# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

January 2009 , Volume 8 , Issue 1

A special issue on
A tribute to Professor Philippe G. Ciarlet on his 70th birthday

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2009, 8(1): 1-4 doi: 10.3934/cpaa.2009.8.1 +[Abstract](2150) +[PDF](110.9KB)
Abstract:
Professor Philippe G. Ciarlet was born on October 14, 1938, in Paris. He obtained his undergraduate degree in 1961 from the celebrated École Polytechnique in Paris, followed by graduate studies (1962-1964) at the École Nationale des Ponts et Chaussées in Paris. Professor Ciarlet received in 1966 his Ph.D. at the Case Institute of Technology, Cleveland, U.S.A., under the guidance of Professor Richard S. Varga. The title of his Ph.D. thesis is Variational Methods for Non-Linear Boundary-Value Problems. He continued with a Doctorat d'État Fonctions de Green Discrètes et Principe du Maximum Discret) at the University of Paris in 1971 and his advisor was Professor Jacques-Louis Lions.

2009, 8(1): 5-36 doi: 10.3934/cpaa.2009.8.5 +[Abstract](2263) +[PDF](432.2KB)
Abstract:
Given a family $A(t)$ of closed unbounded operators on a UMD Banach space $X$ with common domain $W,$ we investigate various properties of the operator $D_{A}:=\frac{d}{dt}-A(\cdot)$ acting from $\mathcal{W}_{per}^{p}:=\{u\in W^{1,p}(0,2\pi ;X)\cap L^{p}(0,2\pi ;W):u(0)=u(2\pi)\}$ into $\mathcal{X} ^{p}:=L^{p}(0,2\pi ;X)$ when $p\in (1,\infty).$ The primary focus is on the Fredholmness and index of $D_{A},$ but a number of related issues are also discussed, such as the independence of the index and spectrum of $D_{A}$ upon $p$ or upon the pair $(X,W)$ as well as sufficient conditions ensuring that $D_{A}$ is an isomorphism. Motivated by applications when $D_{A}$ arises as the linearization of a nonlinear operator, we also address similar questions in higher order spaces, which amounts to proving (nontrivial) regularity properties. Since we do not assume that $\pm A(t)$ generates any semigroup, approaches based on evolution systems are ruled out. In particular, we do not make use of any analog or generalization of Floquet's theory. Instead, some arguments, which rely on the autonomous case (for which results have only recently been made available) and a partition of unity, are more reminiscent of the methods used in elliptic PDE theory with variable coefficients.
2009, 8(1): 37-54 doi: 10.3934/cpaa.2009.8.37 +[Abstract](2076) +[PDF](232.3KB)
Abstract:
We prove that if $n$ closed disks $D_1$,$D_2$,...,$D_n$, of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_1\cap D_2\cap...\cap D_n$ is a complete $K$-spectral set for $A$, with $K\leq n+n(n-1)/\sqrt3$. When $n=2$ and the intersection $X_1\cap X_2$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).
2009, 8(1): 55-81 doi: 10.3934/cpaa.2009.8.55 +[Abstract](2205) +[PDF](366.7KB)
Abstract:
We present models for solid-solid phase transitions with surface energy that allow both smooth and sharp interfaces. The models involve the minimisation of an energy that consists of three terms: the elastic energy (a double-well potential), the smooth-interface surface energy and the sharp-interface surface energy. Existence of solutions is shown in arbitrary dimensions. The second part of the paper deals with the one-dimensional case. For the first 1D model (in which the sharp-interface energy is the same regardless of the size of the jump of the gradient), we study the regime of the parameters (one parameter represents the boundary conditions, one models the energy of the sharp interface, and the third one models the energy of the smooth interfaces) for which the minimiser presents smooth interfaces, sharp interfaces or no interfaces. We also prove that a suitable scaling of the functional $\Gamma$-converges to a pure sharp-interface model, as the parameters penalising the formation of interfaces go to zero. For the second 1D model (in which the sharp-interface energy depends on the size of the jump and can tend to zero as the jump tends to zero), we describe general properties of the minimisers, and show that their gradients have a finite number of discontinuity points.
2009, 8(1): 83-94 doi: 10.3934/cpaa.2009.8.83 +[Abstract](1965) +[PDF](173.4KB)
Abstract:
We consider in this article a class of systems of second order partial differential equations with non-linearity in the first order derivative and zero order term which can be super-quadratic. These problems are motivated by differential geometry and stochastic differential games. Up to now, in the case of systems, only quadratic growth had been considered.
2009, 8(1): 95-121 doi: 10.3934/cpaa.2009.8.95 +[Abstract](2170) +[PDF](350.2KB)
Abstract:
In continuum mechanics problems, we have to work in most cases with symmetric tensors, symmetry expressing the conservation of angular momentum. Discretization of symmetric tensors is however difficult and a classical solution is to employ some form of reduced symmetry. We present two ways of introducing elements with reduced symmetry. The first one is based on Stokes problems, and in the two-dimensional case allows to recover practically all interesting elements on the market. This however is (definitely) not true in three dimensions. On the other hand the second approach (based on a very nice property of several interpolation operators) works for three-dimensional problems as well, and allows, in particular, to prove the convergence of the Arnold-Falk-Winther element with simple and standard arguments, without the use of the Berstein-Gelfand-Gelfand resolution.
2009, 8(1): 123-142 doi: 10.3934/cpaa.2009.8.123 +[Abstract](1963) +[PDF](1697.3KB)
Abstract:
This two-part paper treats the numerical approximation of a tricky quadratic eigenvalue problem arising from the following generalization of the classical Taylor-Couette problem: A viscous incompressible fluid occupies the region between a rigid inner cylinder and a deformable outer cylinder, which we take to be a nonlinearly viscoelastic membrane. The inner cylinder rotates at a prescribed angular velocity ω, driving the fluid, which in turn drives the deformable outer cylinder. The motion of the outer cylinder is not prescribed, but responds to the forces exerted on it by the moving fluid. A steady solution of this coupled fluid-solid system, analogous to the Couette solution of the classical problem, can be found analytically. Its linearized stability is governed by a non-self-adjoint quadratic eigenvalue problem. In Part I, we give a careful formulation of the geometrically exact problem. We compute the eigenvalue trajectories in the complex plane as functions of ω by using a Fourier-finite element method. Computational results show that the steady solution loses its stability by a process suggestive of a Takens-Bogdanov bifurcation. In Part II we prove convergence of the numerical method.
2009, 8(1): 143-160 doi: 10.3934/cpaa.2009.8.143 +[Abstract](2073) +[PDF](261.1KB)
Abstract:
This paper is the second part of a two-part paper treating a non-self-adjoint quadratic eigenvalue problem for the linear stability of solutions to the Taylor-Couette problem for flow of a viscous liquid in a deformable cylinder, with the cylinder modelled as a membrane. The first part formulated the problem, analyzed it, and presented computations. In this second part, we first give a weak formulation of the problem, carefully contrived so that the pressure boundary terms are eliminated from the equations. We prove that the bilinear forms appearing in the weak formulation satisfy continuous inf-sup conditions. We combine a Fourier expansion with the finite element method to produce a discrete problem satisfying discrete inf-sup conditions. Finally, the Galerkin approximation theory for polynomial eigenvalue problems is applied to prove convergence of the spectrum.
