li yau hamilton estimates and bakry emery ricci curvature abstract in this paper we derive cheng yau li yau hamilton estimates for riemannian manifolds with bakry emery ricci curvature bounded from below and also global and local upper bounds in terms of bakry emery ricci curvature for the hessian of positive and bounded solutions of the weighted heat equation on a closed riemannian manifold

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

modica type gradient estimates for an inhomogeneous variant of the normalized p laplacian evolution abstract in this paper we study an inhomogeneous variant of the normalized p laplacian evolution which has been recently treated in banerjee and garofalo 2013 does 2011 manfredi et al 2010 and juutinen 2014 we show that if the initial datum satisfies the pointwise gradient estimate 1 6 a e then the unique solution to the cauchy problem 1 2 satisfies the same gradient estimate a e for all later times see 1 7 a general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of 1 2 was first obtained in caffarelli et al 1994 such estimate generalizes one obtained by l modica for the laplacian and it has connections to a famous conjecture of de giorgi

second order estimate on transition layers abstract in this paper we establish a uniform c 2 estimate for level sets of stable solutions to the singularly perturbed allen cahn equation in dimensions n 10 which is optimal the proof combines two ingredients one is a reverse application of the infinite dimensional lyapunov schmidt reduction method which enables us to reduce the c 2 estimate for these level sets to a corresponding one on solutions of toda system the other one uses a small regularity theorem on stable solutions of toda system to establish various decay estimates which gives a lower bound on distances between different sheets of solutions to toda system or level sets of solutions to allen cahn equation

statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks dynamical stochastic models of single neurons and neural networks often take the form of a system of n 2 coupled stochastic differential equations we consider such systems under the assumption that third and higher order central moments are relatively small in the general case a system of 1 n n13 generally nonlinear coupled ordinary differential equations holds for the approximate means variances and covariances for the general linear system the solutions of these equations give exact results this is illustrated in a simple case generally the moment equations can be solved numerically results are given for a spiking fitzhugh nagumo model neuron driven by a current with additive white noise differential equations are obtained for the means variances and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise s1063 651x 96 03511 8 pacs number s 87 10 1e 05 40 1j 02 50 2r

dynamic least squares kernel density modeling of fokker planck equations with application to neural population the dynamic least squares kernel density lsqkd model c pantano and b shotorban phys rev e 76 066705 2007 is used to solve the fokker planck equations in this model the probability density function pdf is approximated by a linear combination of basis functions with unknown parameters whose governing equations are determined by a global least squares approximation of the pdf in the phase space in this work basis functions are set to be gaussian for which the mean variance and covariances are governed by a set of partial differential equations pdes or ordinary differential equations odes depending on what phase space variables are approximated by gaussian functions three sample problems of univariate double well potential bivariate bistable neurodynamical system g deco and d mart i phys rev e 75 031913 2007 and bivariate brownian particles in a nonuniform gas are studied the lsqkd is verified for these problems as its results are compared against the results of the method of characteristics in nondiffusive cases and the stochastic particle method in diffusive cases for the double well potential problem it is observed that for low to moderate diffusivity the dynamic lsqkd well predicts the stationary pdf for which there is an exact solution a similar observation is made for the bistable neurodynamical system in both these problems least squares approximation is made on all phase space variables resulting in a set of odes with time as the independent variable for the gaussian function parameters in the problem of brownian particles in a nonuniform gas this approximation is made only for the particle velocity variable leading to a set of pdes with time and particle position as independent variables solving these pdes a very good performance by lsqkd is observed for a wide range of diffusivities

symmetry of some entire solutions to the allen cahn equation in two dimensions abstract in this paper we prove even symmetry and monotonicity of certain solutions of allen cahn equation in a half plane we also show that entire solutions with finite morse index and four ends must be evenly symmetric with respect to two orthogonal axes a classification scheme of general entire solutions with finite morse index is also presented using energy quantization

