We study a nonlinear elliptic boundary value problem arising from electrochemistry in the study of heterogeneous electrode surfaces. The boundary condition is of exponential type (Butler–Volmer) and has a periodic structure. We find a limiting or effective problem as the period approaches zero, along with a first order correction. We establish convergence estimates and provide numerical experim...

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size ε≪ 1. In the parent paper [8], we derived a homogenized boundary condition of Navier type as ε→ 0. We show here that for a large class of boundaries, this Navier condition provides a O(ε3/2| ln ε|1/2) approximation in L, instead o...

We study the effect of a periodic roughness on a Neumann boundary condition. We show that, as in the case of a Dirichlet boundary condition, it is possible to approach this condition by a more complex law on a domain without rugosity, called wall law. This approach is however different from that usually used in Dirichlet case. In particular, we show that this wall law can be explicitly written ...

We discuss solutions of the one-dimensional telegrapher’s equation in the presence of boundary conditions. We revisit the case of a radiation boundary condition and obtain an alternative expression for the already known Green’s function. Furthermore, we formulate a backreaction boundary condition, which has been widely used in the context of diffusion-controlled reversible reactions, for a one-...

In this paper we established the Carleman estimate for the two dimensional Lamé system with the zero Dirichlet boundary conditions. Using this estimate we proved the exact controllability result for the Lamé system with with a control locally distributed over a subdomain which satisfies to a certain type of nontrapping conditions.

We present conjectured exact expressions for two types of correlations in the dense O(n = 1) loop model on L×∞ square lattices with periodic boundary conditions. These are the probability that a point is surrounded by m loops and the probability that k consecutive points on a row are on the same or on different loops. The dense O(n = 1) loop model is equivalent to the bond percolation model at ...

We implement the new transform method for solving boundary value problems developed by Fokas for periodic boundary conditions. The approach presented here is not a replacement for classical methods nor is it necessarily an improvement. However, in addition to establishing that periodic problems can indeed be solve by the new transform method (which enhances further its scope and applicability),...

In this paper, we establish a new Lyapunov-type inequality for quasilinear systems with the anti-periodic boundary conditions. It improves some result of Wang [17]. As an application, we also obtain lower bounds for the eigenvalues of corresponding systems.

In this paper we present sufficient conditions for the existence of solutions to the periodic fourth order boundary value problem u(x) = f(x, u(x), u′(x), u′′(x), u′′′(x)) u(a) = u(b), i = 0, 1, 2, 3, for x ∈ [a, b], and f : [a, b] × R4 → R a continuous function. To the best of our knowledge it is the first time where this type of general nonlinearities is considered in fourth order equations w...