the l p robin problem for laplace equations in lipschitz and semi convex domains abstract let n 3 and be a bounded lipschitz domain in r n assume that p 2 and the function b l is non negative where denotes the boundary of denote by the outward unit normal to in this article the authors give two necessary and sufficient conditions for the unique solvability of the robin problem for the laplace equation u 0 in with boundary data u b u f l p respectively in terms of a weak reverse holder inequality with exponent p or the unique solvability of the robin problem with boundary data in some weighted l 2 space as applications the authors obtain the unique solvability of the robin problem for the laplace equation in the bounded semi convex domain with boundary data in weighted l p for any given p 1

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

multistability of complex valued recurrent neural networks with real imaginary type activation functions this paper addresses the multistability problem of n dimensional complex valued recurrent neural networks with real imaginary type activation functions sufficient conditions are proposed for checking the existence of 2 1 2 1 n 1 equilibria under these conditions 1 1 n equilibria are locally exponentially stable and the others are unstable attractive basins of equilibria are also investigated complete attractive basins of equilibria in 1 dimensional complex valued recurrent neural networks are obtained the obtained stability results improve and extend the existing ones two numerical examples are given to illustrate the effectiveness of the obtained results

the neumann problem of laplace s equation in semiconvex domains abstract let n 2 and be a bounded semiconvex domain in r n in this paper we show that the neumann problem for laplace s equation in with boundary data in l p is uniquely solvable for p 1 which improves the well known result for the neumann problem

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

dissipativity analysis of memristor based complex valued neural networks with time varying delays in this paper the problem of dissipativity analysis for memristor based complex valued neural networks mcvnns with time varying delays is investigated extensively dissipativity analysis is an important concept in control system theory and its applications the analysis in the paper employ results from the theory of differential equations with discontinuous right hand side as introduced by filippov by using the framework of filippov solution differential inclusion theory an appropriate lyapunov krasovskii functional and linear matrix inequality lmi technique several new sufficient conditions for global dissipativity global exponential dissipativity and strictly q s r dissipativity are derived in the form of complex valued as well as real valued lmis both real and complex valued lmis guarantee feasibility results for addressed mcvnns these lmis can be solved by using standard available numerical packages moreover the global attractive sets which are positive invariant are obtained finally three numerical examples are established to illustrate the effectiveness of the proposed theoretical results

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

global existence of solutions of the liquid crystal flow for the oseen frank model in r2 abstract in the first part of this paper we establish the global existence of solutions of the liquid crystal gradient flow for the well known oseen frank model the liquid crystal flow is a prototype of equations from the ericksen leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2 sphere the ericksen leslie system is a system of the navier stokes equations coupled with the liquid crystal flow in the second part of this paper we also prove the global existence of solutions of the ericksen leslie system for a general oseen frank model in r 2

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

exponential synchronization of complex valued complex networks with time varying delays and stochastic perturbations via time delayed impulsive control considering the fact that time delays are unavoidable in the control of practical systems this paper considers globally exponential synchronization of complex valued complex dynamical networks with multiple time varying delays and stochastic perturbations by designing a time delayed impulsive control scheme by taking the advantage of lyapunov method in complex field and utilizing an impulsive inequality with delays several synchronization criteria are obtained through strict mathematical proofs our results are general which extend some existing ones concerning impulsive synchronization a numerical example is given to illustrate the effectiveness of theoretical results

exponential stability of complex valued memristive recurrent neural networks in this brief we establish a novel complex valued memristive recurrent neural network cvmrnn to study its stability as a generalization of real valued memristive neural networks cvmrnn can be separated into real and imaginary parts by means of m matrix and lyapunov function the existence uniqueness and exponential stability of the equilibrium point for cvmrnns are investigated and sufficient conditions are presented finally the effectiveness of obtained results is illustrated by two numerical examples

one problem of the navier type for the stokes system in planar domains we study the problem u f u f u g u g in u gu g h h on for a bounded simply connected lipschitz domain in the plane for f 0f 0 g 0g 0 g lp g lp h lq h lq we study a solution in the sense of a nontangential limit for f ws 1 q r2 f ws 1 q r2 g ws q g ws q g wt 1 p p g wt 1 p p h ws 1 q q h ws 1 q q with t s 1t s 1 we prove the existence of a unique solution u wt p r2 ws q u wt p r2 ws q for f bs 1q r r2 g bsq r g bt 1 pp h bs 1 qq r with t s 1t s 1 we prove the existence of a unique solution u btp r2 bsq r for f ck 1 r2 g ck h ck h ck g ck 1 g ck 1 we prove the existence of a unique solution u ck 1 r2 ck

existence uniqueness and exponential stability analysis for complex valued memristor based bam neural networks with time delays this article explores the exponential stability problem of complex valued bidirectional associative memory bam neural networks with time delays this analysis is on the basis of the m matrix approach the differential inclusions theory and the homeomorphism property by constructing a novel lyapunov functional a sufficient criterion for the existence uniqueness and exponential stability for the equilibrium point of the considered system is derived moreover similar results in terms of m matrix are also obtained for the exponential stability problem of delayed complex valued bam neural networks without memristors in the end two numerical examples are provided to demonstrate the availability of the obtained results

