on lp x norms the relation between a banach function space x and its subspaces formed by i those functions with absolutely continuous norm ii those with continuous norm and iii the closure of the set of

double phase problems with variable growth abstract we consider a class of double phase variational integrals driven by nonhomogeneous potentials we study the associated euler equation and we highlight the existence of two different rayleigh quotients one of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues the notion of eigenvalue is understood in the sense of pairs of nonlinear operators as introduced by fucik necas soucek and soucek the analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p x laplace operator the generalized mean curvature operator or the capillarity differential operator with variable exponent the results contained in this paper complement the pioneering contributions of marcellini mingione et al in the field of variational integrals with unbalanced growth

eigenvalue problems for anisotropic elliptic equations an orlicz sobolev space setting abstract the paper studies a class of anisotropic eigenvalue problems involving an elliptic nonhomogeneous differential operator on a bounded domain from r n with a smooth boundary some results regarding the existence or non existence of eigenvalues are obtained in each case the competition between the growth rates of the anisotropic coefficients plays an essential role in the description of the set of eigenvalues

renaming in an asynchronous environment this paper is concerned with the solvability of the problem of processor renaming in unreliable completely asynchronous distributed systems fischer et al prove in 8 that nontrivial consensus cannot be attained in such systems even when only a single benign processor failure is possible in contrast this paper shows that problems of processor renaming can be solved even in the presence of up to t n 2 faulty processors contradicting the widely held belief that no nontrivial problem can be solved in such a system the problems deal with renaming processors so as to reduce the size of the initial name space when only uniqueness of the new names is required we present a lower bound of n 1 on the size of the new name space and a renaming algorithm that establishes an upper bound on n t if the new names are required also to preserve the original order a tight bound of 2 n t 1 1 is obtained

numerical methods for a multicomponent two phase interface model with geometric mean influence parameters in this paper we consider an interface model for multicomponent two phase fluids with geometric mean influence parameters which is popularly used to model and predict surface tension in practical applications for this model there are two major challenges in theoretical analysis and numerical simulation the first one is that the influence parameter matrix is not positive definite the second one is the complicated structure of the energy function which requires us to find out a physically consistent treatment to overcome these two challenging problems we reduce the formulation of the energy function by employing a linear transformation and a weighted molar density and furthermore we propose a local minimum grand potential energy condition to establish the relation between the weighted molar density and mixture compositions from this we prove the existence of the solution under proper conditions and prove the maximum principle of the weighted molar density for numerical simulation we propose a

multiple solutions for a class of p x laplacian equations in involving the critical exponent in this paper we first establish a principle of concentration compactness in then based on this concentration compactness principle we study the existence of solutions for a class of p x lapl

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

an adaptive finite element method for simulating surface tension with the gradient theory of fluid interfaces the gradient theory for the surface tension of simple fluids and mixtures is rigorously analyzed based on mathematical theory the finite element approximation of surface tension is developed and analyzed and moreover an adaptive finite element method based on a physical based estimator is proposed and it can be coupled efficiently with newton s method as well the numerical tests are carried out both to verify the proposed theory and to demonstrate the efficiency of the proposed method

γ convergence of power law functionals with variable exponents abstract convergence results for power law functionals with variable exponents are obtained the main motivation comes from the study of first failure dielectric breakdown some connections with the generalization of the laplace equation to the variable exponent setting are also explored

an energy stable and convergent finite difference scheme for the modified phase field crystal equation we present an unconditionally energy stable finite difference scheme for the modified phase field crystal equation a generalized damped wave equation for which the usual phase field crystal equation is a special degenerate case the method is based on a convex splitting of a discrete pseudoenergy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present a local in time error estimate that ensures the pointwise convergence of the scheme

