Periodic Boundary Data Research Articles

Explicit solution of a Dirichlet problem in nonconvex angle

In the present work, we give an explicit solution of the Dirichlet boundary-value problem for the Helmholtz equation in a nonconvex angle with periodic boundary data. We present uniqueness and existence theorems in an appropriate functional class and we give an explicit formula for the solution in the form of the Sommerfeld integral. The method of complex characteristics [14] is used.

Construction of solutions and asymptotics for the defocusing NLS with periodic boundary data

We study the defocusing nonlinear Schrödinger equation in the quarter-plane with decaying initial datum and Dirichlet and Neumann boundary values approaching periodic single exponentials at large times. By applying Deift-Zhou steepest descent arguments to an associated Riemann-Hilbert problem, we construct solutions and obtain detailed formulas for their long-time asymptotics.

A Variational Approach to the Problem of Continuous Dependence of Solutions to Second Order Periodic System

In the paper a second order differential system with a periodic boundary data is considered. Using some variational methods sufficient conditions for the continuous dependence of trajectories on controls are proved. The obtained results are applied then to the optimal control problem governed by the above system and the cost functional of a Bolza-type. In the end of the paper, an example of periodic optimal control problem demonstrating the applicability of the results is presented.

Exponential decay for a nonlinear model for electrical conduction in biological tissues

We study the asymptotic convergence to a periodic steady state of the solution of a nonlinear system of equations with periodic boundary data modeling electrical conduction in biological tissues, both in the microscopic and in the homogenized version. Such model keeps into account the resistive behavior of the intracellular and extracellular domains and also the capacitive/resistive behavior of the lipidic cellular membrane.The rate of convergence is analyzed and the systems of equations satisfied by the asymptotic limits are exhibited, when the resistive behavior of the membrane is described by a strictly monotone and coercive nonlinear function.The special case of homogeneous boundary conditions is also investigated, where the coercivity assumption can be relaxed.

Periodic boundary data for an integrable model of stimulated Raman scattering: long-time asymptotic behavior

We study the long-time asymptotic behavior of the solution to the initial-boundary value (IBV) problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations of stimulated Raman scattering. We consider the case of zero initial condition and periodic boundary data (p eiωt). Using the steepest descent method for oscillatory matrix Riemann–Hilbert problems we show that the solution of the IBV problem has different asymptotic behavior in different regions. The solution takes the form ofa plane wave of finite amplitude, when 0 < x < ω20t,a modulated elliptic wave of finite amplitude, when ω20t < x < ω2t, anda self-similar vanishing (as t → ∞) wave, when x > ω2t.For the IBV problem with nonzero initial condition and the same periodic boundary data, the solution to this problem is qualitatively similar to that of this study with the only difference that the solitons (of finite amplitude) can appear in the region x > ω2t.

Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π .

A new solution representation for the BBM equation in a quarter plane and the eventual periodicity

The initial-boundary-value problem for the Benjamin–Bona–Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behaviour (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. Towards this end, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator ∂t+α∂x − γ∂xxt defined in the space–time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase. The eventual periodicity of the full BBM equation is obtained under a suitable assumption imposed on its solution.

Example of a displacement convex functional of first order

We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow.

Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data

We study the large time behavior of viscosity solutions of Hamilton–Jacobi equations with periodic boundary data on bounded domains. We establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity.

Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues

We present a macroscopic model of electrical conduction in biological tissues. This model is derived via a homogenization limit by a microscopic formulation based on Maxwell’s equations, taking into account the periodic geometry of the microstructure. We also study the asymptotic behavior of the model for large times. Our results imply that periodic boundary data lead to an asymptotically periodic solution. The model is relevant to applications like electric impedance tomography.

Eventual periodicity for the KdV equation on a half-line

This paper studies the eventual periodicity of solutions to the initial and boundary value problem for the KdV equation on a half-line and with periodic boundary data. We derive a representation formula for solutions to the linearized KdV equation and rigorously establish the eventual periodicity of these solutions. Numerical experiments performed on the full KdV equation indicate that its solutions are also eventually periodic.

Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well-posed in $H^{s}( T^{d} )$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [4]. We use the "$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}(T^{d} )$ threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $T^{d}_\lambda = R^{d}/{\lambda Z^{d}}$, $d=1,2$.

Extremal solutions for nonlinear functional [formula omitted]-Laplacian impulsive equations

In this paper we deal with some nonlinear ϕ -Laplacian second order boundary value problems with impulses, subject to a general type of boundary value conditions which cover Dirichlet and periodic boundary data as particular cases. Our approach is that of upper and lower solutions together with growth restrictions of Nagumo’s type. The impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear. In both cases functional dependence on the solution is allowed.

Existence theory for functional p-Laplacian equations with variable exponents

In this paper we consider the solvability of equations of the form − d dt ϕ(t,u,u(t),u′(t))=f(t,u,u(t),u′(t)) for a.e. t∈I=[a,b] subject to a general type of functional-boundary conditions which cover Dirichlet and periodic boundary data as particular cases. Our approach is that of upper and lower solutions together with growth restrictions of Nagumo's type. An example is provided where a p-Laplacian with variable p is shown to have a solution between given upper and lower solutions.

Landesman-Lazer condition for nonlinear problems with jumping nonlinearities

This paper was motivated by the result of Iannacci and Nkashama ( J. Differential Equations 69 (1987) , 289–309) concerning the existence of at least one solution for the differential equation x″( t) + m 2 x( t) + g( t, x( t)) = e( t) with periodic boundary data x(0) − x(2 π) = x′(0) − x′(2 π) = 0, where m ⩾ 0 is an integer, e is integrable, and g satisfies Carathéodory's conditions. We intend to prove that the Landesman-Lazer type condition is sufficient for solvability of this periodic problem under rather more general assumptions on the growth of the nonlinear term g. Particularly, our hypotheses cover the nonlinearities which may “jump over” the eigenvalues different from m 2 while the assumptions of Iannacci and Nkashama allow g only to asymptotically “touch” the neighbour eigenvalues. Our approach may be used also for two-point boundary value problems. The proofs are based on the Leray-Schauder degree theory and on the shooting method for ordinary differential equations.