2009, 8(1): 161-178 doi: 10.3934/cpaa.2009.8.161 +[Abstract](2515) +[PDF](2474.7KB)
Abstract:
An operator-splitting algorithm is presented for the solution of a partial differential equation arising in the modeling of deposition processes in sand mechanics. Sand piles evolution is modeled by an advection-diffusion equation, with a non-smooth diffusion operator that contains a point-wise constraint on the gradient of the solution. Piecewise linear finite elements are used for the discretization in space. The advection operator is treated with a stabilized SUPG finite element method. An augmented Lagrangian method is proposed for the discretization of the fast/slow non-smooth diffusion operator. A penalization approach, together with a Newton method, is used for the treatment of inequality constraints. Numerical results are presented for the simulation of sand piles on flat and non-flat surfaces, and for extensions to water flows.
2009, 8(1): 179-193 doi: 10.3934/cpaa.2009.8.179 +[Abstract](2286) +[PDF](181.1KB)
Abstract:
In this paper, we consider anitropic singular perturbations of some elliptic boundary value problems. We study the asymptotic behavior as $\varepsilon \rightarrow 0$ for the solution. Strong convergence in some Sobolev spaces is proved and the rate of convergence in cylindrical domains is given.
2009, 8(1): 195-208 doi: 10.3934/cpaa.2009.8.195 +[Abstract](1965) +[PDF](580.2KB)
Abstract:
Pricing options on multiple underlyings or on an underlying modeled with stochastic volatility may involve solving multi-dimensional parabolic partial differential equations (PDE). Computing several such options at once for various moneyness levels can be a numerical challenge. We investigate here the Kolmogorov equation and Dupire or “pre-Dupire" equations to solve the problem faster and we validate the approach numerically. The heart of the method is to use the adjoint of the PDE of the option at the discrete level and to use discrete duality identities to obtain Dupire-like relations. The method works on every linear models. Numerical results are given for a European call option on a basket of two assets.
2009, 8(1): 209-235 doi: 10.3934/cpaa.2009.8.209 +[Abstract](1837) +[PDF](2219.2KB)
Abstract:
The slamming phenomenon is a violent impact of the hull of a ship on the free surface of the sea. This loading case is particularly difficult to modelize for several reasons: first of all, the wet surface of the hull is an unknown; then a coupling with the springing (flexibility of the ship) is very complex and finally the interaction with the waves (even if the eigenfrequencies of the structure and the one of the waves are very different) which can be at the origin of important damage mechanisms, involves pointwise effects. This paper aims at giving a simple mathematical model which enables one to simulate the full coupling between these phenomena.
2009, 8(1): 237-254 doi: 10.3934/cpaa.2009.8.237 +[Abstract](2166) +[PDF](380.7KB)
Abstract:
The aim of this paper is to develop a new class of finite elements on quadrilaterals and hexahedra. The degrees of freedom are the values at the vertices and the approximation is polynomial on each element $K$. In general, with this kind of finite elements, the resolution of second order elliptic problems leads to non-conform approximations.Degrees of freedom are the same than those of isoparametric finite elements. The convergence of the method is analyzed and the theory is confirmed by some numerical results. Note that in the particular case when the finite elements are parallelotopes, the method is conform and coincides with the classical finite elements on structured meshes.
2009, 8(1): 255-274 doi: 10.3934/cpaa.2009.8.255 +[Abstract](1715) +[PDF](587.7KB)
Abstract:
In this work we study, from the numerical point of view, a bone remodeling model. The variational formulation of this problem is written as an elliptic variational equation for the displacement field, coupled with a first-order ordinary differential equation, with respect to the time, to describe the physiological process of bone remodeling. Fully discrete approximations are introduced, based on the finite element method to approximate the spatial variable, and on an Euler scheme to discretize the time derivatives. Error estimates are obtained on the approximate solutions, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some numerical results, involving examples in one, two and three dimensions, are presented to show the accuracy and the performance of the algorithm.
2009, 8(1): 275-293 doi: 10.3934/cpaa.2009.8.275 +[Abstract](1650) +[PDF](661.4KB)
Abstract:
We consider the interaction of a rigid, frictionless, inelastic particle with a rigid boundary that has a corner. Typically, two possible final outcomes can occur: the particle escapes from the corner after experiencing a certain number of collisions with the corner, or the particle experiences an inelastic collapse in which an infinite number of collisions can occur in a finite time interval. For the former case, we determine the number of collisions that the particle will experience with the boundary before escaping the corner. For the latter case, we determine the conditions for which inelastic collapse can occur. For a corner composed of two straight walls, we derive simple analytic solutions and show that for a given coefficient of restitution, there is a critical corner angle above which inelastic collapse cannot occur. We show that as the corner angle tends to the critical corner angle from below, the process of inelastic collapse takes infinitely long. We also show a surprising phenomenon that if the corner has the form of a cusp, the particle can have an infinite number of collisions with the boundary in a finite time interval without losing all of its energy, and eventually escapes from the corner.
2009, 8(1): 295-309 doi: 10.3934/cpaa.2009.8.295 +[Abstract](2477) +[PDF](232.7KB)
Abstract:
In 1999 M. Eastwood has used the general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution to prove, at least in smooth situation, the equivalence of the linear elasticity complex and of the de Rham complex in $\mathbf{R}^{3}$. The main objective of this paper is to study the linear elasticity complex for general Lipschitz domains in $\mathbf{R}^{3}$ and deduce a complete Hodge orthogonal decomposition for symmetric matrix fields in $L^{2}$, counterpart of the Hodge decomposition for vector fields. As a byproduct one obtains that the finite dimensional terms of this Hodge decomposition can be interpreted in homological terms as the corresponding terms for the de Rham complex if one takes the homology with value in $RIG\cong \mathbf{R}^{6}$ as in the (BGG) resolution.
2009, 8(1): 311-333 doi: 10.3934/cpaa.2009.8.311 +[Abstract](2211) +[PDF](297.8KB)
Abstract:
In this paper, we deal with the three-dimensional Boussinesq system. We prove the local exact controllability to the trajectories of this system when the control is supported in a small set.

