Elliptic PDE: Variational Techniques

Abstract

This chapter is the heart of the book. We define the elliptic problems that we will study. A typical example of this is the Laplace problem with homogeneous Dirichlet condition on a regular bounded open set.

We show how the solution of such a partial differential equation is a critical point for a convex functional. The problem then becomes that of looking for a minimal point of a convex functional with good properties on a reflexive Banach space: it tends to +∞ at infinity and is convex and continuous. The convexity and continuity lead to the weak lower semicontinuity of the functional, and the compactness theorems from Chapters 2 and 3 allow us to show that a minimizing sequence admits a subsequence that converges in a suitable sense to a minimum, which by Euler’s equation is the solution of the original partial differential equation.

An important part of this chapter deals with the regularity of the solutions. For example, when the Laplacian of u is a function f belonging to L ², then $u\in W^{2,2}_{\mathrm{loc}} (\Omega)$. In particular, a posteriori, when f is sufficiently regular, the solutions may be of class $\mathcal{C}^{2}$. We thus obtain the classical solutions. We give examples of nonlinear problems such as the p-Laplacian equation and we conclude the chapter with maximum principles and the Hopf principle.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2807-6_8