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

population dynamics under the laplace assumption in this paper we describe a generic approach to modelling dynamics in neuronal populations this approach models a full density on the states of neuronal populations but finesses this high dimensional problem by re formulating density dynamics in terms of ordinary differential equations on the sufficient statistics of the densities considered c f the method of moments the particular form for the population density we adopt is a gaussian density c f the laplace assumption this means population dynamics are described by equations governing the evolution of the population s mean and covariance we derive these equations from the fokker planck formalism and illustrate their application to a conductance based model of neuronal exchanges one interesting aspect of this formulation is that we can uncouple the mean and covariance to furnish a neural mass model which rests only on the populations mean this enables us to compare equivalent mean field and neural mass models of the same populations and evaluate quantitatively the contribution of population variance to the expected dynamics the mean field model presented here will form the basis of a dynamic causal model of observed electromagnetic signals in future work

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

population dynamics variance and the sigmoid activation function this paper demonstrates how the sigmoid activation function of neural mass models can be understood in terms of the variance or dispersion of neuronal states we use this relationship to estimate the probability density on hidden neuronal states using non invasive electrophysiological eeg measures and dynamic casual modelling the importance of implicit variance in neuronal states for neural mass models of cortical dynamics is illustrated using both synthetic data and real eeg measurements of sensory evoked responses

dynamics of a modified hindmarsh rose neural model with random perturbations moment analysis and firing activities abstract in this paper an attempt has been made to understand the activity of mean membrane voltage and subsidiary system variables with moment equations i e mean variance and covariance s under noisy environment we consider a biophysically plausible modified hindmarsh rose h r neural system injected by an applied current exhibiting spiking bursting phenomenon the effects of predominant parameters on the dynamical behavior of a modified h r system are investigated numerically it exhibits period doubling period halving bifurcation and chaos phenomena further a nonlinear system has been analyzed for the first and second order moments with additive stochastic perturbations it has been solved using fourth order runge kutta method and noisy systems by euler s scheme it has been demonstrated that the firing properties of neurons to evoke an action potential in a certain parameter space of the large exact systems can be estimated using an approximated model strong stimulation can cause a change in increase or decrease of the firing patterns corresponding to a fixed set of parameter values the firing behavior and dynamical differences of the collective variables of a large exact and approximated systems are investigated

hamiltonian identities for elliptic partial differential equations abstract new identities for elliptic partial differential equations are obtained several applications are discussed in particular young s law for the contact angles in triple junction formation is proven rigorously structure of level curves of saddle solutions to allen cahn equation are also carefully analyzed

moment analysis of the hodgkin huxley system with additive noise abstract we consider a classical space clamped hodgkin huxley hh model neuron stimulated by a current which has a mean together with additive gaussian white noise of amplitude a system of 14 deterministic first order nonlinear differential equations is derived for the first and second order moments means variances and covariances of the voltage v and the subsidiary variables n m and h the system of equations is integrated numerically with a fourth order runge kutta method as long as the variances as determined by these deterministic equations remain small the latter accurately approximate the first and second order moments of the stochastic hodgkin huxley system describing spiking neurons on the other hand for certain values of when rhythmic spiking is inhibited by larger amplitude noise the solutions of the moment equation strongly overestimate the moments of the voltage a more refined analysis of the nature of such irregularities leads to precise insights about the effects of noise on the hodgkin huxley system for suitable values of which enable rhythmic spiking we analyze by numerical examples from both simulation and solutions of the moment equations the three factors which tend to promote its cessation namely the increasing variance the nature and shape of the basins of attraction of the limit cycle and stable equilibrium point and the speed of the process

positive solutions of an elliptic partial differential equation on rn abstract in this paper we discuss the existence nonexistence and uniqueness of positive solutions of a one parameter family of elliptic partial differential equations on r n n 2 these equations are of interests in mathematical biology and riemannian geometry our approach are based on variational arguments and comparison principles

