a kernel free particle finite element method for hypervelocity impact simulation an improved hybrid particle finite element method has been developed for the simulation of hypervelocity impact problems unlike alternative methods the revised formulation computes the density without reference to any kernel or interpolation functions for either the density or the rate of dilatation this simplifies the state space model and leads to a significant reduction in computational cost the improved method introduces internal energy variables as generalized co ordinates in a new formulation of the thermomechanical lagrange equations example problems show good agreement with exact solutions in one dimension and good agreement with experimental data in a three dimensional simulation copyright 2005 john wiley sons ltd

a priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient abstract we consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient such as f x u d u d 2 u c x u m x d u d u h x in a bounded domain with a dirichlet boundary condition here r c h l p p n 1 c 0 and the matrix m satisfies 0 1 i m 2 i recently this problem was studied in the coercive case c 0 where uniqueness of solutions can be expected and it was conjectured that the solution set is more complex for noncoercive equations this conjecture was verified in 2015 by arcoya de coster jeanjean and tanaka for equations in divergence form by exploiting the integral formulation of the problem here we show that similar phenomena occur for general even fully nonlinear equations in nondivergence form we use different techniques based on the maximum principle we develop a new method to obtain the crucial uniform a priori bounds which permit to us to use degree theory this method is based on basic regularity estimates such as half harnack inequalities and on a vazquez type strong maximum principle for our kind of equations

to the theory of viscosity solutions for uniformly elliptic isaacs equations we show how a theorem about the solvability in c 1 1 c 1 1 of special isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate isaacs equations we apply it also to establish the c 1 c 1 regularity of viscosity solutions and show that finite difference approximations have an algebraic rate of convergence the main coefficients of the isaacs equations are supposed to be in c c with slightly less than 1 2

on fractional order differential equations model for nonlocal epidemics a fractional order model for nonlocal epidemics is given stability of fractional order equations is studied the results are expected to be relevant to foot and mouth disease sars and avian flu

a carleson type inequality for fully nonlinear elliptic equations with non lipschitz drift term abstract this paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term we elaborate on the non homogeneous generalized harnack inequality proved by the second author in 26 to prove a generalized carleson estimate we also prove boundary holder continuity and a boundary harnack type inequality

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

a priori bounds for elliptic equations abstract in this paper we study the relationship existing between the maximum principle and the principal eigenvalue of second order elliptic operators and the expected exit times of the corresponding diffusions a probabilistic approach is discussed that provides a good understanding of classical results established analytically we also obtain an alexandrov bakelman pucci type of estimate using similar methods

an explicit material point finite element method for hyper velocity impact in this paper an explicit material point finite element fe method is proposed and a computer code empfe 3d is developed for simulating hyper velocity impact the material domain is discretized by a mesh of finite elements the momentum equations are solved on a predefined computational grid like the material point method in the large deformation zone and on the fe mesh like the traditional fe method elsewhere the grid may be fixed in space or moved in a predefined way the nodes covered by the grid are treated as material particles and the remaining nodes are treated as fe nodes the proposed method yields the same results as the traditional fe method if the grid vanishes on the other hand it yields the same results as the material point method if the grid covers the entire material domain at all time steps the method combines the advantages of eulerian and lagrangian descriptions of motion while eliminates their drawbacks due to element entanglement and numerical dissipation the method is computationally efficient and can be easily implemented in an existing explicit fe code like dyna3d copyright 2005 john wiley sons ltd

bernstein type techniques for 2d free boundary graphs we prove an a priori bound for the lipschitz constant of a smooth one phase free boundary graph f u in two dimensions the function u satisfies an elliptic equation in its positive side and nabla u 1 on f u

eigenvalues and eigenfunctions of metric measure manifolds in this paper we study the eigenvalues and eigenfunctions of metric measure manifolds we prove that any eigenfunction is c 1 alpha at its critical points and c infty elsewhere moreover the eigenfunction corresponding to the first eigenvalue in the dirichlet problem does not change sign we also discuss the first eigenvalue the sobolev constants and their relationship with the isoperimetric constants 2000 mathematics subject classification 47j05 47j10 53c60 58e05 58c40