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

existence and multiplicity of solutions for dirichlet problem of p x laplacian type without the ambrosetti rabinowitz condition abstract this paper deals with the following variational problem with variable exponents div 1 u p x 1 u 2 p x u p x 2 u f x u in u 0 on where r n n 2 be a bounded domain with the smooth boundary p r is a lipschitz continuous function with 1 p ess inf x p x p x p ess sup x p x n and f c r r is superlinear but does not satisfy the usual ambrosetti rabinowitz type condition under three different superlinear conditions on f at infinity we prove that the above equation has at least a nontrivial solution moreover the existence of infinite many solutions is proved for odd nonlinearity especially some new tricks are introduced to show the boundedness of cerami sequences

new combinatorial topology bounds for renaming the upper bound in the renaming task n 1 processes start with unique input names from a large space and must choose unique output names taken from a smaller name space 0 1 k to rule out trivial solutions a protocol must be anonymous the value chosen by a process can depend on its input name and on the execution but not on the specific process id attiya et al 1990 showed that renaming has a wait free solution when k 2n several algebraic topology proofs of a lower bound stating that no such protocol exists when k

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

an energy stable and convergent finite difference scheme for the phase field crystal equation we present an unconditionally energy stable finite difference scheme for the phase field crystal equation the method is based on a convex splitting of a discrete energy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present local in time error estimates that ensure the convergence of the scheme while this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related swift hohenberg equation as well

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

existence and multiplicity of solutions for a neumann problem involving the p x laplace operator abstract in this paper we study the following nonlinear neumann boundary value problem div u p x 2 u u p x 2 u f x u x t r u v 0 x t r where r n is a bounded domain with smooth boundary u v is the outer unit normal derivative on 0 is a real number p is a continuous function on with inf x p x 1 f r r is a continuous function using the three critical point theorem due to ricceri under the appropriate assumptions on f we establish the existence of at least three solutions of this problem some known results are generalized

existence and multiplicity of solutions for a general p x laplacian neumann problem abstract in this paper we verify that a general p x laplacian neumann problem has at least three weak solutions which generalizes the corresponding result of the reference mihai mihilescu existence and multiplicity of solutions for a neumann problem involving the p x laplace operator nonlinear anal 67 2007 1419 1425

the principle of concentration compactness in lp x spaces and its application abstract in this paper we establish a principle of concentration compactness in l p x spaces and apply it to obtain the existence of solutions for p x laplacian equations with critical growth

eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent in this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power like variable reaction term one of the main features of our work is that the differential operator involves partial derivatives with different variable exponents so that the functional analytic framework relies upon anisotropic sobolev and lebesgue spaces existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations

fast and accurate coarsening simulation with an unconditionally stable time step we present cahn hilliard and allen cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy controlled simulation our stability analysis is based on eyre s theorem and unconditional von neumann stability analysis both of which we present numerical tests confirm the accuracy of the von neumann approach which is straightforward and should be widely applicable in phase field modeling for the cahn hilliard case we show that accuracy can be controlled with an unbounded time step dt that grows with time t as dt t a we develop a classification scheme for the step exponenta and demonstrate that a class of simple linear algorithms gives a51 3 for this class the speedup relative to a fixed time step grows with n the linear size of the system as n lnn with conservative choices for the parameters controlling accuracy and finite size effects we find that an 8192 2 lattice can be integrated 300 times faster than with the euler method

compact trace in weighted variable exponent sobolev spaces w1 p x ω v0 v1 abstract we study trace operators in weighted variable exponent sobolev spaces w 1 p x v 0 v 1 l q x w for sufficiently regular unbounded domain r n n 2 with noncompact boundary where p x is a lipschitz continuous function defined on satisfying 1 p p n we show that when ess inf x n 1 q x n p x 1 0 the trace operators w 1 p x v 0 v 1 l q x w are compact under certain conditions on weight functions v 0 v 1 w

decoupled energy stable schemes for phase field vesicle membrane model we consider the numerical approximations of the classical phase field vesicle membrane models proposed a decade ago in du et al 2004 6 we first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish the energy dissipation law then we develop a stabilized decoupled time discretization scheme for the coupled nonlinear system the scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step stability analysis and ample numerical simulations are presented thereafter

