regularity results for elliptic equations in lipschitz domains abstract we develop a simple variational argument based on the usual nirenberg difference quotient technique to deal with the regularity of the solutions of dirichlet and neumann problems for some linear and quasilinear elliptic equation in lipschitz domains we obtain optimal regularity results in the natural family of sobolev spaces associated with the variational structure of the equations in the linear case we obtain in a completely different way some of the results of d jerison and c e kenig about the laplace equation

error analysis for implicit approximations to solutions to cauchy problems exact error analyses for two implicit approximation schemes to the cauchy problem du t dt mathcal a u t mathrel backepsilon 0 u 0 u 0 in d mathcal a are presented the operator mathcal a is required to be maximal monotone on a hilbert space mathcal h the approximations considered are u h kh i h mathcal a k u 0 and solutions u lambda to the approximating problems du lambda t dt mathcal a lambda u lambda 0 u lambda 0 u 0 where a lambda is the yosida approximation of mathcal a the special case of a being a subgradient is considered and tight estimates of the convergence of both u h and mathcal a h u h or u h and a h u h are given the new results in this case are the linear rates of convergence of u h in l infty 0 infty mathcal h and in the energy norm and the sublinear rate of convergence of a h u h in l 2 0 infty mathcal h applications to both linear and

an abstract existence theorem at resonance and its applications abstract by using brouwer degree theory and a continuation theorem based on mawhin s coincidence degree an abstract existence theorem at resonance for operator equations is developed as applications of this result some sufficient conditions are given for the existence of periodic solutions of semilinear equations at resonance where the kernel of the linear part has dimension n n 2 the results due to nagle and sinkala are generalized improved and unified

homoclinic orbits for a singular second order neutral differential equation using critical point theory and an approximation method we study the existence of homoclinic orbits for a singular second order neutral differential difference equations

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

infinite periodic solutions to a class of second order sturm liouville neutral differential equations abstract by means of variational structure and z 2 group index theory we obtain infinite periodic solutions to a class second order sturm liouville neutral delay equations p t x t s q t x t s f t x t x t x t 2 x t 2 s 0

linking age survival and transit time distributions although the concepts of age survival and transit time have been widely used in many fields including population dynamics chemical engineering and hydrology a comprehensive mathematical framework is still missing here we discuss several relationships among these quantities by starting from the evolution equation for the joint distribution of age and survival from which the equations for age and survival time readily follow it also becomes apparent how the statistical dependence between age and survival is directly related to either the age dependence of the loss function or the survival time dependence of the input function the solution of the joint distribution equation also allows us to obtain the relationships between the age at exit or death and the survival time at input or birth as well as to stress the symmetries of the various distributions under time reversal the transit time is then obtained as a sum of the age and survival time and its properties are discussed along with the general relationships between their mean values the special case of steady state case is analyzed in detail some examples inspired by hydrologic applications are presented to illustrate the theory with the specific results

a posteriori error estimates for variable time step discretizations of nonlinear evolution equations we study the backward euler method with variable time steps for abstract evolution equations in hilbert spaces exploiting convexity of the underlying potential or the angle bounded condition thereby assuming no further regularity we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual these estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps we also prove that they converge to zero with an optimal rate with respect to the regularity of the solution we apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character 2000 john wiley sons inc

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

regularity theorems and optimal error estimates for linear parabolic cauchy problems we prove some regularity results for the solution of a linear abstract cauchy problem of parabolic type as an application we study the approximation of the solution by means of an implicit euler discretization in time which is stable with respect to a wide class of galerkin approximation methods in space the error is evaluated in norms of typel 2 0 l 2 andl 2 0 v h 00 1 2 0 h h 1 0 v wherev h v are hilbert spaces the embeddings are supposed to be dense and continuous we prove error estimates which are optimal with respect to the regularity assumptions on the right hand side of the equation

existence of periodic solutions to second order hamiltonian systems with potential indefinite in sign we have presented some new results for periodic solutions to a class of hamiltonian systems with potential indefinite in sign these results improve and correct some known results in the literature

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

on the probabilistic structure of water age the age distribution of water in hydrologic systems has received renewed interest recently especially in relation to watershed response to rainfall inputs the purpose of this contribution is first to draw attention to existing theories of age distributions in population dynamics fluid mechanics and stochastic groundwater and in particular to the mckendrick von foerster equation and its generalizations and solutions a second and more important goal is to clarify that when hydrologic fluxes are modeled by means of time varying stochastic processes the age distributions must themselves be treated as random functions once their probabilistic structure is obtained it can be used to characterize the variability of age distributions in real systems and thus help quantify the inherent uncertainty in the field determination of water age we illustrate these concepts with reference to a stochastic storage model which has been used as a minimalist model of soil moisture and streamflow dynamics

qualitative theory of compartmental systems dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations many nonlinear that are called compartmental systems in this paper the authors define compartmental systems specify their relations to other nonnegative systems and discuss examples of applications the authors review the qualitative results on linear and nonlinear compartmental systems including their relation to cooperative systems they review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems providing a framework for unifying them under a few general theorems in the course of that they complete the solution of a problem posed by bellman and show that closed nonlinear autonomous n compartment systems can show the full gamut of possible behaviors of systems of odes finally to provide additional structure to this study the authors show how to partiti

multiplicity results of periodic solutions for a class of second order delay differential systems abstract this paper is concerned with the periodic boundary value problem 1 u t u t f t u t a e t 0 2 u 0 u 2 u 0 u 2 0 where 0 is a given constant r is a parameter the variational principle is given and some multiplicity results of periodic solutions of 1 are obtained directly via variational methods

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

transit time and age distributions for nonlinear time dependent compartmental systems many processes in nature are modeled using compartmental systems reservoir pool box systems usually they are expressed as a set of first order differential equations describing the transfer of matter across a network of compartments the concepts of age of matter in compartments and the time required for particles to transit the system are important diagnostics of these models with applications to a wide range of scientific questions until now explicit formulas for transit time and age distributions of nonlinear time dependent compartmental systems were not available we compute densities for these types of systems under the assumption of well mixed compartments assuming that a solution of the nonlinear system is available at least numerically we show how to construct a linear time dependent system with the same solution trajectory we demonstrate how to exploit this solution to compute transit time and age distributions in dependence on given start values and initial age distributions furthermore we derive equations for the time evolution of quantiles and moments of the age distributions our results generalize available density formulas for the linear time independent case and mean age formulas for the linear time dependent case as an example we apply our formulas to a nonlinear and a linear version of a simple global carbon cycle model driven by a time dependent input signal which represents fossil fuel additions we derive time dependent age distributions for all compartments and calculate the time it takes to remove fossil carbon in a business as usual scenario

existence of periodic solutions for p t laplacian systems abstract in this paper by using minimax methods in critical point theory we obtain some existence theorems for periodic solutions of p t laplacian system u t p t 2 u t f t u t 0 u 0 u t u 0 u t 0 and our results generalize some existed theorems