regularity of free boundaries of two phase problems for fully nonlinear elliptic equations of second order i lipschitz free boundaries are c1 α in this paper we study an extension of a c1 regularity theory developed by l caffarelli in 2 to some fully nonlinear elliptic equations of second order in fact we investigate a two phase free boundary problem in which a fully nonlinear elliptic equation of second order is verified by the solution in the positive and the negative domains assuming the free boundary is locally a lipschitz graph we have established the c1 regularity of the free boundary 2000 john wiley sons inc

maximum principle for fully nonlinear equations via the iterated comparison function method we present various versions of generalized aleksandrov bakelman pucci abp maximum principle for l p viscosity solutions of fully nonlinear second order elliptic and parabolic equations with possibly superlinear growth gradient terms and unbounded coefficients we derive the results via the iterated comparison function method which was introduced in our previous paper koike and wiech in nonlin diff eq appl 11 491 509 2004 for fully nonlinear elliptic equations our results extend those of koike and wiech in nonlin diff eq appl 11 491 509 2004 and fok in comm partial diff eq 23 5 6 967 983 in the elliptic case and of crandall et al in indiana univ math j 47 4 1293 1326 1998 comm partial diff eq 25 1997 2053 2000 wang in comm pure appl math 45 27 76 1992 and crandall and wiech in lecture notes in pure and applied mathematics vol 234 dekker new york 2003 in the parabolic case

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

pucci s conjecture and the alexandrov inequality for elliptic pdes in the plane the inequality of alexandrov bakel man and pucci is a basic tool in the theory of linear elliptic partial differential equations pdes which are not in divergence form as well as in the more general theory of nonlinear elliptic pdes here in two dimensions we prove the sharp form of the maximum principle as conjectured by pucci in 1966 give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting these results make use of the substantial recent advances in the planar theory of quasiconformal mappings astala was supported in part by the academy of finland projects 34082 and 41933 iwaniec by the national science foundation grants dms 0301582 and dms 0244297 martin by the n z marsden fund and the n z royal society james cook fellowship ams 1991 classification primary 30c60 35j15 35j70

directional ellipticity on special domains weak maximum and phragmen lindelof principles abstract we prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains r n of cylindrical type the main structural assumption is the uniform ellipticity of the operator along the bounded directions of with possible degeneracy along the unbounded directions

two phase problems for a class of fully nonlinear elliptic operators lipschitz free boundaries are c 1 γ we prove c 1 regularity of lipschitz free boundaries of two phase problems for a class of homogeneous fully nonlinear elliptic operators f d 2 u x x with holder dependence on x containing convex concave operators

reproducing kernel particle methods for large deformation analysis of non linear structures abstract large deformation analysis of non linear elastic and inelastic structures based on reproducing kernel particle methods rkpm is presented the method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis the current formulation considers hyperelastic and elasto plastic materials since they represent path independent and path dependent material behaviors respectively in this paper a material kernel function and an rkpm material shape function are introduced for large deformation analysis the support of the rkpm material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation the essential boundary conditions are introduced by the use of a transformation method the transformation matrix is formed only once at the initial stage if the rkpm material shape functions are employed the appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions in transient problems with an explicit time integration method the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles several hyperelasticity and elasto plasticity problems are studied to demonstrate the effectiveness of the method the numerical results indicated that rkpm handles large material distortion more effectively than finite elements due to its smoother shape functions and consequently provides a higher solution accuracy under large deformation unlike the conventional finite element approach the nodal spacing irregularity in rkpm does not lead to irregular mesh shape that significantly deteriorates solution accuracy no volumetric locking is observed when applying non linear rkpm to nearly incompressible hyperelasticity and perfect plasticity problems further model adaptivity in rkpm can be accomplished simply by adding more points in the highly deformed areas without remeshing

