Introduction

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-

ods.

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

iuk}

T- ' **

*s eQua^y °ften a

l

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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http://dx.doi.org/10.1090/cbms/074/01