a halanay type inequality approach to the stability analysis of discrete time neural networks with delays in this paper we consider the stability of discrete time neural networks with time varying delays a new discrete time halanay type inequality is presented based on which we derive sufficient conditions for the globally asymptotic and exponential stability of discrete time delayed neural networks two kinds of neural networks without and with impulsive effects are investigated numerical examples and simulations are given to illustrate the effectiveness of the obtained results

fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

phase transition with the line tension effect we make the connection between the geometric model for capillarity with line tension and the cahn hilliard model of two phase fluids to this aim we consider the energies f eps u eps int o du 2 1 over eps int o w u lambda int bd o v u where u is a scalar density function and w and v are double well potentials we show that the behaviour of f e in the limit e 0 and depends on the limit of e log if this limit is finite and strictly positive then the singular limit of the energies f e leads to a coupled problem of bulk and surface phase transitions and under certain assumptions agrees with the relaxation of the capillary energy with line tension these results were announced in abs1 and abs2

markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

exponential stability of continuous time and discrete time cellular neural networks with delays convergence characteristics of continuous time cellular neural networks with discrete delays are studied by using lyapunov functionals we obtain delay independent sufficient conditions for the networks to converge exponentially toward the equilibria associated with the constant input sources halanay type inequalities are employed to obtain sufficient conditions for the networks to be globally exponentially stable it is shown that the estimates obtained from the halanay type inequalities improve the estimates obtained from the lyapunov methods discrete time analogues of the continuous time cellular neural networks are formulated and studied it is shown that the convergence characteristics of the continuous time systems are preserved by the discrete time analogues without any restriction imposed on the uniform discretization step size

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

maximizing expected terminal utility of an insurer with high gain tax by investment and reinsurance abstract this paper investigates optimal investment and proportional reinsurance policies for an insurer who subjects to pay high gain tax the surplus process of the insurer and the return process of the financial market are both modulated by the external macroeconomic environment the dynamic of the external macroeconomic environment is specified by a markov chain with finite states once the insurer s accumulated profits attain a new maximum they have to pay high gain tax the objective of the insurer is to maximize the expected terminal utility by investment and reinsurance the controlled wealth process of the insurer turned out to be a controlled jump diffusion process with reflections and markov regime switching by the weak dynamic programming principle wdpp we prove that the value function is the unique viscosity solution to the coupled hamilton jacob bellman hjb equations with first derivative boundary constraints by the markov chain approximating method for the hjb equations we construct a numerical scheme for approximating the viscosity solution to the coupled hjb equations two numerical examples are presented to illustrate the impact of both high gain tax and regime switching on the optimal policies

on the first and second variations of a nonlocal isoperimetric problem we consider a nonlocal perturbation of an isoperimetric variational prob lem the problem may be viewed as a mathematical paradigm for the ubiquitous phenom enon of energy driven pattern formation associated with competing short and long range interactions in particular it arises as a g limit of a model for microphase separation of diblock copolymers in this article we establish precise conditions for criticality and stability i e we explicitly compute the first and second variations we also present some applications

controllability for a class of linear time varying impulsive systems with time delay in control input in this note the controllability for a class of linear time varying impulsive systems with time delay in control input is studied sufficient and necessary conditions for the controllability of such systems are established specializing the obtained results to the cases of linear time varying impulsive systems and linear time invariant impulsive systems with time delay some simple criteria with respect to coefficient matrices which are less conservative than some existing results are derived it is also shown that the time delay in control input contributes to achieve the controllability of impulsive system even if the system is not controllable two examples are worked out to illustrate our results

power penalty method for solving hjb equations arising from finance nonlinear hamilton jacobi bellman hjb equation commonly occurs in financial modeling implicit numerical scheme is usually applied to the discretization of the continuous hjb so as to find its numerical solution since it is generally difficult to obtain its analytic viscosity solution this type of discretization results in a nonlinear discrete hjb equation we propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term under some mild conditions we give the unique solvability of the penalized equation and show its convergence to the original discrete hjb equation moreover we establish a sharp convergence rate of the power penalty method which is of an exponential order with respect to the power of the penalty term we further develop a damped newton algorithm to iteratively solve the lower order penalized equation finally we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method we also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

