Sobolev Spaces

Sobolev spaces are, roughly speaking, spaces of p-integrable functions whose derivatives are also p-integrable. There are two basic types of Sobolev spaces we wish to consider. In the first situation, we have a bounded domain Ω ⊂ ℝN and we consider functions u : Ω → ℝ for which u ≡ 0 on ∂Ω. In the second, we look at functions u : [0, T] → ℝN satisfying u(0) = u(T). The only problem in straightforwardly defining these spaces is that p-integrable functions need not be differentiate and restricting our attention to those that are differentiate does not provide us with what we want — the resulting spaces are not complete. We must weaken our notion of differentiability.

Definition A.1. In the following two cases we define an appropriate notion of weak differentiability.

Eigenvectors, geodesies, minimal surfaces, harmonic maps, conformal metrics with prescribed curvature, subharmonics of Hamiltonian systems, solutions of semilinear elliptic partial differential equations and Yang-Mills fields are all critical points of some functional on an appropriate manifold. This is not surprising since many of the laws of mathematics and physics can be formulated in terms of extremum principles.

Finding such points by minimization is as ancient as the least action principle of Fermat and Maupertuis, and the calculus of variations has been an active field of mathematics for almost three centuries. For more general, unstable extrema, the methods have a more recent history. Two, not unrelated, theories are available for dealing with the existence of such points: Morse theory and the min-max methods (or the calculus of variations in the large) introduced by G. Birkhoff and later developed by Ljusternik and Schnirelmann in the first half of this century. Currently, both theories are being actively refined and extended in order to overcome the limitations to their applicability in the theory of partial differential equations: limitations induced by the infinite dimensional nature of the problems and by the prohibitive regularity and non-degeneracy conditions that are not satisfied by present-day variational problems.

Weak Solutions

In the variational approach to solving problems, we associate with the given problem a C1 functional, ϕ, on a suitable Banach space (more generally, a manifold) in such a way that the critical points of ϕ are exactly the solutions to the original problem. Before proceeding with the variational formulation of some differential equations, we need to settle on an appropriate notion of differentiability for functions defined on a Banach space.

Definition B.1. Suppose that X is a Banach space and a function ϕ : X → ℝ is given.

In this chapter, we present two results of Ghoussoub and Maurey: the first is concerned with linear perturbations and is applicable in reflexive Banach spaces. It will be used in Theorems 2.12 and 2.13 below to get generic minimization results in the case of critical exponents and non-zero data. The second one deals with the possibility of finding plurisubharmonic perturbations and can be applied to spaces as “bad” as L1. The proofs essentially boil down to showing that these new classes of functions are, in the terminology of Chapter 1, admissible cones of perturbations. However, the methods here are slightly more involved than the ones used earlier. Moreover, the minima for the perturbed functionals are not necessarily close to any given minimizing sequence of the original function, and the perturbations we are seeking here, can never be bounded on the whole Banach space.

Actually, our main goal for this chapter, is to introduce the reader to new and different methods for proving perturbed variational principles and especially, to the martingales techniques used in Theorem 2.17.

A minimization principle with linear perturbations

We shall first consider a situation where the perturbations can be taken to be linear.