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

existence and non existence results for a logistic type equation on manifolds we study the steady state solutions of a generalized logistic type equation on a complete riemannian manifold we provide sufficient conditions for existence respectively non existence of positive solutions which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold which is mostly taken into account by means of conditions on the volume growth of geodesic balls

liouville theorem for the nonlinear poisson equation on manifolds abstract in this note we study a modica type gradient estimate for smooth solutions to general non linear poisson equation u f u 0 in m n u m n r where m g is a complete riemannian manifold with bounded geometry and non negative ricci curvature and f is the derivative of the non negative smooth function f u on r then we use this gradient estimate to conclude a liouville theorem

three circles theorems for harmonic functions we proved two three circles theorems for harmonic functions on manifolds in integral sense as one application on manifold with nonnegative ricci curvature whose tangent cone at infinity is the unique metric cone with unique conic measure we showed the existence of nonconstant harmonic functions with polynomial growth this existence result recovered and generalized the former result of ding and led to a complete answer of ni s conjecture furthermore in similar context combining the techniques of estimating the frequency of harmonic functions with polynomial growth which were developed by colding and minicozzi we confirmed their conjecture about the uniform bound of frequency

an optimization problem with volume constraint for a degenerate quasilinear operator abstract we consider the optimization problem of minimizing u p d x with a constraint on the volume of u 0 we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

two results on entire solutions of ginzburg landau system in higher dimensions in this short article we prove two results on the ginzburg landau system of equations u u u 2 1 where u c2 rn rm n m 1 first we prove a liouville type theorem which asserts that every solution u satisfying rn u 2 1 2 is constant and of unit norm provided n 4 here m 1 in our second result we give an answer to a question raised by brezis open problem 3 of proceedings of the symposium on pure mathematics vol 65 american mathematical society providence ri 1999 about the symmetry for the ginzburg landau system in the case n m 3 we also formulate three open problems concerning the classification of entire solutions of the ginzburg landau system in any dimension

on the maximum of solutions for a semilinear elliptic problem on etudie certaines proprietes d un probleme aux valeurs propres semilineaires elliptique avec second membre non defini on montre que chaque solution doit avoir son maximum dans un intervalle specifie j

the space clamped hodgkin huxley system with random synaptic input inhibition of spiking by weak noise and analysis with moment equations we consider a classical space clamped hodgkin huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by ornstein uhlenbeck processes using numerical solutions of the stochastic model system obtained by an euler method it is found that with excitation only there is a critical value of the steady state excitatory conductance for repetitive spiking without noise and for values of the conductance near the critical value small noise has a powerfully inhibitory effect for a given level of inhibition there is also a critical value of the steady state excitatory conductance for repetitive firing and it is demonstrated that noise in either the excitatory or inhibitory processes or both can powerfully inhibit spiking furthermore near the critical value inverse stochastic resonance was observed when noise was present only in the inhibitory input process the system of deterministic differential equations for the approximate first and second order moments of the model is derived they are solved using runge kutta methods and the solutions are compared with the results obtained by simulation for various sets of parameters including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex the mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels in the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells the assumptions underlying the moment equation approach are not satisfied the simulation results show that noisy synaptic input of either an excitatory or inhibitory character or both may lead to the suppression of firing in neurons operating near a critical point and this has possible implications for cortical networks although suppression of firing is corroborated for the system of moment equations there seem to be substantial differences between the dynamical properties of the original system of stochastic differential equations and the much larger system of moment equations

regularity of the free boundary in an optimization problem related to the best sobolev trace constant in this paper we study the regularity properties of a free boundary problem arising in the optimization of the best sobolev trace constant in the immersion h 1 omega hookrightarrow l q partial omega for functions that vanish in a subset of omega this problem is also related to a minimization problem for steklov eigenvalues

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

a priori estimates uniqueness and existence of positive solutions of yamabe type equations on complete manifolds abstract we study the asymptotic behaviour of non negative solutions of yamabe type equations on a complete riemannian manifold then we provide a comparison result based on a form of the weak maximum principle at infinity which together with the a priori estimates previously obtained yields uniqueness under very general ricci assumptions the paper ends with an existence result and an application to the non compact yamabe problem