sharp dimension free quantitative estimates for the gaussian isoperimetric inequality we provide a full quantitative version of the gaussian isoperimetric inequality the difference between the gaussian perimeter of a given set and a half space with the same mass controls the gap between the norms of the corresponding barycenters in particular it controls the gaussian measure of the symmetric difference between the set and the half space oriented so to have the barycenter in the same direction of the set our estimate is independent of the dimension sharp on the decay rate with respect to the gap and with optimal dependence on the mass

discrete halanay type inequalities and applications in this paper we derive new discrete halanay type inequalities and show some applications in the investigation of the asymptotic behavior of nonlinear di4erence equations in particular di4erence equations involving the maximum functional are considered 2003 elsevier ltd all rights reserved

optimal investment consumption strategy with liability and regime switching model under value at risk constraint abstract this paper considers the optimal investment consumption problem with liability subject to a maximum value at risk denoted by mvar constraint the model also contains regime switching market modes whose states are interpreted as the states of the economy in each state of economy state we constrain a var value for the portfolio in a short time duration and mvar is defined as the maximum value of the vars in all economy states we suppose that both the price dynamics of the risky asset and the liability value process are governed by a markov modulated geometric brownian motion with the objective of maximizing the discounted utility of consumption we obtain a system of hjb equations corresponding to the economy states by using the dynamic programming principle then by adopting the techniques of chen et al 2010 we get explicit expressions of value functions moreover with the help of lagrange multiplier method we derive the optimal investment and the optimal consumption finally a numerical example is investigated and the effects of many parameters on the optimal investment on the optimal consumption and on var value are studied furthermore we also explore how var value affects the optimal investment and the optimal consumption

global exponential stability of impulsive discrete time neural networks with time varying delays this paper studies the problem of global exponential stability and exponential convergence rate for a class of impulsive discrete time neural networks with time varying delays firstly by means of the lyapunov stability theory some inequality analysis techniques and a discrete time halanay type inequality technique sufficient conditions for ensuring global exponential stability of discrete time neural networks are derived and the estimated exponential convergence rate is provided as well the obtained results are then applied to derive global exponential stability criteria and exponential convergence rate of impulsive discrete time neural networks with time varying delays finally numerical examples are provided to illustrate the effectiveness and usefulness of the obtained criteria

optimal dynamic trading strategies with risk limits value at risk var has emerged in recent years as a standard tool to measure and control the risk of trading portfolios yet existing theoretical analysis of the optimal behavior of a trader subject to var limits has produced a negative view of var as a risk control tool in particular var limits have been found to induce increased risk exposure in some states and an increased probability of extreme losses however these conclusions are based on models that are either static or dynamically inconsistent in this paper we formulate a dynamically consistent model of optimal portfolio choice subject to var limits and show that the concerns expressed in earlier papers do not apply if consistently with common practice the var limit is reevaluated dynamically in particular we find that the optimal risk exposure of a trader subject to a var limit is always lower than that of an unconstrained trader and that the probability of extreme losses is also lower we also consider risk limits formulated in terms of tail conditional expectation tce a coherent risk measure often advocated as an alternative to var and show that in our dynamic setting it is always possible to transform a tce limit into an equivalent var limit and conversely

hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations this paper is concerned with the problem of asymptotic synchronization for a class of uncertain complex networks with delays and general uncertain perturbations in order to cope with the bad effects generated by the uncertain perturbations a novel hybrid adaptive and impulsive controller is designed such that the complex network can be asymptotically synchronized onto an isolate chaotic system with uncertain perturbations all the perturbations can be different from each other on the basis of a new lemma squeezing rule and lyapunov krasovskii functionals several sufficient conditions guaranteeing the realization of the synchronization goal are derived it is shown that the designed hybrid controllers exhibit powerful robustness some existing results are improved and extended numerical simulations verify the effectiveness of the theoretical results and the robustness of the new controller

