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Norms of functions

Suppose we consider functions defined on some nice set , for example . Then it is useful to have an idea how large the functions are
Definition 1:

A norm on a vector space V is a real valued function on V, satisfying

a)
for all , and .
b)
for all .
c)
for all and .

Definition 2:

A space is called a metric space provided there is defined a real-valued distance between every , such that

a)
for all
b)
for every ,
c)
whenever
d)
for all
on many spaces we can take if there is a suitable norm
Definition 3:

(limits in a normed space)

Definition 4:

A sequence is called a Cauchy sequence in if an only if for every there exists an integer such that

``In second year calculus we learn that every Cauchy sequence converges, but that is for sequences in , not any old space!'' M.A. McKiernan (Applied Math., Waterloo)
Definition 5:

is a complete metric space if every Cauchy sequence converges to a limit in
Definition 6:

A normed linear space that is complete when considered as a metric space is called a Banach space
Definition 7:

Let X denote a linear space over (or over ). An inner product is a mapping satisfying

  1. where ``--------'' denotes complex conjugate
  2. and iff x = 0.
Exercise:
Show that the function defines a norm, so we may write
Definition 8:

A Hilbert space is an inner product space which, with the norm , is a Banach space
Definition 9:

A Sobolev norm is a norm of the form

or similar involving higher derivatives (the idea is that the derivatives are involved)
Remark: if p = 2 this is an inner product norm
Definition 10:

A Sobolev space is a Banach space where the metric is given by where is a Sobolev norm
Pages 22-26 Sobolev: key points

Cauchy problem: Lu = 0 in (infinite)

Dirichlet problem: Lu = 0 in

Neumann problem: Lu = 0 in

Pages 26-32 discuss what is special about these types of boundary conditions, and define the concepts of continuous dependence on the boundary conditions, and of a well-posed problem (in the book this is called ``correctly formulated'' but this is bad translation). We may as well work with some generality now. Suppose our PDE is

which holds when , and the boundary conditions are given on certain manifolds by

where is a given function defined on the manifold . We now consider Sobolev norms on functions defined on the , and a Sobolev norm on functions (in a class u) defined on .

and

For u we take the class of functions with continuous derivatives up to order p in , and for we take the class of functions with continuous derivatives (with respect to the parameters of the manifolds ) up to order k. For the Sobolev norms we take, eg.,

and

means all possible j order derivatives
(this is the -norm case of the norms given before). Then our problem depends continuously on the data provided that for all there exists a such that if

then

where solves Lu = 0 in , on
and u solves Lu = 0 in , on . A problem is called well-posed if

(a)
a solution exists
(b)
the solution is unique
(c)
the solution depends continuously on the data.

Remark: This notion is justly famous. It is a very powerful idea. A problem that is not well-posed is called ill-posed. Ill-posed problems are very difficult to deal with: arbitrarily small perturbations in the input data may cause great changes in the solution. However, many important practical problems are ill-posed: inverse problems in geophysics (ore body detection), medical imaging (eg. F. Beardwood's work), etc. One has to try to regularize the problem in some physically meaningful way, to restrict the class of allowable perturbations in the data, for example. This is a deep and important subject. On the other hand, continuous dependence isn't always as strong as we need it to be, either! It allows the phenomenon of Sensitive Dependence on Initial Conditions (S.I.C.). A problem that has bounded solutions that are S.I.C. is called chaotic. The difficulty is that the required may well be exponentially small with respect to . Sobolev was perfectly well aware of this (2 paragraph, p. 32). There is now a new notion in the literature (alas, not too much noticed) that addresses this by considering perturbations to the problem, ie. the PDE as well as to the boundary conditions. We say that a problem is well-enough conditioned if there exists some physically meaningful function and a moderate constant K such that if solves a problem with then (it is not necessary that be small --- P can be the Lyapunov exponent, or the dimension of the attractor, eg.)
Exercise:
Consider the problem (ODE)

and the neighbouring problems

and discuss whether it is likely that this could be well-enough conditioned. (It is, like all smooth IVP for ODE, well-posed.)





next up previous
Next: Hadamard's example Up: No Title Previous: No Title



Robert Corless
Wed Feb 4 14:46:04 EST 1998