integral operators on the halfspace in generalized lebesgue spaces lp part ii in this paper we generalize a version of the classical calderon zygmund theorem on principle value integrals in generalized lebesgue spaces lp proved in j reine angew math 563 2003 197 220 to kernels which do not satisfy standard estimates on rd 1 this result will be used in part ii of this paper to prove the classical theorem on halfspace estimates of agmon douglis and nirenberg comm pure appl math 12 1959 623 727 for generalized lebesgue spaces lp

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

existence of solutions for systems arising in electromagnetism abstract in this paper we study the following p x curl systems u p x 2 u a x u p x 2 u f x u g x u u 0 in u p x 2 u n 0 u n 0 on where r 3 is a bounded simply connected domain with a c 1 1 boundary denoted by p 1 is a continuous function a l f g r 3 r 3 are caratheodory functions and are two parameters using variational arguments based on fountain theorem and dual fountain theorem we establish some existence and non existence results for solutions of this problem our main results generalize the results of xiang et al 2017 41 bahrouni and repovs 2018 9 and ge and lu 2019 22

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

quasistatic analysis on configuration of two phase flow in y shaped tubes we investigate the two phase flow in a horizontally placed y shaped tube with different young s angle and width in each branch by using a quasistatic approach we can determine the specific contact position and the equilibrium contact angle of fluid in each branch based on the minimization problem of the free energy of the system the wettability condition and the width of the two branches play important roles in the distribution of fluid in each branch we also consider the effect of gravity some fluid in the upper branch will be pulled down due to the competition of the surface energy and the gravitational energy the result provides some insights on the theory of two phase flow in porous media in particular it highlights that the inhomogeneous wettability distribution affects the direction of the fluid penetrating a given porous medium domain it also sheds light on the current debate whether relative permeability may be considered as a full tensor rather than a scalar

harnack s inequality for p harmonic functions with unbounded exponent p abstract we study properties of the function u lim u where u is the solution of the min p laplacian dirichlet problem with bounded sobolev boundary function here p n is a variable exponent such that 1 p is lipschitz continuous we derive bloch type estimates and using them we prove harnack s inequality in cases of unbounded but finite exponent

entire solutions of multivalued nonlinear schrodinger equations in sobolev spaces with variable exponent abstract we establish the existence of an entire solution for a class of stationary schrodinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows up at infinity the abstract framework is related to lebesgue sobolev spaces with variable exponent the proof is based on the critical point theory in the sense of clarke and we apply chang s version of the mountain pass lemma without the palais smale condition for locally lipschitz functionals our result generalizes in a nonsmooth framework a result of rabinowitz p h rabinowitz on a class of nonlinear schrodinger equations z angew math phys zamp 43 1992 270 291 on the existence of ground state solutions of the nonlinear schrodinger equation

stabilized semi implicit numerical schemes for the cahn hilliard like equation with variable interfacial parameter abstract we present stabilized semi implicit schemes for the mmc tdgl equation a cahn hilliard like equation with variable interfacial parameter by introducing two stabilization terms we decompose the 4th order variable coefficient pde into two decoupled helmholtz type equations with the constant coefficients the proposed scheme is proved to be stable which means the decay of energy is preserved due to the periodic boundary condition the fast fourier transformation method is applied to accelerate the computations to improve the stability of the proposed scheme we present a nonlinear stabilized semi implicit scheme which can be simulated efficiently with fixed point iteration for the nonlinear terms some numerical experiments are carried out to simulate the phase transition to verify the total energy decay and to observe the influence of some parameters the strictly energy stable scheme saves much cpu time due to the constant coefficients with fft method without losing the accuracy it is the key novelty