stability analysis of quaternion valued neural networks decomposition and direct approaches in this paper we investigate the global stability of quaternion valued neural networks qvnns with time varying delays on one hand in order to avoid the noncommutativity of quaternion multiplication the qvnn is decomposed into four real valued systems based on hamilton rules ij ji k jk kj i ki ik j i 2 j 2 k 2 ijk 1 with the lyapunov function method some criteria are respectively presented to ensure the global mu stability and power stability of the delayed qvnn on the other hand by considering the noncommutativity of quaternion multiplication and time varying delays the qvnn is investigated directly by the techniques of the lyapunov krasovskii functional and the linear matrix inequality lmi where quaternion self conjugate matrices and quaternion positive definite matrices are used some new sufficient conditions in the form of quaternion valued lmi are respectively established for the global mu stability and exponential stability of the considered qvnn besides some assumptions are presented for the two different methods which can help to choose quaternion valued activation functions finally two numerical examples are given to show the feasibility and the effectiveness of the main results

regularity of flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

remark on a nonlocal isoperimetric problem abstract we consider isoperimetric problem with a nonlocal repulsive term given by the newtonian potential we prove that regular critical sets of the functional are analytic this optimal regularity holds also for critical sets of the ohta kawasaki functional we also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set

optimal portfolios with maximum value at risk constraint under a hidden markovian regime switching model this paper studies an optimal portfolio selection problem in the presence of the maximum value at risk mvar constraint in a hidden markovian regime switching environment the price dynamics of n risky assets are governed by a hidden markovian regime switching model with a hidden markov chain whose states represent the states of an economy we formulate the problem as a constrained utility maximization problem over a finite time horizon and then reduce it to solving a hamilton jacobi bellman hjb equation using the separation principle the mvar constraint for n risky assets plus one riskless asset is derived and the method of lagrange multiplier is used to deal with the constraint a numerical algorithm is then adopted to solve the hjb equation numerical results are provided to demonstrate the implementation of the algorithm

a new fuzzy impulsive control of chaotic systems based on t s fuzzy model in this paper fuzzy impulsive control is used for stabilization of chaotic systems based on the takagi sugeno t s model the stability issue of the general nonlinear impulsive control system is first investigated via comparison criterion then a novel impulsive control scheme is presented for chaotic systems based on the t s fuzzy model some sufficient conditions are given to stabilize the t s fuzzy model our results are proven to be less conservative theoretically and numerically moreover we have also estimated the stable region of the impulsive interval finally the proposed fuzzy impulsive control scheme is successfully applied to stabilize rossler s system and chua s circuit the numerical simulations demonstrate the effectiveness and advantage of our main results

symmetric properties of positive solutions for fully nonlinear nonlocal system abstract in this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain bounded or unbounded in r n via using a direct method of moving planes this extends the results in wang and niu 2017 and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order

regularity of radial minimizers and extremal solutions of semilinear elliptic equations we consider a special class of radial solutions of semilinear equations u g u in the unit ball of rn it is the class of semi stable solutions which includes local minimizers minimal solutions and extremal solutions we establish sharp pointwise lq and wk q estimates for semi stable radial solutions our regularity results do not depend on the specific nonlinearity g among other results we prove that every semi stable radial weak solution u h01 is bounded if n 9 for every g and belongs to h3 w3 2 in all dimensions n for every g increasing and convex the optimal regularity results are strongly related to an explicit exponent which is larger than the critical sobolev exponent

on the best constant of weighted poincare inequalities abstract we compute the optimal constant for some weighted poincare inequalities obtained by fausto ferrari and enrico valdinoci in f ferrari e valdinoci some weighted poincare inequalities indiana univ math j 58 4 2009 1619 1637

existence and characterization of regions minimizing perimeter under a volume constraint inside euclidean cones we study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a euclidean cone our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half space this allows us to give some criteria ensuring existence of isoperimetric regions for instance local convexity of the cone at some boundary point we also characterize which are the stable regions in a convex cone i e second order minima of perimeter under a volume constraint from this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone

augmented lagrangian method applied to american option pricing the american option pricing problem is originally formulated as a stochastic optimal stopping time problem it is also equivalent to a variational inequality problem or a complementarity problem involving the black scholes partial differential operator in this paper the corresponding variational inequality problem is discretized by using a fitted finite volume method based on the discretized form an algorithm is developed by applying augmented lagrangian method alm to the valuation of the american option convergence properties of alm are considered by empirical numerical experiments we conclude that alm is more effective than penalty method and lagrangian method and comparable with the projected successive overrelaxation method psor furthermore numerical results show that alm is more robust in terms of computation time under changes in market parameters interest rate and volatility