estimates for solutions to equations of p laplace type in ahlfors regular nta domains this thesis consists of six scientific papers an introduction and a summary all six papers concern the boundary behavior of non negative solutions to partial differential equations paper i concerns solutions to certain p laplace type operators with variable coefficients suppose that u is a non negative solution that vanishes on a part of an ahlfors regular nta domain we prove among other things that the gradient du of u has non tangential limits almost everywhere on the boundary piece and that log du is a bmo function on the boundary furthermore for ahlfors regular nta domains that are uniformly n r0 approximable by lipschitz graph domains we prove a boundary harnack inequality provided that is small enough paper ii concerns solutions to a p laplace type operator with lower order terms in reifenberg flat domains we prove that the ratio of two non negative solutions vanishing on a part of the boundary is holder continuous provided that is small enough furthermore we solve the martin boundary problem provided is small enough in paper iii we prove that the boundary type riesz measure associated to an a capacitary function in a reifenberg flat domain with vanishing constant is asymptotically optimal doubling paper iv concerns the boundary behavior of solutions to certain parabolic equations of p laplace type in lipschitz cylinders among other things we prove an intrinsic carleson type estimate for the degenerate case and a weak intrinsic carleson type estimate in the singular supercritical case in paper v we are concerned with equations of p laplace type structured on hormander vector fields we prove that the boundary type riesz measure associated to a non negative solution that vanishes on a part of an x nta domain is doubling on paper vi concerns a one phase free boundary problem for linear elliptic equations of non divergence type assume that we know that the positivity set is an nta domain and that the free boundary is a graph furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction we prove that the graph giving the free boundary is lipschitz continuous

decay of heat semigroups in l and applications to nonlinear parabolic problems in unbounded domains we characterize all domains of rn such that the heat semigroup decays in l l or l l1 as t namely we prove that this property is equivalent to the poincare inequality and that it is also equivalent to the solvability of u f in l for all f l in particular under mild regularity assumptions on these properties are equivalent to the geometric condition that has finite inradius next we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains first we consider the quenching problem for singular parabolic equations we prove that the solution in quenches in finite time no matter how small the nonlinearity is if and only if does not fulfill the poincare inequality second for the semilinear heat equation with a power nonlinearity we prove roughly speaking that the trivial solution is stable in l or in l1 if and only if has finite inradius

paradox of enrichment a fractional differential approach with memory abstract the paradox of enrichment poe proposed by rosenzweig m rosenzweig the paradox of enrichment science 171 1971 385 387 is still a fundamental problem in ecology most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels poe resolution via these two organizational levels can be interpreted as a microscopic and macroscopic level solution fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes the derivatives can be physically interpreted via two time scales that are considered simultaneously the ideal equably flowing homogeneous local time and the cosmic inhomogeneous non local time several mechanisms and theories have been proposed to resolve the poe problem but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non local effects which capture memory here we formulate the fractional counterpart of the rosenzweig model and analyze the stability behavior of a system we conclude that there is a threshold for the memory effect parameter beyond which the rosenzweig model is stable and may be used as a potential agent to resolve poe from a new perspective via fractional differential equations

material point method applied to multiphase flows the particle in cell method pic especially the latest version of it the material point method mpm has shown significant advantage over the pure lagrangian method or the pure eulerian method in numerical simulations of problems involving large deformations it avoids the mesh distortion and tangling issues associated with lagrangian methods and the advection errors associated with eulerian methods its application to multiphase flows or multi material deformations however encounters a numerical difficulty of satisfying continuity requirement due to the inconsistence of the interpolation schemes used for different phases it is shown in section 3 that current methods of enforcing this requirement either leads to erroneous results or can cause significant accumulation of errors in the present paper a different numerical method is introduced to ensure that the continuity requirement is satisfied with an error consistent with the discretization error and will not grow beyond that during the time advancement in the calculation this method is independent of physical models its numerical implementation is quite similar to the common method used in eulerian calculations of multiphase flows examples calculated using this method are presented

a mesh grading material point method and its parallelization for problems with localized extreme deformation abstract as a kind of meshless method material point method mpm applies an eulerian background grid served as a finite element mesh in each time step and therefore its accuracy and efficiency are mainly dependent on the cell size setting of background grid however the conventional mpm commonly uses a regular background grid with uniform cells which is not apposite for localized extreme deformation problems from the view point of computation efficiency where in fact a local refined background grid is preferable hence a mesh grading material point method mgmpm is proposed here for such problems to supply mpm with the ability for local refinement simulation the edge displacement continuity associated with mesh grading is embedded in the nodal shape functions besides the truss element is incorporated into mgmpm to model the steel reinforcement bars in reinforced concrete impacting problems based on our previous work furthermore the proposed method is parallelized using openmp open multi processing to take advantage of pc power with multi core and hyper threading technologies for large scale engineering problems where both loop level parallelism and code block parallelism are used several numerical examples including stress wave propagation taylor bar impact and penetration problems are studied which show that the efficiency of mgmpm is much higher than that of conventional mpm and with lower memory requirement

