fields everywhere in the matrix and inside the inclusions numerical examples are included to demonstrate the effectiveness of the approach

on the problem of linear elasticity for an infinite region containing a finite number of non intersecting spherical inhomogeneities abstract the problem of linear elasticity for an infinite number of non intersecting spherical and more generally ellipsoidal inhomogeneities is attacked the approach taken does not misrepresent geometry of inhomogeneities although the continuity conditions at the intefaces are only approximately satisfied the principal idea of the approach is to extend the method of kachanov 1985 int j fracture28 r11 r19 1987 int j solids structures23 23 43 for interacting cracks to the realm of the eshelby equivalent inclusion method the application to a test problem for two spherical cavities suggests that the approach can be useful for predictions of the overall response of composite materials and interfacial stress concentrations

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

blow up rates of large solutions for elliptic equations in this paper we mainly study the boundary behavior of solutions to boundary blow up elliptic problems for more general nonlinearities f which may be rapidly varying at infinity u b x f u x u where is a bounded domain with smooth boundary in rn and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary further when f s sp f1 s for s sufficiently large where p 1 and f1 is normalized regularly varying at infinity with index p1 0 p we show the influence of the geometry of on the boundary behavior for solutions to the problem we also give the existence and uniqueness of solutions

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

asymptotic boundary estimates to infinity laplace equations with γ varying nonlinearity abstract in a previous paper of the authors wang et al 2014 40 the asymptotic estimates of boundary blow up solutions were established to the infinity laplace equation u b x f u in r n with the nonlinearity 0 f c 0 regularly varying at and the weighted function b c positive in and vanishing on the boundary the present paper gives a further investigation on the asymptotic behavior of boundary blow up solutions to the same equation by assuming f to be varying note that a varying function grows faster than any regularly varying function we first quantitatively determine the boundary blow up estimates with the first expansion relying on the decay rate of b near the boundary and the growth rate of f at infinity and further characterize these results via examples possessing various decay rates for b and growth rates for f in particular we pay attention to the second order estimates of boundary blow up solutions it was observed in our previous work that the second expansion of solutions to the infinity laplace equation is independent of the geometry of the domain quite different from the classical laplacian the second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow up solutions between the infinity laplacian and the classical laplacian

a galerkin boundary integral method for multiple circular elastic inclusions the problem of an infinite isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered the analysis procedure is based on the use of a complex singular integral equation the unknown tractions at each circular boundary are approximated by a truncated complex fourier series a system of linear algebraic equations is obtained by using the classical galerkin method and the gauss seidel algorithm is used to solve the system several numerical examples are considered to demonstrate the effectiveness of the approach copyright 2001 john wiley sons ltd

l variational problems and aronsson equations in this paper we prove that viscosity solutions of aronsson equations are absolute minimizers in certain l variational problems

boundary integral equation method for general viscoelastic analysis from basic assumptions of viscoelastic constitutive relations and weight residual techniques a boundary element procedure is achieved for both kelvin and boltzmann models imposing spatial approximations and adopting convenient kinematical relations for strain velocities a system of time differential equations is achieved this system is solved adopting linear approximations for displacements resulting in a time marching methodology this approach avoids the use of relaxation functions and makes easier changes in boundary conditions along time natural or essential an important feature of the resulting technique is the absence of domain discretizations which simplify the treatment of problems involving infinite domains tunnels and cavities inside the soil some examples are shown in order to demonstrate the accuracy and stability of the technique when compared to analytical solutions

counterexample to c1 boundary regularity of infinity harmonic functions abstract in this short note we construct a planar infinity harmonic function u in a c 1 domain with smooth boundary data g however d u is not continuous on the boundary changyou wang and yifeng yu 2012 proved the c 1 boundary regularity of planar infinity harmonic functions provided that and g are c 2 they asked the question whether the result holds when and g are assumed to be c 1 our counterexample answered this question negatively our construction of the counterexample is strongly inspired by the counterexample of dongsheng li and lihe wang 2006 for uniformly elliptic equations