connectivity of phase boundaries in strictly convex domains we consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem inf int omega u dx m int omega frac 1 varepsilon w u frac varepsilon 2 big nabla u big 2 dx here w is a double well potential and omega subset r n is a strictly convex domain for e small this is closely related to the problem of partitioning into two subdomains of fixed volume where the subdomain boundaries correspond to the transitional boundary between phases motivated by this geometry problem we show that in a strictly convex domain stable critical points of the original variational problem have a connected thin transition layer separating the two phases this relates to work in gm where special geometries such as cylindrical domains were treated and is analogous to the results in cho which show that in a convex domain stable critical points of the corresponding unconstrained problem are constant the proof of connectivity employs tools from geometric measure theory including the co area formula and the isoperimetric inequality on manifolds the thinness of the transition layer follows from a separate calculation establishing spatial decay of solutions to the pure phases

uncertain exit time multi period mean variance portfolio selection with endogenous liabilities and markov jumps this paper considers an uncertain exit time multi period mean variance portfolio selection problem with endogenous liabilities in a markov jump market where assets and liabilities of the balance sheet are simultaneously optimized the random returns of assets and liabilities depend on the states of the financial market by applying the lagrange duality method the khatri rao matrix product technique and the dynamic programming approach the explicit expressions for the mean variance efficient strategy and efficient frontier are derived in addition the optimal balance sheet structures in both cases with and without boundary constraints are studied moreover some degenerate cases are discussed and some results in the literature are recovered as degenerate cases under our setting furthermore a numerical example based on real data from the chinese stock market is provided to illustrate the results obtained in this paper and some interesting findings are presented

exponential synchronization of linearly coupled neural networks with impulsive disturbances this brief investigates globally exponential synchronization for linearly coupled neural networks nns with time varying delay and impulsive disturbances since the impulsive effects discussed in this brief are regarded as disturbances the impulses should not happen too frequently the concept of average impulsive interval is used to formalize this phenomenon by referring to an impulsive delay differential inequality we investigate the globally exponential synchronization of linearly coupled nns with impulsive disturbances the derived sufficient condition is closely related with the time delay impulse strengths average impulsive interval and coupling structure of the systems the obtained criterion is given in terms of an algebraic inequality which is easy to be verified and hence our result is valid for large scale systems the results extend and improve upon earlier work as a numerical example a small world network composing of impulsive coupled chaotic delayed nn nodes is given to illustrate our theoretical result

optimal switching for linear quadratic problem of switched systems in discrete time the optimal switching problem is attracting plenty of attention this problem can be considered as a special type of discrete optimization problem and is np complete in this paper a class of optimal switching problem involving a family of linear subsystems and a quadratic cost functional is considered in discrete time where only one subsystem is active at each time point by deriving a precise lower bound expression and applying the branch and bound method a computational method is developed for solving this discrete optimization problem numerical examples have been implemented to demonstrate the efficiency and effectiveness of the proposed method

entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

markowitz s mean variance portfolio selection with regime switching from discrete time models to their continuous time limits we study a discrete time version of markowitz s mean variance portfolio selection problem where the market parameters depend on the market mode regime that jumps among a finite number of states the random regime switching is delineated by a finite state markov chain based on which a discrete time markov modulated portfolio selection model is presented such models either arise from multiperiod portfolio selections or result from numerical solution of continuous time problems the natural connections between discrete time models and their continuous time counterpart are revealed since the markov chain frequently has a large state space to reduce the complexity an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a two time scale problem we prove that the process of interest yields a switching diffusion limit using weak convergence methods next based on the optimal control of the limit process obtained from our recent work we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

uniqueness of stable capillary hypersurfaces in a ball in this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical our result also provides a proof of a conjecture proposed by sternberg and zumbrun j reine angew math 503 63 85 1998 we also prove a heintze karcher ros type inequality for hypersurfaces with free boundary in a ball which together with the new minkowski formula yields a new proof of alexandrov s theorem for embedded cmc hypersurfaces in a ball with free boundary

a pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature abstract we prove a pointwise gradient bound for bounded solutions of u f u 0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature we also obtain some rigidity results when equality in the bound holds at some point

