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) A k W = 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 e Qua^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. é http://dx.doi.org/10.1090/cbms/074/01
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