the neumann problem and helmholtz decomposition in convex domains we show that the neumann problem for laplace s equation in a convex domain with boundary data in lp is uniquely solvable for 1 p as a consequence we obtain the helmholtz decomposition of vector fields in lp rd

stability analysis of fractional order complex valued neural networks with both leakage and discrete delays in this paper the problem of uniform stability for fractional order complex valued neural networks with both leakage and discrete delays is considered base on the contracting mapping principle a sufficient condition is proposed for the existence and uniqueness of the equilibrium point of the addressed neural networks by employing analysis technique some delay dependent criteria are established for checking the global uniform stability of the fractional order complex valued neural networks two numerical examples are presented to demonstrate the validity and feasibility of the proposed results

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

stability analysis of complex valued impulsive systems with time delay in this paper the global exponential stability of complex valued impulsive systems is addressed some new sufficient conditions are obtained to guarantee the global exponential stability by the lyapunov razumikhin theory which extend and improve most of recent results moreover the obtained razumikhin conditions are very simple and efficient to verify in real problems and helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation finally a numerical example is given to show the effectiveness of the obtained results

exponential stability of complex valued memristor based neural networks with time varying delays abstract in this paper we propose a new type of complex valued memristor based neural networks with time varying delays and discuss their exponential stability firstly by using a matrix measure method the halanay inequality and some analytic techniques we derive a sufficient condition for the global exponential stability of this type of neural networks then we build a lyapunov functional and utilize the halanay inequality to establish several criteria for the exponential stability of such networks with time varying delays finally we show two numerical simulations to demonstrate the theoretical results

superconvergence analysis for the explicit polynomial recovery method a new recovery technique explicit polynomial recovery epr is analyzed for finite element methods epr reconstructs the value at edge centers by solving a local problem in combination with the finite element solution at the vertex a quadratic approximation is constructed besides improving the accuracy it can also be applied in building the epr based error estimator for the poisson equation the element center is a superconvergent point of the gradient of the epr recovered function on an equilateral triangulation numerical examples are presented to verify the theoretical results and to show the performance of the epr in the adaptive finite element method

necessary and sufficient conditions for the solvability of the l p dirichlet problem on lipschitz domains we study the homogeneous elliptic systems of order 2 ell with real constant coefficients on lipschitz domains in mathbb r n n ge 4 for any fixed p 2 we show that a reverse holder condition with exponent p is necessary and sufficient for the solvability of the dirichlet problem with boundary data in l p we also obtain a simple sufficient condition as a consequence we establish the solvability of the l p dirichlet problem for n ge 4 and 2 epsilon p frac 2 n 1 n 3 epsilon the range of p is known to be sharp if ell ge 2 and 4 le n le 2 ell 1 for the polyharmonic equation the sharp range of p is also found in the case n 6 7 if ell 2 and n 2 ell 2 if ell ge 3

the neumann problem in lp on lipschitz and convex domains abstract given p 2 and a bounded lipschitz domain we obtain a necessary and sufficient condition for the solvability of the neumann problem for laplace s equation u 0 in with boundary data in l p as a result we show that the l p neumann problem on convex domains in r d is solvable for 1 p if d 2 for 1 p 4 if d 3 and for 1 p 3 e if d 4

stability of complex valued impulsive stochastic functional differential equations on networks with markovian switching abstract this paper investigates the stability problem of complex valued impulsive stochastic functional differential equations on networks with markovian switching cisfnm markovian switching and impulsive effects are considered into complex valued systems for the first time by virtue of complex generalized ito s formula the stability of cisfnm can be analyzed effectively without splitting the real and imaginary parts then based on razumikhin technique graph theoretical technique as well as average dwell time approach several novel stability criteria are derived which mainly depend on integral average value of time varying coefficients compared with some existing results our results gain not only less conservativeness but also lower calculation complexity subsequently the stability of complex valued impulsive stochastic networks with time varying delays and markovian switching is studied as an application complex valued coupled oscillators are considered and numerical simulations are also presented to show the effectiveness of our main results

global stability of complex valued recurrent neural networks with time delays since the last decade several complex valued neural networks have been developed and applied in various research areas as an extension of real valued recurrent neural networks complex valued recurrent neural networks use complex valued states connection weights or activation functions with much more complicated properties than real valued ones this paper presents several sufficient conditions derived to ascertain the existence of unique equilibrium global asymptotic stability and global exponential stability of delayed complex valued recurrent neural networks with two classes of complex valued activation functions simulation results of three numerical examples are also delineated to substantiate the effectiveness of the theoretical results

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

matrix measure strategies for exponential synchronization and anti synchronization of memristor based neural networks with time varying delays this paper is concerned with exponential synchronization and anti synchronization of memristor based neural networks under the framework of filippov systems and a linear controller the exponential synchronization and anti synchronization criteria for memristor based neural networks can be guaranteed by the matrix measure and halanay inequality the criteria are very simple to implement in practice finally two numerical examples are given to demonstrate the correctness of the theoretical results it is shown that the matrix measure can increase the exponential convergence rate and decrease the feedback gain effectively