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

on the superlinear problems involving the p x laplacian and a non local term without ar condition abstract in this paper we consider a kind of nonlinear eigenvalue problem without ambrosetti rabinowitz condition that is u div u p x 2 u f x u a e in u 0 on where r n is a bounded domain p 1 is a continuous function u is a non local term defined by the following relation u 2 1 p x u p x d x p p 1 1 p x u p x d x p p 1 here 0 is a parameter and f x u is p 2 p superlinear at infinity existence of nontrivial solution is established for arbitrary 0 we first prove the existence of nontrivial solutions of the system for almost every parameter 0 by using mountain pass theorem and then we consider the continuation of the solutions

tight bounds for asynchronous renaming this article presents the first tight bounds on the time complexity of shared memory renaming a fundamental problem in distributed computing in which a set of processes need to pick distinct identifiers from a small namespace we first prove an individual lower bound of k process steps for deterministic renaming into any namespace of size subexponential in k where k is the number of participants the bound is tight it draws an exponential separation between deterministic and randomized solutions and implies new tight bounds for deterministic concurrent fetch and increment counters queues and stacks the proof is based on a new reduction from renaming to another fundamental problem in distributed computing mutual exclusion we complement this individual bound with a global lower bound of k log k c on the total step complexity of renaming into a namespace of size ck for any c 1 this result applies to randomized algorithms against a strong adversary and helps derive new global lower bounds for randomized approximate counter implementations that are tight within logarithmic factors on the algorithmic side we give a protocol that transforms any sorting network into a randomized strong adaptive renaming algorithm with expected cost equal to the depth of the sorting network this gives a tight adaptive renaming algorithm with expected step complexity o log k where k is the contention in the current execution this algorithm is the first to achieve sublinear time and it is time optimal as per our randomized lower bound finally we use this renaming protocol to build monotone consistent counters with logarithmic step complexity and linearizable fetch and increment registers with polylogarithmic cost

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

bifurcation results for a class of p x laplacian equations abstract in this paper we study the following p x laplacian equation div u p x 2 u u p x 2 u f x u x u x 0 x where r n is bounded 0 under suitable assumptions we obtain the existence of global branches of solutions for the above problem via the subsolution supersolution method

hardy s inequality in power type weighted lp 0 spaces abstract in this article our aim is to prove hardy s inequality in power type weighted l p 0 spaces by obtaining regularity condition on the exponents p q and defined at every point of the domain of test function with u 0 0 or vanishing at infinity under the log holder continuity conditions at the origin and infinity

more choices allow more faults set consensus problems in totally asynchronous systems abstract we define the k set consensus problem as an extension of the consensus problem where each processor decides on a single value such that the set of decided values in any run is of size at most k we require the agreement condition that all values decided upon are initial values of some processor we show that the problem has a simple k 1 resilient protocol in a totally asynchronous system in an attempt to come up with a matching lower bound on the number of failures we study the uncertainty condition which requires that there must be some initial configuration from which all possible input values can be decided we prove using a combinatorial argument that any k resilient protocol for the k set agreement problem would satisfy the uncertainty condition while this is not true for any k 1 resilient protocol this result seems to strengthen the conjecture that there is no k resilient protocol for this problem we prove this result for a restricted class of protocols our motivation for studying this problem is to test whether the number of choices allowed to the processors is related to the number of faults we hope that this will provide intuition towards achieving better bounds for more practical problems that arise in distributed computing e g the renaming problem the larger goal is to characterize the boundary between possibility and impossibility in asynchronous systems given multiple faults

the existence of radial solutions for differential inclusion problems in rn involving the p x laplacian abstract in this paper we consider a differential inclusion in r n involving a p x laplacian of the type p p x u e x u p x 2 u j x u x in r n u w 1 p x r n where p r n r is a continuous function satisfying some given assumptions the approach used in this paper is the variational method for locally lipschitz functions more precisely on the basis of the weierstrass theorem and the mountain pass theorem we prove that there exist at least two nontrivial solutions when p finally we obtain the existence of at least one nontrivial solution when p

