边界元方法的一些研究进展

本文旨在综述我们小组近二十年来在边界元方法这一领域的一些研究成果,在简要介绍边界元方法的基本思想后,主要介绍了一类非线性界面问题的有限元-边界元耦合方法、求解电磁散射问题的有限元-边界元耦合方法和超奇异积分的一类计算方法.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

Sharp estimates for the anisotropic torsional rigidity and the principal frequency

In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic nonlinear operators.     

A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature

We prove a pointwise gradient bound for bounded solutions of Δu+F^′(u)=0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature.We also obtain some rigidity results when equality in the bound holds at some point.     

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = (σ+m)n/(n-σ-2) is its critical exponent provided max{-1, [(1-m)n-2]/(n+1)} σ n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Furthermore, we demonstrate that if max{1, σ + m} p ≤ pc, then every positive solution of the equations blows up in finite time; whereas for ppc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n ≤σ+2.     

On Some Elliptic Problems in Unbounded Domains

The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution.     

Competition between ferroelasticity and ferroelectricity

In the crystals of the KH₂PO₄ family, the properties of ferroelasticity and ferroelectricity strongly interact. It is for example possible to change by application of an electric field the orientation of the phase front between high temperature and low temperature phases, and to observe the rotation of reticular planes during an electric hysteresis cycle drawing. These phenomena, as the existence of a quasi constant domain width illustrate the particular competition between ferroelasticity and ferroelectricity in these compounds.     

Quantum and Classical Evolutions of a Nonautonomous Dynamical System—A Comparison

The influence of moving boundaries on the stability of quantum Hamiltonian systems, in particular on the dynamics of quantum versions of the classical Pustilnikov model, is investigated (the latter consists of a masspoint bouncing above an oscillating plate under the influence of constant gravity.) It is shown that, in contrast to the classical Pustilnikov model, generic time-periodic boundary conditions (including the Dirichlet condition) on the quantum models do not allow unlimited energy gain (speeding up) of these systems.     

D-Domains of attraction of stable measures on stratified Lie groupsof attraction of stable measures on stratified Lie groups

LetG be a stratified Lie group and (_t)_{t 0} be a continuous convolution semigroup of probability measures onG. A probability measurev is said to belong to the -domain of attraction of ₁, if there exists a sequence (a _n ) of positive real numbers such that weakly, where ₁ denotes the natural dilation onG. We prove convergence criteria for discrete convolution semigroups. These are used to obtain a simple necessary and sufficient condition for the existence of sucha _n if (_t)_{t 0} has no Gaussian component. For the proof we introduce the notion of regularly varying measures onG and develop the necessary theory of regular variation.