on multiple circular inclusions in plane magnetoelasticity abstract a general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper by the use of complex variable theory and laurent series expansion method the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived expanding the definition of the airy s stress function of pure elastic field into the magnetoelastic field and applying the superposition method the general expression then can be reduced to a set of linear algebraic equations and solved in a series form an approximate closed form solution for the case of two arbitrarily located inclusions is also provided for illustrating the effect of the pertinent parameters the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form

a galerkin boundary integral method for multiple circular elastic inclusions with uniform interphase layers abstract the paper presents a numerical method for solving the problem of an infinite isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with uniform interphase layers the bonds between the inclusions and the interphases as well as between the interphases and the matrix are assumed to be perfect in general the inclusions may have different elastic properties and sizes the thicknesses of the interphases and their elastic properties are arbitrary the analysis is based on a numerical solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex fourier series the galerkin technique is used to obtain a system of linear algebraic equations the resulting numerical method allows one to calculate the elastic fields everywhere in the matrix and inside the inclusions and the interphases using the assumption of macro isotropic behavior in a plane section one can find the effective elastic moduli for an equivalent homogeneous material the method can be viewed as an extension of our previous work int j solids struct 39 2002 4723 where simpler spring like interface conditions were modeled the problem of overlapping of the fibers and matrix inherent to spring type interface is discussed in the context of the present model numerical examples are included to demonstrate the effectiveness of the new approach

elastic fields of interacting elliptic inhomogeneities abstract this paper presents a method for the calculation of two dimensional elastic fields in a solid containing any number of inhomogeneities under arbitrary far field loadings the method called pseudo dislocations method is illustrated for the solution of interacting elliptic inhomogeneities it reduces the interacting inhomogeneities problem to a set of linear algebraic equations numerical results are presented for a variety of elliptic inhomogeneity arrangements including the special cases of elliptic holes cracks and circular inhomogeneities all these complicated problems can be solved with high accuracy and efficiency

solution of periodic group circular hole problems by using the series expansion variational method this paper investigates the periodic group circular holes composed of infinite groups with numbering from j 2 1 0 1 2 to j placed periodically in an infinite plate the same loading condition and the same geometry are assumed for holes in all groups the series expansion variational method sevm is used for the solution of the periodic group circular hole problems after using the sevm the boundary value problem is then reduced to an algebraic equation for the undetermined coefficients in the series expansion form which is formulated on the central group the influences on the central group from central group itself and many neighbouring groups are evaluated exactly the influences on the central group from remote groups from j m 2 m 1 m 1 m 2 to j are approximately summed up into one term this suggested technique is called the remainder estimation technique ret hereafter it is proved from the computed results that the ret is very effective for the solution of the periodic group hole problems finally several numerical examples are given and the interaction between the groups is addressed comparison between various sources of computation is presented in the uniaxial tension in y direction the stacking effect of the stacked groups is studied copyright 2006 john wiley sons ltd

on the use of somigliana s formula and fourier series for elasticity problems with circular boundaries this paper considers the problem of an infinite isotropic elastic plane containing an arbitrary number of non overlapping circular holes and isotropic elastic inclusions the holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can if desired be different the analysis is based on the two dimensional version of somigliana s formula which gives the displacements at a point inside a region v in terms of integrals of the tractions and displacements over the boundary s of this region we take v to be the infinite plane and s to be an arbitrary number of circular holes within this plane any or all of the holes can contain an elastic inclusion and we assume for simplicity that all inclusions are perfectly bonded to the material matrix the displacements and tractions on each circular boundary are represented as truncated fourier series and all of the integrals involved in somigliana s formula are evaluated analytically an iterative solution algorithm is used to solve the resulting system of linear algebraic equations several examples are given to demonstrate the accuracy and efficiency of the numerical method copyright 2003 john wiley sons ltd

