We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacian (that is, unmodified). Our main results contain the study of a "continuity" in those effective models with respect to the volume fraction of the holes and some new convergence rates for homogenization in the dilute setting. Our method explores the classical two-scale expansion ansatz and relies on asymptotic analysis of the rescaled cell problems using layer potential theory.

We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) , where f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) stands for the density of unique measure-valued solution f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) satisfying\[\int| v|^{2k-2+ } dF_0 (v)<\infty, 0\leq < 2, k= 2, 3, 4,\cdots\] provided that f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) is not a single Dirac mass.

A reformulation of the planetary geostrophic equations (PGEs) with the inviscid balance equation is proposed and the existence of global weak solutions is established, provided that the mechanical force satisfies an integral constraint. There is only one prognostic equation for the temperature field, and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. Furthermore, the velocity profile can be accurately represented as a function of the temperature gradient. In particular, the vertical velocity depends only on the first-order derivative of the temperature. As a result, the bound for the L ∞ (0, t 1 ; L 2 ) ∩ L 2 (0, t 1 ; H 1 ) norm of the temperature field is sufficient to show the existence of the weak solution.

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.

Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).