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

stochastic dissipativity and passivity analysis for discrete time neural networks with probabilistic time varying delays in the leakage term this paper deals with the dissipativity and passivity analysis for discrete time stochastic neural networks with probabilistic time varying delays the main contribution of this paper is to reduce the conservatism of the dissipativity conditions for the considered neural networks by utilizing the reciprocally convex combination approach this approach is proposed to bound the forward differences of the double and triple summable terms taken in the lyapunov functional by introducing a stochastic variable with a bernoulli distribution the information of probability distribution of the time varying delays are considered and transformed into one with deterministic time varying delays by employing lyapunov functional approach sufficient conditions are derived in terms of linear matrix inequalities to guarantee that the considered neural networks to be strictly q s r dissipative and passive finally numerical examples are given to demonstrate the effectiveness of the obtained results

on cmc free boundary stable hypersurfaces in a euclidean ball in this note we prove that if b is the unit ball centred in the origin in the euclidean space with dimension n 1 n ge 2 then a cmc free boundary stable hypersurface sigma in b satisfies i begin aligned frac nh 2 2 int sigma 1 x 2 dvol sigma na le l le na left 1 h right end aligned where l a and h denote the length of partial sigma the area of sigma and the mean curvature of sigma respectively and the orientation of sigma is in a such way that h ge 0 the left side of i is an equality if and only if sigma is a totally geodesic disk or a spherical cap consequently if the boundary partial sigma of sigma is embedded then sigma must be either totally geodesic or starshaped with respect to the center of the ball this result is a slightly improvement of a theorem proved by ros and vergasta in particular if n 2 in this case its not necessary to assume the boundary partial sigma is embedded the only cmc free boundary stable surfaces in b are the totally geodesic disks or the spherical caps this classification result was proved very recently by nunes using an extended stability result and a modified hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces we don t use that modified hersch type argument however we use a nunes stability type lemma and a crucial result due to ros and vergasta our technique considering a nunes stability type lemma can be applied to study sets which are stable for the volume constrained least area problem within the unit ball and provide a proof for the sternberg zumbrun s conjecture

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

rigidity results in diffusion markov triples abstract we consider stable solutions of semilinear equations in a very general setting the equation is set on a polish topological space endowed with a measure and the linear operator is induced by a carre du champs equivalently the equation is set in a diffusion markov triple under suitable curvature dimension conditions we establish that stable solutions with integrable carre du champs are necessarily constant weaker conditions characterize the structure of the carre du champs and carre du champ itere the proofs are based on a geometric poincare formula in this setting from the general theorems established several previous results are obtained as particular cases and new ones are provided as well

optimal portfolios with regime switching and value at risk constraint we consider the optimal portfolio selection problem subject to a maximum value at risk mvar constraint when the price dynamics of the risky asset are governed by a markov modulated geometric brownian motion gbm here the market parameters including the market interest rate of a bank account the appreciation rate and the volatility of the risky asset switch over time according to a continuous time markov chain whose states are interpreted as the states of an economy the mvar is defined as the maximum value of the vars of the portfolio in a short time duration over different states of the chain we formulate the problem as a constrained utility maximization problem over a finite time horizon by utilizing the dynamic programming principle we shall first derive a regime switching hamilton jacobi bellman hjb equation and then a system of coupled hjb equations we shall employ an efficient numerical method to solve the system of coupled hjb equations for the optimal constrained portfolio we shall provide numerical results for the sensitivity analysis of the optimal portfolio the optimal consumption and the var level with respect to model parameters these results are also used to investigating the effect of the switching regimes

synchronization of a complex dynamical network with coupling time varying delays via sampled data control abstract in this paper the synchronization problem of a complex dynamical network with coupling time varying delays via delayed sampled data controller is investigated in order to make full use of the sawtooth structure characteristic of the sampling input delay a discontinuous lyapunov functional is proposed based on the extended wirtinger inequality from a convex representation of the sector restricted nonlinearity in system dynamics the stability condition based on lyapunov stability theory is obtained by utilization of linear matrix inequality formulation to find the controller which achieves the synchronization of a complex dynamical network with coupling time varying delay finally two numerical examples are given to illustrate the effectiveness of the proposed methods