a particle method for history dependent materials abstract a broad class of engineering problems including penetration impact and large rotations of solid bodies causes severe numerical problems for these problems the constitutive equations are history dependent so material points must be followed this is difficult to implement in a eulerian scheme on the other hand purely lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables a process which can introduce errors proposed here is an extension of the particle in cell method in which particles are interpreted to be material points that are followed through the complete loading process a fixed eulerian grid provides the means for determining a spatial gradient because the grid can also be interpreted as an updated lagrangian frame the usual convection term in the acceleration associated with eulerian formulations does not appear with the use of maps between material points and the grid the advantages of both eulerian and lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced also it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity

on the alexandroff bakelman pucci estimate and the reversed holder inequality for solutions of elliptic and parabolic equations the constant appearing in the classical alexandroff bakelman pucci estimate for subsolutions of second order uniformly elliptic equations in nondivergence form was known to depend on the diameter of the domain using the krylov safonov boundary weak harnack inequality due to trudinger we show that the dependence on the diameter may be replaced by dependence on a more precise geometric quantity of the domain as a consequence we get dependence on the measure instead of the diameter we also give new bounds for subsolutions in some unbounded domains such as domains contained in cones we apply the fabes and stroock reversed holder inequality for the green s function to improve our estimates we also give a new proof of the reversed holder inequality for the green s function based on the krylov safonov harnack inequality finally we find new bounds for subsolutions of uniformly parabolic equations in cylindrical and noncylindrical domains the constant in the alexandroff bakelman pucci krylov tso estimate was known to depend on the diameter of the base of the cylinder we get dependence either on the measure of the base or on the height of the cylinder we also give bounds for subsolutions in noncylindrical domains 1995 john wiley sons inc

an adaptive finite element material point method and its application in extreme deformation problems abstract taking advantages of both lagrangian and eulerian methods material point method mpm is suitable for modeling problems with extreme deformation however mpm is less accurate and less efficient than finite element method fem for small deformation problems due to particle quadrature and mappings between particles and background grid applied in mpm to take advantages of both fem and mpm an adaptive finite element material point method is developed for modeling the dynamic behavior of material under extreme loading bodies are initially discretized by finite elements and then the elements with large strain are adaptively converted into mpm particles based on their degree of distortion or plastic strain during the solution process the interaction between the remaining finite elements and mpm particles is implemented based on the background grid in mpm framework several numerical examples are presented to validate the efficiency and accuracy of the proposed method and the numerical results are in good agreement with experiments while the efficiency of the method is higher than that of both mpm and fem

a new aleksandrov bakelman pucci maximum principle for p laplacian operator abstract we present a new aleksandrov bakelman pucci maximum principle for p laplacian operator our proof is based on the solution of an obstacle problem

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

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

on the aleksandrov bakel man pucci estimate for some elliptic and parabolic nonlinear operators in this work we prove the aleksandrov bakel man pucci estimate for possibly degenerate nonlinear elliptic and parabolic equations of the form rm div left f left nabla u x right right f left x right quad rm in omega subset mathbb r n and u t x t rm div left f left nabla u x t right right f left x t right quad rm in q subset mathbb r n 1 for f a fancyscript c 1 monotone field under some suitable conditions examples of applications such as the p laplacian and the mean curvature flow are considered as well as extensions of the general results to equations that are not in divergence form such as the m curvature flow

a harnack inequality approach to the regularity of free boundaries part ii flat free boundaries are lipschitz soit u une solution faible d un probleme aux limites libres du type u 0 sur u 0 et sur u 0 et soit le long de la frontiere libre f la relation u v g u x est satisfaite au sens faible alors si f est proche du graphe d une fonction de lipschitz dans la boule unite b 1 de r n f est le graphe d une fonction de lipschitz dans b 1 2