The main objective of this paper is to present a new method to control systems associated to equations of fluid dynamics. This method consists of controlling the same system with an additional control acting on the divergence condition in a first step and lifting this condition in a second step. In this paper, we have chosen to apply this technique to the Boussinesq system.
2009, 8(1): 335-359 doi: 10.3934/cpaa.2009.8.335 +[Abstract](2294) +[PDF](332.7KB)
Abstract:
In this article, we establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension $2$. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of the boundary. Two cases are considered, the subcritical and supercritical modes depending on the fact that the frequency mode is less or greater than the ratio between the reference stratified flow (around which we linearized) and the buoyancy frequency. The problem of boundary layers for the LPEs is of a new type since the corresponding limit problem displays a set of (unusual) nonlocal boundary conditions.
2009, 8(1): 361-382 doi: 10.3934/cpaa.2009.8.361 +[Abstract](2134) +[PDF](647.9KB)
Abstract:
In this paper, we address an inverse problem of reconstruction of the initial temperature in a heat conductive system when some measurement data of temperature are available, which may be observed in a subregion inside or on the boundary of the physical domain, along a time period which may start at some point, possibly far away from the initial time. A conditional stability estimate is first achieved by the Carleman estimate for such reconstruction. Numerical reconstruction algorithm is proposed based on the output least-squares formulation with the Tikhonov regularization using the finite element discretization, and the existence and convergence of the finite element solution are presented. Numerical experiments are carried out to demonstrate the applicability and effectiveness of the proposed method.
2009, 8(1): 383-404 doi: 10.3934/cpaa.2009.8.383 +[Abstract](4710) +[PDF](826.7KB)
Abstract:
Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, pre-defined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of defining the best sample of points is nevertheless rather complex and is in general open. In this paper we propose a way to derive such sets of points. We do not claim that the points resulting from the construction explained here are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive.
2009, 8(1): 405-418 doi: 10.3934/cpaa.2009.8.405 +[Abstract](1942) +[PDF](226.7KB)
Abstract:
The known theory on the one-side exact boundary controllability for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions at the non-control side (see [5],[10]). In this paper we illustrate, with an inspiring example, that the one-side exact boundary controllability can still be realized by means of a suitable coupling among the unknown variables in the quasilinear hyperbolic system itself instead of in boundary conditions.
2009, 8(1): 419-433 doi: 10.3934/cpaa.2009.8.419 +[Abstract](2605) +[PDF](276.7KB)
Abstract:
Nested boundary layers mean that one boundary layer lies inside another one. In this paper, we consider one such problem, namely,

$\varepsilon^3xy''(x)+x^2y'(x)- (x^3+\varepsilon)y(x)=0$ with $0 < x <1$, $y(0) = 1$ and $y(1) = \sqrt{e}$.

An asymptotic solution, which holds uniformly for $x\in [0,1]$, is constructed rigorously. This result also provides an explicit formula for the exponentially small leading term in the interval where the exact solution exhibits such behavior. This henomenon has never been mentioned in the existing literature.

2009, 8(1): 435-456 doi: 10.3934/cpaa.2009.8.435 +[Abstract](2184) +[PDF](247.5KB)
Abstract:
This manuscript aims at characterizing energy densities and constitutive laws of transversely isotropic materials, orthotropic elastic materials and materials with non orthogonal families of fibers. It makes explicit references to results that are scattered over the literature and, although said to be well-known, are not always easy to locate. Direct proofs that are thought to be new and simplified expressions of constitutive laws for materials with two preferred directions are given.
2009, 8(1): 457-471 doi: 10.3934/cpaa.2009.8.457 +[Abstract](2414) +[PDF](283.5KB)
Abstract:
This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation. This issue has been extensively addressed in the literature both in the case of complete and approximate insensitization (see [19] and [1], respectively).
But in the context of pure insensitization there are very few results identifying the class of initial data that can be insensitized. This is a delicate issue which is related to the fact that insensitization turns out to be equivalent to suitable observability estimates for a coupled system of heat equations, one being forward and the other one backward in time. The existing Carleman inequalities techniques can be applied but they only give interior information of the solutions, which hardly allows identifying the initial data because of the strong irreversibility of the equations involved in the system, one of them being an obstruction at the initial time $t=0$ and the other one at the final one $t=T$.
In this article we consider different geometric configurations in which the subdomains to be insensitized and the one in which the external control acts play a key role. We show that, under rather restrictive geometric restrictions, initial data in a class that can be characterized in terms of a summability condition of their Fourier coefficients with suitable weights, can be insensitized. But, the main result of the paper, which might seem surprising, shows that this fails to be true in general, so that even the first eigenfunction of the system can not be insensitized. This result is similar to those obtained in the context of the null controllability of the heat equation in unbounded domains in [14] where it is shown that smooth and compactly supported initial data may not be controlled.
Our proofs combine the existing observability results for heat equations obtained by means of Carleman inequalities, energy and gaussian estimates and Fourier expansions.
2009, 8(1): 473-491 doi: 10.3934/cpaa.2009.8.473 +[Abstract](2397) +[PDF](266.3KB)
Abstract:
The existence of stationary solution to an exterior domain of the Boltzmann equation was first studied by S. Ukai and K. Asano in [25, 27] and was recently generalized by S. Ukai, T. Yang, and H. J. Zhao in [29] to more general boundary conditions. We note, however, that the results obtained in [25, 29] require that the temperature of the far field Maxwellian is the same as the one of the Maxwellian preserved by the boundary conditions. The main purpose of this paper is to discuss the case when these two temperatures are different. The analysis is based on some new estimates on the linearized collision operator and the method introduced in [25, 27, 29].

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