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

a special boundary element formulation for multiple circular hole problems in an infinite plate abstract a special boundary element formulation is proposed to solve the multiple circular hole problems in an infinite plate the main ideas of the proposed formulation are followed from two directions the first formulation in which all the unknowns are taken from the boundary tractions is like the boundary element method in the second all the derivations of the solution are derived from the complex variable method of plane elasticity using the principle of superposition a system of linear algebraic equations for the unknowns is established finally several numerical examples are given

weakly coupled systems of the infinity laplace equations we derive the weakly coupled systems of the infinity laplace equations via a tug of war game introduced by peres schramm sheffield and wilson 2009 we establish existence uniqueness results of the solutions and introduce a new notion of generalized cones for systems by using generalized cones we analyze blow up limits of solutions

on boundary blow up solutions to equations involving the laplacian abstract given a non negative continuous function h on r such that h x 0 0 for all x h x t 0 in 0 and h x t non decreasing in t for each x we study the boundary value problem u h x u in u on where r n n 2 is a bounded domain and is the laplacian a degenerate elliptic operator we provide sufficient conditions on h under which the above problem admits a solution or fails to admit a solution a necessary and sufficient condition on f is given for a solution to exist in the special case when h x t b x f t in the latter case an asymptotic boundary behavior of solutions will be studied as an application a sufficient condition on f will be given to ensure the uniqueness of solutions in case b is a constant

on the evolution governed by the infinity laplacian we investigate the basic properties of the degenerate and singular evolution equation ut d 2 u du du du du which is a parabolic version of the increasingly popular infinity laplace equa tion we prove existence and uniqueness results for both dirichlet and cauchy problems establish interior and boundary lipschitz estimates and a harnack inequality and also provide numerous explicit solutions

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

boundary behaviour for solutions of boundary blow up problems in a borderline case we investigate boundary blow up solutions of the equation u f u in a bounded domain rn under the condition that f t has a relatively slow growth as t goes to infinity we show how the mean curvature of the boundary appears in the asymptotic expansion of the solution u x in terms of the distance of x from

multiple void crack interaction abstract this paper presents a method for fracture analysis of a general two dimensional system containing multiple holes or voids and cracks the superposition technique is used to break the problem into a number of single hole and single crack problems each hole problem is modeled using the method of pseudo tractions and each crack problem is modeled by a distribution of dislocations an integral equation approach is developed based on two types of fundamental solutions one due to point loads in a solid with a hole and the other due to point dislocations in an infinite elastic body the resulting integral equations present cauchy type singularities only on the crack part of the multiple hole crack problem the results in terms of stress intensity factors sifs are presented for a variety of hole and crack arrangements relative sizes of cracks and holes spacings and crack orientations the amplification and retardation effects on sifs are investigated it is found that the hole crack arrangements have significant effects on the nature of the amplification or retardation in the fractured porous elastic medium modeled as a crack surrounded by holes amplification or retardation can occur depending on the relative size of the holes and cracks and the spacing between them very strong retardation exists as the spacing becomes small some optimal retardations void toughening are achieved by adjusting the geometry parameters an array of periodical crack hole structure is examined as a numerical example

analysis of stress concentrations in plates with rectangular openings by a combined conformal mapping finite element approach plates with rectangular openings develop stress concentrations under bending while these stresses can be determined using finite elements in many problems this would be difficult because a high density mesh would be needed in the neighborhood of every opening corner in this paper it is shown how a complex variable conformal mapping approach can be numerically coupled with the finite element method to analyze these corner stresses this approach can be used even with relatively coarse meshes where the finite element results by themselves do not resolve the stress concentrations in essence the method relies on finite element analysis to obtain information on the stress field in a region surrounding the plate opening this information is subsequently used to set up the parameters of the conformal mapping approach to obtain the near field stresses at the opening corners

boundary behavior for large solutions to elliptic equations with singular weights abstract in this paper we analyze the boundary behavior of large positive solutions to some semilinear elliptic equations which include a singular weight the most important point is that the growth of the solutions can be determined in terms of the solution to a one dimensional first order equation we also consider the questions of existence and uniqueness of positive solutions

uniqueness and nonuniqueness of viscosity solutions to aronsson s equation for a bounded domain omega subset mathbb r n and g in c delta omega assume that h in c 2 mathbb r n is convex and coercive and that p in mathbb r n h p rm min mathbb r n h has no interior points then we establish the uniqueness of viscosity solutions to the dirichlet problem of aronsson s equation begin aligned mathcal a u h p nabla u otimes h p nabla u nabla 2 u 0 rm in omega u g rm on delta omega end aligned for h h p x depending on x we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to aronsson s equation and that of the hamilton jacobi equation h nabla u x c

