Discretizations

Abstract

The W1,2 functional setting for the action functional A, introduced in Chapter 3, presents several drawbacks. First of all, the regularity that we can expect for A is only C1,1, at least if we work with a general convex quadratic-growth Lagrangian. This prevents the applicability of all those abstract results that require more smoothness, for instance the Morse lemma from critical point theory. Moreover, the W ^1,2 topology is sometimes uncomfortable to work with.