The paper deals with the spectral structure of the operator $ H=-\nabla\cdot b \nabla $ in $\mathbb R^n$ where $b$ is a stratified matrix-valued function. Using a partial Fourier transform, it is represented as a direct integral of a family of ordinary differential operators $H_p,\, p\in \mathbb{R}^n.$ Every operator $H_p$ has two thresholds and the kernels are studied in their (spectral) neighborhoods, uniformly in compact sets of $p$. As in [3], such estimates lead to a limiting absorption principle for $H$. Furthermore, estimates of the resolvent of $H$ near the bottom of its spectrum ("low energy" estimates) are obtained.

We consider a class of nonlinear beam equations in the whole space $\mathbb R^n$. Using previous important work due to Levandovsky and Strauss we prove that, locally, the $H^1$-norm of a strong solution approaches zero as $t \to +\infty$ as long as the spatial dimension $n \ge 6$. The problem remains open for dimensions $1 \le n \le 5$.

The purpose of this article is to show that a weak solution to the Navier-Stokes equations (NSE) in an infinite channel or in a spherical domain is actually regular up to the boundary if one component of the velocity is smooth enough. As a novelty, the 3D NSE are viewed as a perturbation of the 2D NSE in a sense to be explained below.

In this paper we consider the initial-boundary value problem for linear symmetric hyperbolic systems with non-homogeneous, maximally non-negative and characteristic boundary condition. We prove the existence of regular solutions in suitable functions spaces which take into account the loss of regularity in the normal direction to the characteristic boundary.

In this paper we investigate the solvability of the nonlinear Neumann problem (1.1) involving a critical Sobolev nonlinearity in an exterior domain. We examine the common effect of the shape of the graph of the weight function, the mean curvature of the boundary and a nonlinear perturbation of lower order on the existence of solutions of problem (1.1).

We study here the almost periodic solutions of first-order differential equations. We give sufficient conditions for the existence and uniqueness. The method relies on penalization and a priori estimates. One of the main difficulties consists of verifying that the limit of the sequence of perturbed solutions remains almost periodic. We introduce the notions of minimal/maximal solutions.