on a family of torsional creep problems involving rapidly growing operators in divergence form let n n 2 be a bounded domain with smooth boundary and p n be a sequence of real numbers converging to as n for each integer n 1 we define the function varphi n t p n vert t vert p n 2 te vert t vert p n for all t and we prove the existence of a unique nonnegative variational solution for the problem div n u x u x u x n 1 when x subject to the homogeneous dirichlet boundary condition next we establish the uniform convergence in of the sequence of solutions for the above family of equations to the distance function to the boundary of our result complements the earlier developments on the topic obtained by payne and philippin 26 kawohl 21 bhattacharya dibenedetto and manfredi 2 perez llanos and rossi 27 and bocea and mih ilescu 4

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

an axiomatic approach to scalar data interpolation on surfaces we discuss possible algorithms for interpolating data given on a set of curves in a surface of 3 we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolutely minimizing lipschitz extension model amle is singled out and studied in more detail we study the correctness of our numerical approach and we show experiments illustrating the interpolation of data on some simple test surfaces like the sphere and the torus

on multiple circular inclusions in plane thermoelasticity a general series solution to the problem of interacting circular inclusions in plane thermoelasticity is provided in this paper based upon the complex variable theory and the use of laurent series expansion the general expression of the stress functions is derived explicitly for the circular inclusion problem under remote uniform heat flow by applying the use of the superposition the problem dealing with any number of arbitrarily located inclusions can be then reduced to a set of linear algebraic equations which are solved with the aid of a perturbation technique for illustrating the use of the present approach an approximate closed form solution of the stress functions is derived explicitly for the problem containing two arbitrarily located inclusions numerical results of the interfacial stresses around a rigid circular inclusion or hoop stress along a circular hole due to the presence of an elastic inclusion are provided to demonstrate the dependence of the solution upon the pertinent parameters

a hybrid micro macro bem formulation for micro crack clusters in elastic components local analysis schemes capable of detailed representations of the micro features of a problem are integrated with a macro scale bem technique capable of handling complex finite geometries and realistic boundary conditions the micro scale effects are introduced into the macro scale bem analysis through an augmented fundamental solution obtained from an integral equation representation of the micro scale features the proposed hybrid micro macro bem formulation allows decomposition of the complete problem into two sub problems one residing entirely at the micro level and the other at the macro level this allows for investigations of the effects of the micro structural attributes while retaining the macro scale geometric features and actual boundary conditions for the component or structure under consideration as a first attempt elastic fracture mechanics problems with interacting cracks at close spacings are considered the numerical results obtained from the hybrid bem analysis establish the accuracy and effectiveness of the proposed micro macro computational scheme for this class of problems the proposed micro macro bem formulation can easily be extended to investigate the effects of other micro features e g interfaces short or continuous fibre reinforcements voids and inclusions in the context of linear elasticity on macroscopic failure modes observed in structural components

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

a sharp uniqueness result for a class of variational problems solved by a distance function we consider the minimization problem for an integral functional j possibly nonconvex and non coercive in w 1 1 0 where r n is a bounded smooth set we prove su cient conditions in order to guarantee that a suitable minkowski distance is a minimizer of j the main result is a necessary and su cient condition in order to have the uniqueness of the minimizer we show some application to the uniqueness of solution of a system of pdes of monge kantorovich type arising in problems of mass transfer theory

on the behaviour of harmonic functions near isolated points abstract in this work we study the behaviour of harmonic functions near isolated exceptional points we discuss two situations i when points are in the interior and ii when points are on a flat portion of the boundary on which a solution vanishes we consider both bounded and unbounded solutions under the assumptions of boundedness solutions are well behaved in either case we show that unbounded solutions in i change sign and provide some lower bounds for growth rates valid for both i and ii in ii if the solution has one sign then we can also find an upper bound we also look at the case of the ball