a geometric inequality in the heisenberg group and its applications to stable solutions of semilinear problems in the heisenberg group framework we obtain a geometric inequality for stable solutions of delta mathbb h u f u in a domain omega subseteq mathbb h more precisely if we denote the horizontal intrinsic hessian by hu the mean curvature of a level set by h its imaginary curvature by p the intrinsic normal by and the unit tangent by we have that int limits omega nabla mathbb h phi 2 nabla mathbb h u 2 quad geq int limits omega cap nabla mathbb h u ne0 left hu 2 langle hu 2 nu nu rangle mathbb h 2 left tyuxu txuyu right right phi 2 quad int limits omega cap nabla mathbb h u ne0 nabla mathbb h u 2 big h 2 big p frac langle huv nu rangle mathbb h nabla mathbb h u big 2 2 langle t nu v rangle mathbb h big phi 2 for any phi in c infty 0 omega stable solutions in the entire mathbb h satisfying a suitably weighted energy growth and such that langle t nu v rangle mathbb h ge0 are then shown to have level sets with vanishing mean curvature

w1 q estimates for the extremal solution of reaction diffusion problems abstract we establish a new w 1 2 n 1 n 2 estimate for the extremal solution of u f u in a smooth bounded domain of r n which is convex for arbitrary positive and increasing nonlinearities f c 1 r satisfying lim t f t t

flattening results for elliptic pdes in unbounded domains with applications to overdetermined problems semilinear elliptic pdes are dealt with either in the half space or in overdetermined settings we obtain several new results and also give new proofs of celebrated theorems by exploiting some geometric analysis of level sets a geometric inequality and a pointwise gradient estimate

equations involving fractional laplacian operator compactness and application abstract in this paper we consider the following problem involving fractional laplacian operator 1 u u 2 2 e u u in u 0 on where is a smooth bounded domain in r n e 0 2 2 0 1 2 2 n n 2 and is either the spectral fractional laplacian or the restricted fractional laplacian we show for problem 1 with the spectral fractional laplacian that for any sequence of solutions u n of 1 corresponding to e n 0 2 2 satisfying u n h c in the sobolev space h defined in 1 2 u n converges strongly in h provided that n 6 and 0 the same argument can also be used to obtain the same result for the restricted fractional laplacian an application of this compactness result is that problem 1 possesses infinitely many solutions under the same assumptions

on application of an alternating direction method to hamilton jacobin bellman equations this paper presents a numerical method for the approximation of viscosity solutions to a hamilton jacobi bellman hjb equation governing a class of optimal feedback control problems the first order hjb equation is first perturbed by adding a diffusion term with a singular perturbation parameter the time and spatial variables in the resulting equation are then discretized respectively by an implicit modified method of characteristics and the alternating direction ad scheme we show that the ad procedure s perturbation error is virtually negligible due to the small perturbation parameter and the efficient ad scheme can be applied to our hjb equation without generating splitting error numerical results performed to verify the usefulness of the method show that the method gives accurate approximate solutions to both of the control and the state variables

a geometric inequality and a symmetry result for elliptic systems involving the fractional laplacian we study the symmetry properties for solutions of elliptic systems of the type s1u f1 u v s2v f2 u v where f cloc1 1 r2 s1 s2 0 1 and the operator s is the so called fractional laplacian we obtain some poincare type formulas for the harmonic extension in the half space that we use to prove a symmetry result both for stable and for monotone solutions

semilinear pdes in the heisenberg group the role of the right invariant vector fields abstract some geometric features of the stable solutions of semilinear pdes in the heisenberg group are dealt with a nonexistence result is given the analysis is carried out by investigating the role of the right invariant vector fields the monotonicity along these vector fields implies the stability of the solution and it naturally gives rise to geometrically interesting quantities

the stability of stochastic coupled systems with time delays and time varying coupling structure this paper is concerned with the pth moment exponential stability of stochastic coupled systems with time delays and time varying coupling structure sctvcs the coupling structure is time varying first second the method of graph theory has been generalized from a system established on a digraph with constant coupling structure to a digraph with time varying coupling structure successfully combining graph theory with lyapunov method a systematic method is given to construct a lyapunov function for sctvcs several pth moment exponential stability criteria are presented including a lyapunov type theorem and some sufficient criteria in the form of coefficients furthermore the theoretical conclusions are applied to the stochastic coupled oscillators with time varying coupling structure and two stability criteria are obtained finally we give two numerical examples to illustrate the effectiveness of our results