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

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

extended method of moments for deterministic analysis of stochastic multistable neurodynamical systems the analysis of transitions in stochastic neurodynamical systems is essential to understand the computational principles that underlie those perceptual and cognitive processes involving multistable phenomena like decision making and bistable perception to investigate the role of noise in a multistable neurodynamical system described by coupled differential equations one usually considers numerical simulations which are time consuming because of the need for sufficiently many trials to capture the statistics of the influence of the fluctuations on that system an alternative analytical approach involves the derivation of deterministic differential equations for the moments of the distribution of the activity of the neuronal populations however the application of the method of moments is restricted by the assumption that the distribution of the state variables of the system takes on a unimodal gaussian shape we extend in this paper the classical moments method to the case of bimodal distribution of the state variables such that a reduced system of deterministic coupled differential equations can be derived for the desired regime of multistability

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

a minimum problem with free boundary in orlicz spaces abstract we consider the optimization problem of minimizing g u u 0 d x in the class of functions w 1 g with u 0 w 0 1 g for a given 0 0 and bounded w 1 g is the class of weakly differentiable functions with g u d x the conditions on the function g allow for a different behavior at 0 and at we prove that every solution u is locally lipschitz continuous that it is a solution to a free boundary problem and that the free boundary u 0 is a regular surface also we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the lipschitz regularity of the weak solutions and the c 1 regularity of their free boundaries near flat free boundary points

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

optimal boundary holes for the sobolev trace constant in this paper we study the problem of minimizing the sobolev trace rayleigh quotient u w1 p p u lq p among functions that vanish in a set contained on the boundary of given boundary measure we prove existence of extremals for this problem and analyze some particular cases where information about the location of the optimal boundary set can be given moreover we further study the shape derivative of the sobolev trace constant under regular perturbations of the boundary set

finite size dynamics of inhibitory and excitatory interacting spiking neurons the dynamic mean field approach we recently developed is extended to study the dynamics of population emission rates nstd for a finite network of coupled excitatory sed and inhibitory sid integrate and fire if neurons the power spectrum of nstd in an asynchronous state is computed and compared to simulations we calculate the interpopulations transfer functions and show how synaptic interaction modulates the otherwise low pass filter with resonances which go well beyond the filter s cut sv nd allowing efficient information transmission on very short time scales determined by spike transmission delays the saddle node instability of the asynchronous state is studied and a simple exact dependence of the stability condition on the current to rate gain functions is derived by which self couplings ee and ii decrease stability while mutual interaction ei and ie favor stability realistic models of interacting spiking neurons must ultimately encompass multiple interacting modules for which the intermodule and intramodule connectivity pattern and synaptic couplings define the architecture of interest in a specific setting even at the crude modeling level of integrateand fire if current driven neurons with instantaneous synaptic input predicting the dynamical properties of multiple interacting neural populations is a difficult task it has been increasingly recognized that noisy regimes in which the collective firing of a population of neurons exhibits important fluctuations provide a key for the description of biologically relevant dynamic phenomena e g fast collective oscillations with frequencies largely exceeding the single neuron firing rate 1 3 the diffusion approximation has been widely adopted for the description of the noisy dynamics of a single neuron or a homogeneous population of if neurons a few attempts have dealt so far with multiple populations of interacting if neurons in this brief report by extending an approach introduced in 4 to two self and mutually interacting populations of excitatory and inhibitory neurons we attempt a further step towards the analytical and quantitative description of biologically interesting architectures besides this generic interest the approach taken here has some implications on the problem of information transmission in complex neural systems 5 6 it is known that a network of spiking neurons firing asynchronously can encode signals covering a frequency band extending well beyond the emission rate of the individual neurons in the second part of the present paper we briefly focus on the input output properties of the neural populations as linear systems characterized by suitable transfer functions both the intramodule and intermodule synaptic interactions produce resonances in the frequency response which select optimal bands for the information transmission the present work illustrates how one can analytically put in relatively simple terms the expected role of synaptic couplings in shaping the signal to noise transmission properties before describing the results we summarize the formalism referring for details to 4 where we introduced a dynamic mean field emission rate equation describing the time dependent firing activity nstd of a finite number of interacting if neurons in noisy regimes with a distribution of spike transmission delays and instantaneous synaptic currents in the diffusion approximation 7 the stochastic dynamics of an if neuron s membrane potential is described by the langevin equation v fsvd msv td ssv tdgstd where fsvd is the leakage term and msv td and s 2 sv td are the contributions to the infinitesimal mean and variance ofv due to the afferent current gstd is a white noise with zero mean and unit variance the associated fokker planck fp equation 8 for the probability psv tddv of having vstd p fv v dvg is