time discontinuous material point method for transient problems abstract this paper presents a time discontinuous material point method tdmpm for transient problems such as the wave propagation and impact responses in solids by dividing the continuous time domain into discrete time intervals the weak form of the tdmpm is established by considering the discrete grid based governing equations constraint and discontinuity conditions the displacement and velocity fields in a time interval are interpolated with the piecewise cubic and linear functions respectively by substituting the assumed displacement and velocity fields into the weak form a novel computational framework for the grid displacements and velocities at the discrete time instants is constructed in the new formulations the displacement field at each time instant remains to be continuous whereas the velocity field at the time instant becomes discontinuous these unique features ensure the tdmpm could properly capture the discontinuous characteristics and control the spurious numerical oscillations two numerical examples under the impact loading are used to verify the proposed method two representative impact problems are then presented for further verification and demonstration besides the corresponding contact algorithm adopted in the tdmpm is shown to be capable of capturing the correct contact behavior with higher fidelity and less computational cost than the mpm the presented results illustrate that the tdmpm could successfully control the spurious numerical oscillations associated with transient simulations

fully nonlinear singularly perturbed equations and asymptotic free boundaries in this paper we study one phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials e u with e 0 we establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non degenerated for problems governed by concave equations we establish uniform weak geometric properties of approximating level surfaces we also provide a thorough analysis of the free boundary problem obtained as a limit as the e parameter term goes to zero we find the precise jumping condition of limiting solutions through the phase transition which involves a subtle homogenization process of the governing fully nonlinear operator in particular for rotational invariant operators f d2u we show the normal derivative of limiting function is constant along the interface smoothness properties of the free boundary are also addressed

numerical simulation for coupled systems of nonlinear fractional order integro differential equations via wavelets method abstract in this paper a new method for solving coupled systems of nonlinear fractional order integro differential equations is proposed the idea is to use bernoulli wavelets and operational matrix the main purpose of the technique is to transform the studied systems of fractional order integro differential equations into systems of algebraic equations which can be solved easily illustrative examples and comparisons with haar wavelets and legendre wavelets are included to reveal the effectiveness of the method and the accuracy of the convergence analysis

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

an evaluation of explicit time integration schemes for use with the generalized interpolation material point method the stability and accuracy of the generalized interpolation material point gimp method is measured directly through carefully formulated manufactured solutions over wide ranges of cfl numbers and mesh sizes the manufactured solutions are described in detail the accuracy and stability of several time integration schemes are compared via numerical experiments the effect of various treatments of particle size are also considered the hypothesis that gimp is most accurate when particles remain contiguous and non overlapping is confirmed by comparing manufactured solutions with and without this property

liouville property and regularity of a hessian quotient equation we are concerned with properties of convex solutions to the hessian quotient equation sn c d2 p 1 k n as our first main result we prove some regularity of strong solutions while the second states that for p 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

material point method enhanced by modified gradient of shape function a numerical scheme of computing quantities involving gradients of shape functions is introduced for the material point method mpm so that the quantities are continuous as material points move across cell boundaries the noise and instability caused by cell crossing of the material points are then eliminated in this scheme the formulas used to compute these quantities can be expressed in the same forms as in the original material point method but with the gradient of the shape function modified for one dimensional cases the gradient of the shape function used in the generalized interpolation material point gimp method is a special case of the modified gradient if the characteristic function of a material point is introduced the characteristic function of a material point is not otherwise needed in this scheme therefore difficulties in tracking its evolution are avoided although the support of the modified gradient of a shape function is enlarged from the cell containing the material point to also include the immediate neighbor cells all the non local effects of a material point can be accounted for by two consecutive local operations therefore this scheme can be used in calculations with unstructured grids this scheme is proved to satisfy mass and momentum conservations exactly the error in energy conservation is shown to be second order on both spatial and temporal discretizations although the error in energy conservation is the same order as that in the original material point method numerical examples show that this scheme has significantly better energy conservation properties than those of the original material point method

regularity for shape optimizers the nondegenerate case we consider minimizers of f 1 n 1 where f is a function strictly increasing in each parameter and k is the kth dirichlet eigenvalue of our main result is that the reduced boundary of the minimizer is composed of c1 graphs and exhausts the topological boundary except for a set of hausdorff dimension at most n 3 we also obtain a new regularity result for vector valued bernoulli type free boundary problems 2018 wiley periodicals inc

harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations abstract we present a new harnack inequality for non negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices this estimate applies to all supersolutions and has the harnack constant depending on the graph distance on lattices for the proof we modify the proof of the weak harnack inequality applying the same idea to elliptic equations in a euclidean space we also derive a harnack type inequality for non negative viscosity supersolutions

optimal control of impulsive volterra equations with variable impulse times we obtain necessary conditions of optimality for impulsive volterra integral equations with switching and impulsive controls with variable impulse time instants the present work continues and complements our previous work on impulsive volterra control with fixed impulse times

gradient estimates for viscosity solutions of singular fully nonlinear elliptic equations abstract we establish lipschitz regularity for solutions to a family of non isotropic fully nonlinear partial differential equations of elliptic type in general such a regularity is optimal no sign constraint is imposed on the solution thus limiting free boundaries may have two phases our estimates are then employed in combination with fine regularizing techniques to prove existence of viscosity solutions to singular nonlinear pdes

two phase problems for linear elliptic operators with variable coefficients lipschitz free boundaries are c 1 γ we prove c 1 regularity of lipschitz free boundaries of two phase problems for linear elliptic operators with holder continuous coefficients

regularity of the free boundary in two phase problems for linear elliptic operators abstract in this paper we complete the study of the regularity of the free boundary in two phase problems for linear elliptic operators started in m c cerutti f ferrari s salsa two phase problems for linear elliptic operators with variable coefficients lipschitz free boundaries are c 1 arch ration mech anal 171 2004 329 348 in particular we prove that lipschitz and flat free boundaries in a suitable sense are smooth as byproduct we prove that lipschitz free boundaries are smooth in the case of quasilinear operators of the form div a x u u with lipschitz coefficients

bolza problems with discontinuous lagrangians and lipschitz continuity of the value function we study the local lipschitz continuity of the value function v associated with a bolza problem in the presence of a lagrangian l x q convex and uniformly superlinear in q but only borel measurable in x under these assumptions the associated integral functional is not lower semicontinuous with respect to the suitable topology which ensures the existence of minimizers so all results known in the literature fail to apply yet the lipschitz regularity of v does not depend on the existence of minimizers in fact it is enough to control the derivatives of quasi minimal curves but the problem is nontrivial due to the general growth conditions assumed here on l x cdot we propose a new approach based on suitable reparameterization arguments to obtain suitable a priori estimates on the lipschitz constants of quasi minimizers as a consequence of our analysis we derive the lipschitz continuity of v and a compactness result for value functions associated with sequences of locally equibounded discontinuous lagrangians

on lp viscosity solutions of bilateral obstacle problems with unbounded ingredients the global equi continuity estimate on l p viscosity solutions of bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous the existence of l p viscosity solutions is established via an approximation of given data the local holder continuity estimate on the first derivative of l p viscosity solutions is shown when the obstacles belong to c 1 beta and p n

regularity of lipschitz free boundaries in two phase problems for the p laplace operator in this paper we continue the study in ln6 concerning the regularity of the free boundary in a general two phase free boundary problem for the p laplace operator by proving regularity of the free boundary assuming that the free boundary is close to a lipschitz graph

revisited fisher s equation in a new outlook a fractional derivative approach abstract the well known fisher equation with fractional derivative is considered to provide some characteristics of memory embedded into the system the modified model is analyzed both analytically and numerically a comparatively new technique residual power series method is used for finding approximate solutions of the modified fisher model a new technique combining sinc collocation and finite difference method is used for numerical study the abundance of the bird species phalacrocorax carbois considered as a test bed to validate the model outcome using estimated parameters we conjecture non diffusive and diffusive fractional fisher equation represents the same dynamics in the interval memory index 0 8384 0 9986 we also observe that when the value of memory index is close to zero the solutions bifurcate and produce a wave like pattern we conclude that the survivability of the species increases for long range memory index these findings are similar to fisher observation and act in a similar fashion that advantageous genes do