the combinatorial structure of wait free solvable tasks this paper presents a self contained study of wait free solvable tasks a new necessary condition for wait free solvability based on a restricted set of executions is proved this set of executions induces a very simple to understand structure which is used to prove tight bounds for k set consensus and renaming the framework is based on topology but uses only elementary combinatorics and in contrast to previous works does not rely on algebraic or geometric arguments

a multiplicity result for p x laplacian problem in rn abstract in this paper we discuss the solutions of the following p x laplacian problem div u p x 2 u u p x 2 u f x u x r n where 1 p p x p n based on the theory of the variable exponent sobolev spaces w 1 p x r n we get that there exist at least two non trivial weak solutions

the robin eigenvalue problem for the p x laplacian as p abstract we study the asymptotic behavior as p of the first eigenvalues and the corresponding eigenfunctions for the p x laplacian with robin boundary conditions in an open bounded domain r n with smooth boundary we obtain uniform bounds for the sequence of first eigenvalues suitably rescaled and we prove that the positive first eigenfunctions converge uniformly in to a viscosity solution of a problem involving the laplacian subject to appropriate boundary conditions

a finite element method for the numerical solution of the coupled cahn hilliard and navier stokes system for moving contact line problems in this paper a semi implicit finite element method is presented for the coupled cahn hilliard and navier stokes equations with the generalized navier boundary condition for the moving contact line problems in our method the system is solved in a decoupled way for the cahn hilliard equations a convex splitting scheme is used along with a p1 p1 finite element discretization the scheme is unconditionally stable a linearized semi implicit p2 p0 mixed finite element method is employed to solve the navier stokes equations with our method the generalized navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way the efficiency and capacity of the present method are well demonstrated with several numerical examples

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

unconditionally stable methods for simulating multi component two phase interface models with peng robinson equation of state and various boundary conditions in this paper we consider multi component dynamic two phase interface models which are formulated by the cahn hilliard system with peng robinson equation of state and various boundary conditions these models can be derived from the minimum problems of helmholtz free energy or grand potential in the realistic thermodynamic systems the resulted cahn hilliard systems with various boundary conditions are fully coupled and strongly nonlinear a linear transformation is introduced to decouple the relations between different components and as a result the models are simplified from this we further propose a semi implicit unconditionally stable time discretization scheme which allows us to solve the cahn hilliard system by a decoupled way and thus our method can significantly reduce the computational cost and memory requirements the mixed finite element methods are employed for the spatial discretization and the approximate errors are also analyzed for both space and time numerical examples are tested to demonstrate the efficiency of our proposed methods

multiplicity results for some elliptic problems with nonlinear boundary conditions involving variable exponents in this paper we analyze an elliptic partial differential equation involving variable exponent growth conditions coupled with a nonlinear boundary condition we show the existence of infinitely many bounded weak solutions provided there is a suitable oscillatory behavior of the nonlinearity either at infinity or at zero our proofs rely on a method due to saint raymond

a topological treatment of early deciding set agreement the k set agreement problem consists for a set of n processes to agree on less than k among n possibly different values each initially known to only one process the problem is at the heart of distributed computing and generalizes the celebrated consensus problem this paper considers the k set agreement problem in a synchronous message passing distributed system where up to t processes can fail by crashing we determine the number of communication rounds needed for all correct processes to reach a decision in a given run as a function of the degree of coordination k and the number of processes that actually fail in the run email protected t we prove that for any integer email protected k

compact imbedding theorems with symmetry of strauss lions type for the space w1 p x ω abstract in this paper we extend some compact imbedding theorems of strauss lions type to the space w 1 p x when the domain has some symmetric properties and p x satisfies some conditions

