Recent Advances on Elliptic and Parabolic Issues

PREFACE.

CONTENTS.

In this paper we are interested with a strongly coupled system of partial differential equations that modelizes free convection in a two-dimensional bounded domain filled with a fluid saturated porous medium. This model is inspired by the one of free convection near a semi-infinite impermeable vertical flat plate embedded in a fluid saturated porous medium. We establish the existence and uniqueness of the solution for small data in some unusual spaces.

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest order nonlinear terms. The general theorems are illustrated by a number of model problems.

An existence theory for solutions of a parabolic problem u_t – div(A(x)∇u) + q(x)u = f(x, t) for x ∈ D and t ∈ (0,T] under a dynamical boundary condition σ(x)u_t + ∇u^T A(x)n = g(x, t) for x ∈ ∂D and t ∈ (0, T] is developed and a spectral representation formula is derived. It extends the results of [2] to problems with variable coefficients. We are interested in the case where the dynamical coefficient σ is a sign-changing or negative function. The one-dimensional parabolic problem is well-posed in the space C([0,T], H¹(D)). This is not true in higher dimensions. Our approach is based on the spectral theory of an associated elliptic problem with the eigenvalue parameter both in the equation and the boundary condition. By means of the theory of compact operators the spectrum is analyzed. Qualitative properties of the eigenfunctions are derived, e.g. strict positivity of the principal two eigenfunctions follows from a Harnack-type inequality. An interesting phenomenon is the "parameter-resonance", where for a specific value of the mean of the dynamical coefficient σ(x), two eigenvalues of the elliptic problem cross.

The goal of this paper is to provide a simple introduction to the theory of nonlocal elliptic problems.

We deal with generalized notions of convexity for sets. Namely, the polyconvexity, quasiconvexity, rank one convexity and separate convexity. The question has its origin in the calculus of variations. We try to systematize the results concerning these generalized notions imitating as much as possible the classical approach of convex analysis. Throughout the article, we will discuss the relations between the different convexities, separation and Carathéodory type theorems, the notion of hull of a set and extremal points.

The motion of closed polygonal curves in the plane moving under anisotropic crystalline curvature flows is investigated. It is known that the flow develops a singularity in finite time. We will discuss various kinds of situations of singularities, especially for convex curves. For nonconvex curves, a simple but suggestive example of singularity will be showed.

In this paper it is discussed whether reaction and linear diffusion bring about a effect of nonlinear diffusion or not. It is proved that a cross–diffusion system for two competitive species is realized in a singular limit of a reaction–diffusion system with a small parameter under some assumptions.

The uniqueness and the multiplicity of radial solutions to the Brezis-Nirenberg equation on the unit ball with the Robin condition are discussed. The scalar-field equation is also treated and the unified approach is presented. Moreover, depending on the Robin condition parameter, the difference between the structure of solutions in the three dimension and that in a higher dimension is shown.

We consider the Ginzburg-Landau functional in a 3-dimensional loop which has thin cross section. It is formally shown that this functional is approximated by a reduced functional in a 1-dimensional ring. In this article we rigorously prove that if the reduced functional has a nondegenerate local minimizer and if the cross section is sufficiently thin, there exists a local minimizer of the 3-dimensional one.

No abstract received.

No abstract received.

We address some convergence issues regarding the solution of elliptic problems when the data are periodic and the size of the domain becomes unbounded.

No abstract received.