a priori bounds for an indefinite superlinear elliptic equation with exponential growth this paper considers positive solutions to problem u u h x eu x r2 where is a parameter h x c2 r2 r changes sign uniform a priori bounds are obtained using less restrictive conditions from previous works

a qualitative phragmen lindelof theorem for fully nonlinear elliptic equations abstract we establish qualitative results of phragmen lindelof type for upper semicontinuous viscosity solutions of fully nonlinear partial differential inequalities of the second order in general unbounded domains of r n

on the uniqueness of a solution of a two phase free boundary problem in this paper we study the uniqueness problem of a two phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data we show that in general the comparison principle between the sub and super solutions does not hold and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter examples in various cases in any dimension in one dimension a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed in fact the critical case signifies the change from uniqueness to non uniqueness of a solution of the free boundary problem non uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes such as melting of ice or solidification of water even with the same prescribed boundary data however we prove that a uniqueness theorem is true for the initial boundary value problem of an e evolutionary problem which is the smoothed two phase parabolic free boundary problem

global positivity estimates and harnack inequalities for the fast diffusion equation abstract we investigate local and global properties of positive solutions to the fast diffusion equation u t u m in the range d 2 d m 1 corresponding to general nonnegative initial data for the cauchy problem posed in the whole euclidean space r d we prove sharp local positivity estimates weak harnack inequalities and elliptic harnack inequalities we use them to derive sharp global positivity estimates and a global harnack principle for the mixed initial and boundary value problem posed in a bounded domain of r d with homogeneous dirichlet condition we prove weak and elliptic harnack inequalities our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow

optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations in this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients more precisely we deal with coefficients and right hand sides rhs in lq with q geq n this extends the result of caffarelli on the c loc 1 1 regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded rhs moreover we also provide a regularity result with estimates for omega semiconvex functions that are supersolutions to the same type of equations with unbounded rhs i e rhs in l q q geq n by a completely different method our results here extend the recent regularity results obtained by braga et al adv math 334 184 242 2018 for q n as far as fully nonlinear pdes are concerned these results include in particular the apriori estimate obtained by caffarelli et al commun pure appl math 38 2 209 252 1985 on the modulus of continuity of the gradient of omega semiconvex supersolutions for linear equations and bounded rhs that have a holder modulus of semiconvexity

on symmetry properties of parabolic equations in bounded domains abstract we study symmetry properties of nonnegative bounded solutions of fully nonlinear parabolic equations on bounded domains with dirichlet boundary conditions we propose sufficient conditions on the equation and domain which guarantee asymptotic symmetry of solutions

memory effects and macroscopic manifestation of randomness it is shown that due to memory effects the complex behavior of components in a stochastic system can be transmitted to macroscopic evolution of the system as a whole within the markov approximation widely used in ordinary statistical mechanics memory effects are neglected as a result a time scale separation between the macroscopic and the microscopic level of description exists the macroscopic differential picture is not a consequence of microscopic nondifferentiable dynamics on the other hand the presence of complete memory in a system means that all its components have the same behavior if the memory function has no characteristic time scales the correct description of the macroscopic evolution of such systems has to be in terms of the fractional calculus

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method qacm ii a novel strategy for developing low order membrane elements with analytical element stiffness matrices is proposed first some complete low order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method qacm ii then these solutions are taken as the trial functions for developing new membrane elements thus the interpolation formulae for displacement fields naturally possess second order completeness in physical space cartesian coordinates finally by introducing nodal conforming conditions new 4 node and 5 node membrane elements with analytical element stiffness matrices are successfully constructed the resulting models denoted as qac atf4 and qac atf5 have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added these two elements exhibit excellent performance in various bending problems with mesh distortion it is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method and the qacm ii is a powerful tool for constructing high performance quadrilateral finite element models copyright 2008 john wiley sons ltd

maximum principle in some unbounded domains for an elliptic operator with unbounded drift abstract we study the maximum principle for wg domain which can be unbounded the wg condition was introduced by a vitolo and the maximum principle was proved for the uniformly elliptic operator with bounded coefficients using safonov s growth lemma we treat the operator when the first order coefficients belong to n integrable lebesgue space some examples including infinite cones various unbounded coefficients are presented

