My intention in these notes is to explain systematically, and I hope clearly,
many of the most important techniques set forth in recent years for studying
nonlinear partial differential equations (PDE's) via weak convergence meth-
The basic issue, in its most abstract, cleanest form, is this: suppose we
wish to solve some nonlinear PDE, which we write symbolically as
(0.1) A[u] = f,
A[] denoting a given nonlinear Operator, / a given function, and u the
unknown. To establish the existence of a Solution u of (0.1), an obvious idea
is first to invent an appropriate collection of nicer, approximating problems,
which we can in fact solve. These we write abstractly as
(0.2) AkW = fk ( * = 1 , 2 , . . . ) ,
where Ak[ · ] represents a nonlinear Operator which is somehow close to A[ · ]
for large k, fkis close to / , and uk is a Solution. The hope ncw is that
the functions {uk}°°_ will converge to a Solution u of (0.1).
This proposed procedure is of course very general, too general to be of
specific guidance for any particular problem, and consequently practical Im-
plementation usually demands great care in the choice of approximations.
In practice, the Operators Ak[·] may represent finite-dimensional projec-
tions, singular regularizations, discretizations, gradients of approximate en-
ergy functionals, Systems collapsing in the limit to a Single equation, etc.
Indeed, given a nonlinear PDE like (0.1), it is usually not particularly diffi-
cult to dream up some reasonable seeming and solvable approximation: the
trick is to demonstrate that Solutions of (0.2) really do converge to a Solution
of (0.1). The overall impediment is of course the nonlinearity. Whereas it
is often the case that certain uniform estimates can be had for the family
T- ' **
*s eQua^y °ften a
s o t r u e
^a t these best available bounds are
none too strong. With such relatively poor estimates in hand, we can con-
sequently usually show only that the functions {uk} °°_ (or a subsequence)
converge weakly in some function space to a limit u :
(0.3) uk u as k oo.
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