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

bifurcation in lp rn for a semilinear elliptic equation keywords lsp p bifurcation constrained minimization sobolev spaces nonlinear eigenvalue problem semilinear spectrum linearisation variational structure reference ana article 1988 001doi 10 1112 plms s3 57 3 511 record created on 2008 12 10 modified on 2016 08 08

the dirichlet problem of higher order quasilinear elliptic equation abstract the paper is concerned with the dirichlet problem of higher order quasilinear elliptic equation m 1 d a x m u g f x u x d u 0 x m 1 where r n is a bounded domain 1 p p n m g is a bounded linear functional on w 0 m p x and a x m u f x u satisfy caratheodory condition and some p x growth conditions respectively we show that there exists at least one non trivial solution for the above problem in w 0 m p x

nonlinear elliptic equations with variable exponent old and new abstract in this survey paper by using variational methods we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type div a x u u q x 2 u in u 0 on where is a bounded or an exterior domain of r n and q is a continuous positive function the results presented in this paper extend several contributions concerning the lane emden equation and we focus on new phenomena which are due to the presence of variable exponents

wait free k set agreement is impossible the topology of public knowledge in the classical consensus problem each of n processors receives a private input value and produces a decision value which is one of the original input values with the requirement that all processors decide the same value a central result in distributed computing is that in several standard models including the asynchronous shared memory model this problem has no deterministic solution the k set agreement problem is a generalization of the classical consensus proposed by chaudhuri inform and comput 105 1993 pp 132 158 where the agreement condition is weakened so that the decision values produced may be different as long as the number of distinct values is at most k for n k geq 2 it was not known whether this problem is solvable deterministically in the asynchronous shared memory model in this paper we resolve this question by showing that for any k n there is no deterministic wait free protocol for n processors that solves the k set agreement problem the proof technique is new it is based on the development of a topological structure on the set of possible processor schedules of a protocol this topological structure has a natural interpretation in terms of the knowledge of the processors of the state of the system this structure reveals a close analogy between the impossibility of wait free k set agreement and the brouwer fixed point theorem for the k dimensional ball

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

stabilized linear semi implicit schemes for the nonlocal cahn hilliard equation abstract comparing with the well known classic cahn hilliard equation the nonlocal cahn hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials on the other hand it evidently brings more computational costs in numerical simulations thus efficient and accurate time integration schemes are highly desired in this paper we propose two energy stable linear semi implicit methods with first and second order temporal accuracies respectively for solving the nonlocal cahn hilliard equation the temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly while the spatial discretization is carried out by the fourier collocation method with fft based fast implementations the energy stabilities are rigorously established for both methods in the fully discrete sense numerical experiments are conducted for a typical case involving gaussian kernels we test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one in addition long time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay

the neumann boundary value problem of higher order quasilinear elliptic equation abstract this paper is concerned with the generalized solution to the neumann boundary value problem of higher order quasilinear elliptic equation 1 m 1 d a x m u f x u x b x m u g x u x we show that there exists at least one non trivial solution for the above problem in w m p x

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved

two phase fluid simulation using a diffuse interface model with peng robinson equation of state in this paper two phase fluid systems are simulated using a diffusive interface model with the peng robinson equation of state eos a widely used realistic eos for hydrocarbon fluid in the petroleum industry we first utilize the gradient theory of thermodynamics and variational calculus to derive a generalized chemical equilibrium equation which is mathematically a second order elliptic partial differential equation pde in molar density with a strongly nonlinear source term to solve this pde we convert it to a time dependent parabolic pde with the main interest in its final steady state solution a lagrange multiplier is used to enforce mass conservation the parabolic pde is then solved by mixed finite element methods with a semi implicit time marching scheme convex splitting of the energy functional is proposed to construct this time marching scheme where the volume exclusion effect of an eos is treated implicitly while the pairwise attraction effect of eos is calculated explicitly this sch

removable sets for holder continuous solutions of elliptic equations involving variable exponent abstract we prove the existence and continuity of solutions to obstacle problems with p x type nonstandard growth and establish the conditions of removability of closed sets for holder continuous solutions of the elliptic equations in the framework of variable exponent sobolev spaces