on the maximum principle for complete second order elliptic operators in general domains abstract this paper is concerned with the maximum principle for second order linear elliptic equations in a wide generality by means of a geometric condition previously stressed by berestycki nirenberg varadhan cabre was very able to improve the classical abp estimate obtaining the maximum principle also in unbounded domains such as infinite strips and open connected cones with closure different from the whole space now we introduce a new geometric condition that extends the result to a more general class of domains including the complements of hypersurfaces as for instance the cut plane the methods developed here allow us to deal with complete second order equations where the admissible first order term forced to be zero in a preceding result with cafagna depends on the geometry of the domain

estimates in the generalized campanato john nirenberg spaces for fully nonlinear elliptic equations we establish estimates in bmo and campanato john nirenberg spaces bmo for the second derivatives of solutions to the fully nonlinear elliptic equation f d2u x f x

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

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

positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains abstract we consider the dirichlet problem for the semilinear equation u f u 0 on a bounded domain r n we assume that is convex in a direction e and symmetric about the hyperplane h x r n x e 0 it is known that if n 2 and is of class c 2 then any nonzero nonnegative solution is necessarily strictly positive and consequently it is reflectionally symmetric about h and decreasing in the direction e on the set x x e 0 in this paper we prove the same result for a large class of nonsmooth planar domains in particular the result is valid if any of the following additional conditions on holds i is convex not necessarily symmetric in the direction perpendicular to e ii is strictly convex in the direction e iii is piecewise c 1 1

liouville type theorems monotonicity results and a priori bounds for positive solutions of elliptic systems this paper is a contribution to the study of boundary value problems for systems of elliptic partial differential equations

equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains we consider the dirichlet problem for a class of semilinear parabolic equations on a bounded domain which is reflectionally symmetric about a hyperplane h the equations consist of a symmetric time autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity in our first theorem we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem the novelty of this result is that the solutions considered are not assumed uniformly positive which prevents one from applying common techniques based on the method of moving hyperplanes in our second main theorem we classify the positive entire solutions of the unperturbed time autonomous problems in particular we characterize all entire solutions which are not symmetrically decreasing in the direction orthogonal to h as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set

inhomogeneous hopf oleĭnik lemma and regularity of semiconvex supersolutions via new barriers for the pucci extremal operators abstract in this paper we construct new barriers for the pucci extremal operators with unbounded rhs the geometry of these barriers is given by a harnack inequality up to the boundary type estimate under the possession of these barriers we prove a new quantitative version of the hopf ole nik lemma for quasilinear elliptic equations with g laplace type growth finally we prove sharp regularity for semiconvex supersolutions for some nonlinear pdes these results are new even for second order linear elliptic equations in nondivergence form moreover these estimates extend and improve a classical a priori estimate proven by l caffarelli j j kohn j spruck and l nirenberg in 13 in 1985 as well as a more recent result on the c 1 1 regularity for convex supersolutions obtained by c imbert in 33 in 2006

landesman lazer type results for second order hamilton jacobi bellman equations we study the boundary value problem f d2u du u x u f x u in u 0on where the second order differential operator f is of hamilton jacobi bellman type f is sub linear in u at infinity and rn is a regular bounded domain we extend the well known landesman lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator we provide conditions under which the solution branches extend globally along the eigenvalue gap we also present examples illustrating the results and hypotheses

on the phragmen lindelof principle for second order elliptic equations abstract this paper is concerned with the maximum principle for subsolutions of second order elliptic equations in non divergence form in unbounded domains eventually the zero order term can change sign and the involved functions can be unbounded at infinity with an admissible growth depending on the geometric properties of the domain following gilbarg and hopf we also show a phragmen lindelof principle in angular sectors and give an example of an interesting field of application to nonlinear equations deriving comparison principles for quasi linear operators

nondivergent elliptic equations on manifolds with nonnegative curvature we consider a class of second order linear elliptic operators intrinsically defined on riemannian manifolds that correspond to nondivergent operators in euclidean space under the assumption that the sectional curvature is nonnegative we prove a global krylov safonov harnack inequality and as a consequence a liouville theorem for solutions of such equations from the harnack inequality we obtain alexandroff bakelman pucci estimates and maximum principles for subsolutions 1997 john wiley sons inc

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations