nach oben

2009 | Buch

Kapitel lesen Erstes Kapitel lesen

Elliptic Equations: An Introductory Course

verfasst von: Michel Chipot

Verlag: Birkhäuser Basel

Buchreihe : Birkhäuser Advanced Texts / Basler Lehrbücher

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues.

The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.

Inhaltsverzeichnis

Frontmatter

Basic Techniques

Frontmatter

Chapter 1. Hilbert Space Techniques

Abstract

The goal of this chapter is to collect the main features of the Hilbert spaces and to introduce the Lax-Milgram theorem which is a key tool for solving elliptic partial differential equations.

Chapter 2. A Survey of Essential Analysis

Abstract

We recall here some basic techniques regarding L^p-spaces in particular those involving approximation by mollifiers.

Chapter 3. Weak Formulation of Elliptic Problems

Abstract

Solving a partial differential equation is not an easy task. We explain here the weak formulation method which allows to obtain existence and uniqueness of a solution “in a certain sense” which coincides necessarily with the solution in the usual sense if it exists. First let us explain briefly in which context second-order elliptic partial differential equations do arise.

Chapter 4. Elliptic Problems in Divergence Form

Abstract

The goal of this chapter is to introduce general elliptic problems extending those already addressed in the preceding chapter and putting them all in the same framework.

Chapter 5. Singular Perturbation Problems

Abstract

This theory is devoted to study problems with small diffusion velocity. One is mainly concerned with the asymptotic behaviour of the solution of such problems when the diffusion velocity approaches 0. Let us explain the situation on a classical example (see [68],[69]).

Chapter 6. Asymptotic Analysis for Problems in Large Cylinders

Abstract

Let us denote by Ω the rectangle in ℝ² defined by

$$ \Omega _\ell = \left( { - \ell ,\ell } \right) \times \left( { - 1,1} \right). $$

(6.1)

Chapter 7. Periodic Problems

Abstract

In the previous chapter we have addressed the issue of the convergence of problems set in cylinders becoming large in some directions and for data independent of these directions. To be independent of one direction could also be interpreted as to be periodic in this direction for any period. Then the question arises to see if periodic data can force at the limit the solution of a periodic problem to be periodic. This is the kind of issue that we would like to address now.

Chapter 8. Homogenization

Abstract

The theory of homogenization is a theory which was developed in the last forty years. Its success lies in the fact that practically every partial differential equation can be homogenized. Also during the last decades composite materials have invaded our world. To explain the principle of this theory the composite materials are indeed very well adapted. Suppose that we built a “composite*”i.e., a material made of different materials by juxtaposing small identical cells containing the different type of materials for instance a three material composite — see Figure 8.1 below. Assuming that we make the cells smaller and smaller at the limit we get a new material — a composite — which inherits some properties which can be very different from the ones of the materials composing it. For instance mixing three materials with different heat conductivity in the way above leads at the limit to a new material for which we can study the conductivity by cutting a piece of it — say Ω. This is such an issue which is addressed by homogenization techniques (see [14],[40],[41],[72],[46],[47],[61],[87],[92]).

Chapter 9. Eigenvalues

Abstract

Suppose to simplify that Ωis the interval (-a,a), a > 0.

Chapter 10. Numerical Computations

Abstract

Suppose that Ωis a bounded open subset of ℝ². We would like to compute numerically the solution of the Dirichlet problem

$$ \left\{ \begin{gathered} - \Delta u = f in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} \right. $$

(10.1)

For simplicity we restrict ourselves to the two-dimensional case but the methods extend easily to higher dimensions (see [38]).

More Advanced Theory

Frontmatter

Chapter 11. Nonlinear Problems

Abstract

The goal of this chapter is to introduce nonlinear problems — i.e., problems whose solution does not depend linearly on the data. We will restrict ourselves to simple techniques of existence or uniqueness.

Chapter 12. L∞-estimates

Abstract

In general any quantity appearing in nature is finite. Thus a natural question is to derive some conditions on our data that will assure the solution of our elliptic equation to be bounded.

Chapter 13. Linear Elliptic Systems

Abstract

Sometimes, in many physical situations, one has not only to look for a scalar u but for a vector u = (u¹,... ,u^m). (In what follows a bold letter will always denote a vector.) This could be a position in space, a displacement, a velocity... So one is in need to introduce systems of equations. The simplest one is of course the one consisting in m copies of the Dirichlet problem, that is to say

$$ \left\{ \begin{gathered} - \Delta u^1 = f^1 in \Omega , \hfill \\ \cdots \cdots \cdots \cdots \hfill \\ - \Delta u^m = f^m in \Omega , \hfill \\ u = \left( {u^1 , \ldots u^m } \right) = 0 on \partial \Omega . \hfill \\ \end{gathered} \right. $$

(13.1)

Chapter 14. The Stationary Navier—Stokes System

Abstract

Let Ώbe a bounded open set of ℝⁿ. Like for the Stokes problem one is looking for a couple

$$ \left( {u,p} \right) $$

(14.1)

representing respectively the velocity of a fluid and its pressure such that

$$ \left\{ \begin{gathered} - \mu \Delta u + \left( {u \cdot \nabla } \right)u + \nabla p = f in \Omega , \hfill \\ div u = 0 in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} \right. $$

(14.2)

μ is the viscosity of the fluid. Note that here one is taking into account the nonlinear effect. The operator u · ∇ is defined as

$$ u^i \partial _{x_i } $$

(14.3)

with the summation convention in i. We will restrict ourselves to the physical relevant cases - i.e., n = 2 or 3. We refer the reader to [45], [56] for a physical background on the problem (see also [51], [52], [68]). Eliminating the pressure as we did in the preceding chapter we are reduced to find u such that

$$ \left\{ \begin{gathered} u \in \widehat{\mathbb{H}_0^1 }\left( \Omega \right), \hfill \\ \mu \int\limits_\Omega {\nabla u \cdot \nabla vdx} + \int\limits_\Omega {\left( {u \cdot \nabla } \right)u \cdot vdx} = \int\limits_\Omega {f \cdot v \forall v \in \widehat{\mathbb{H}_0^1 }\left( \Omega \right)} , \hfill \\ \end{gathered} \right. $$

(14.4)

where $ \widehat{\mathbb{H}_0^1 } $(Ώ) is defined by (13.38). In order to do that we introduce the trilinear form defined as

$$ t\left( {w;u,v} \right) = \int\limits_\Omega {\left( {w \cdot \nabla } \right)u \cdot vdx} = \int\limits_\Omega {w^i \partial _{x_i } u^j v^j dx} $$

(14.5)

(with the summation convention in i, j). One should notice that if

$$ w^i ,\partial _{x_i } u^j ,v^j \in L^2 \left( \Omega \right) $$

it is not clear that the product above is integrable. This will follow however for w,v ∈ ℍ ₀ ¹ (Ώ) from the Sobolev embedding theorem.

Chapter 15. Some More Spaces

Abstract

Let Ωbe a bounded open set of ℝⁿ. Suppose that we would like to solve the following nonlinear Dirichlet problem:

$$ \left\{ \begin{gathered} - \partial _{x_i } \left( {\left| {\nabla u} \right|^{p - 2} \partial _{x_i } u} \right) = f in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} \right. $$

(1)

Chapter 16. Regularity Theory

Abstract

We have seen in Chapter 3 that when u is a regular or strong solution of the Dirichlet problem

$$ \left\{ \begin{gathered} - \Delta u = f in \Omega , \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right. $$

(1)

then u is a weak solution of it. This was an argument to introduce weak solutions since strong solutions could not slip through our existence theory. Having obtained a weak solution to (16.1) one can then ask oneself if this solution also satisfies (16.1) in the usual sense. This requires of course some assumptions on f: recall that we know how to solve (16.1) for very general f ∈ H^-1(Ω). Let us try to get some insight through the one-dimensional problem

$$ \left\{ \begin{gathered} - u'' = f in \Omega = \left( {0,1} \right), \hfill \\ u\left( 0 \right) = u\left( 1 \right) = 0. \hfill \\ \end{gathered} \right. $$

(1)

Chapter 17. The p-Laplace Equation

Abstract

In this chapter we would like to consider in more details the Dirichlet problem for the p-Laplace operator, namely, if Ωis an open subset of ℝⁿ, 1 < p < +∞ the problem

(1)

Chapter 18. The Strong Maximum Principle

Abstract

Since we have seen that weak solutions can be smooth (see Chapter 16) in this chapter we would like to introduce some tools and properties appropriate for smooth solutions of elliptic problems (see also [15], [17], [44], [48]–[50], [54], [55], [57], [80]–[82]).

Chapter 19. Problems in the Whole Space

Abstract

We would like to consider here elliptic equations set in the whole Rⁿ. For instance if A is a positive definite matrix and a a nonnegative function we are going to consider the equation

$$ - div\left( {A\left( x \right)\nabla u} \right) + a\left( x \right)u = 0 in \mathbb{R}^n . $$

(1)

Backmatter

Titel: Elliptic Equations: An Introductory Course
verfasst von: Michel Chipot
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-9982-5
Print ISBN: 978-3-7643-9981-8
DOI: https://doi.org/10.1007/978-3-7643-9982-5