harmonic maps from noncompact riemannian manifolds with non negative ricci curvature outside a compact set in this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete noncompact riemannian manifold m g with sobolev constant s2 m 0 and ricci curvature ric m 0 outside some compact subset into a complete manifold of nonpositive curvature or a regular ball in particular we prove the uniqueness and existence of bounded harmonic functions on m g

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

on a capacity for modular spaces the purpose of this article is to define a capacity on certain topological measure spaces x with respect to certain function spaces v consisting of measurable functions in this general theory we will not fix the space v but we emphasize that v can be the classical sobolev space w1 p the classical orlicz sobolev space w1 the hajlasz sobolev space m1 p the musielak orlicz sobolev space or generalized orlicz sobolev space and many other spaces of particular interest is the space v w 1 p given as the closure of w1 p cc in w1 p in this case every function u v a priori defined only on has a trace on the boundary which is unique up to a capp polar set

an optimization problem with volume constraint in orlicz spaces we consider the optimization problem of minimizing g u dx in the class of functions w1 g with a constraint on the volume of u 0 the conditions on the function g allow for a different behavior at 0 and at we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

orlicz sobolev boundary trace embeddings we prove a sharp trace embedding theorem for orlicz sobolev spaces into orlicz spaces on the boundary an optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established

some nonlocal optimal design problems abstract in this paper we study two optimal design problems associated to fractional sobolev spaces w s p then we find a relationship between these two problems and finally we investigate the convergence when s 1

fractional order orlicz sobolev spaces abstract in this paper we define the fractional order orlicz sobolev spaces and prove its convergence to the classical orlicz sobolev spaces when the fractional parameter s 1 in the spirit of the celebrated result of bourgain brezis mironescu we then deduce some consequences such as convergence of the modulars and convergence of solutions for some fractional versions of the g operator as the fractional parameter s 1

diffusion type operators liouville theorems and gradient estimates on complete manifolds abstract we study liouville theorems and gradient estimates for solutions of eq 1 1 with the help of a diffusion operator and the related geometry

positive solutions for logistic type quasilinear elliptic equations on rn in this paper we consider positive solutions of the logistic type p laplacian equation pu a x u p 2u b x u q 1u x rn n 2 we show that under rather general conditions on a x and b x for large x the behavior of the positive solutions for large x can be determined this is then used to show that there is a unique positive solution our results improve the corresponding ones in j london math soc 2 64 2001 107 124 and j anal math in press

symmetry of traveling wave solutions to the allen cahn equation in mathbb r 2 in this paper we prove even symmetry of monotone traveling wave solutions to the balanced allen cahn equation in the entire plane related results for the unbalanced allen cahn equation are also discussed

an optimization problem with volume constraint we consider the optimization problem of minimizing dirichlet integral int omega nabla u 2 with volume constraint on the set u 0 we use a penalization method which for small values of the penalization parameter leads to a solution of the original problem

a gradient bound and a liouville theorem for nonlinear poisson equations soit f c 2 r une fonction non negative et u c 3 r n une solution dans tout r n de l equation u f u ou f f est la derivee 1ere de f si u est borne sur r n et s il existe x 0 r n tel que f u x 0 0 alors u est constante

liouville theorems for harmonic maps we prove several liouville theorems for harmonic maps between certain classes of riemannian manifolds in particular the results can be applied to harmonic maps from the euclidean space r m g 0 to a large class of riemannian manifolds our assumptions on the harmonic maps concern the asymptotic behavior of the maps at