Variational Methods in Differential Equations

Abstract

This chapter concerns classical variational methods in boundary value problems and a free boundary problem, with a special emphasis on how to view a differential equation as a variational problem. Variational methods are simple, but very powerful analytical tools for differential equations. In particular, the unique solvability of a differential equation reduces to a minimization problem, for which a minimizer is shown to be a solution to the original equation. As a model problem, the Poisson equation with different types of boundary conditions is considered. We begin with the derivation of the equation in the context of potential theory, and then show successful applications of variational methods to these boundary value problems. Finally, we study a free boundary problem by developing the idea to a minimization problem with a constraint.

Appendix

Here, the basic inequalities in analysis known as Hölder’s inequality and the Poincaré inequality are supplied for the sake of completeness.

Hölder’s inequality (or the Cauchy-Schwarz inequality) states that

$$\begin{aligned} \left| \int \limits _{\Omega }uv\,\mathrm{{d}}x\right| \le \Vert u\Vert _{L^2(\Omega )}\Vert v\Vert _{L^2(\Omega )} \end{aligned}$$

(25)

holds for $u,v\in L^2(\Omega )$. This is a natural generalization of the inequality $|x\cdot y|\le |x||y|$ for $x,y\in \mathbb {R}^n$, since the left-hand side is the inner product $(u,v)_{L^2(\Omega )}$. In particular, equality holds in (25) if and only if $u=\alpha v$ for some scalar $\alpha \in \mathbb {R}$.

The Poincaré inequality states that, for a bounded domain $\Omega$, there is a constant $C>0$ such that

$$\begin{aligned} \Vert u\Vert _{L^2(\Omega )}\le C\Vert \nabla u\Vert _{L^2(\Omega )} \end{aligned}$$

holds for $u\in H_0^1(\Omega )$. The boundedness of $\Omega$ can be relaxed to some extent; however, the inequality does not hold, in general, for unbounded domains. Moreover, $H_0^1(\Omega )$ cannot be replaced by $H^1(\Omega )$ for the inequality to hold. Indeed, the constant function $u\equiv 1$ violates the inequality.