Weak Convergence Methods for Nonlinear Partial Differential ...

Chapter 1
Introduction
Writing a general partial differential equation (PDE for short) as
A(u) = f, (1.1)
where A is a (nonlinear) partial differential operator (PDO), one is often interested
in solving a appropriate approximate problem
A ε (u ε ) = f ε , (1.2)
This way one hopes, e.g., if it is too hard to show that (1.1) does have solutions,
to first solve the easier problem (1.2) and then find a solution of (1.1) in the
limit ε ց 0. Another motivation is dictated by numerics: In order to solve (1.1)
numerically, one needs to discretize the equation, thus arriving at an approximate
problem of the form (1.2). But also the reverse point of view is interesting in
applications. Many (physical) systems involve some small parameter (or scale)
ε. Then (1.2) desribes a complicated system which we would like to approximate
by a simpler equation of the form (1.1) for small ε.
In any case, the main task is to show that solutions u ε of (1.2) do converge
to solutions u of (1.1), possibly up to extracting subsequences. Now typically
one does not have much knowledge about the sequence (u ε ). In particular, if k
is a PDO of order k, there is no hope that (u ε ) converges strongly in C k or W k,p
at all! However, suitable a priori estimates may guarantee at least that (u ε ) is
bounded and so—up to subsequences—weakly convergent to some function u.
Now the notion of weak convergence is taylored so as to give convergent
quantityies under linear operations. Ususally, if (u ε ) converges to u weakly (write
u ε ⇀ u) and A is nonlinear, one cannot deduce that A(u ε ) ⇀ A(u), let alone
A ε (u ε ) ⇀ A(u). In order to succeed we will therefore have to use the only
available piece of information, namely, that u ε solves the PDE (1.2), in a crucial
way. The core theme of this course will be how this fact ‘compensates’ for the
lack of strong compactness of (u ε ).
In this sense, we are basically investigating weak continuity properties of
nonlinear operations on function spaces. A strongly related question arises in
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