12.1 The Dirichlet principle
Let \(\Omega \) be a bounded region with smooth boundary, \(f\) continuous on \(\overline \Omega \), \(g\) given on \(\partial \Omega \), and define the energy \[ E[w] = \int _\Omega \Bigl (\frac 12|\nabla w|^2 - fw\Bigr )dx \] on the admissible class \(\mathcal A\) of functions \(w \in C^1(\overline \Omega )\) with \(w = g\) on \(\partial \Omega \). For an elastic membrane under load \(f\), \(\frac 12|\nabla w|^2\) is the stretching energy and \(-fw\) the potential energy of the load.
Theorem 12.1.1 (Dirichlet principle). A function \(u \in \mathcal A \cap C^2(\Omega )\) solves \(-\Delta u = f\) in \(\Omega \) if and only if \(E[u] = \min _{w \in \mathcal A}E[w]\).
Proof. For any \(w \in \mathcal A\), \(v = w - u\) vanishes on \(\partial \Omega \), and \[ E[u + v] = E[u] + \int _\Omega (\nabla u\cdot \nabla v - fv)\,dx + \frac 12\int _\Omega |\nabla v|^2\,dx . \] If \(-\Delta u = f\), integration by parts gives \(\int \nabla u\cdot \nabla v = \int (-\Delta u)v = \int fv\), the middle term vanishes, and \(E[w] = E[u] + \frac 12\int |\nabla v|^2 \geq E[u]\). Conversely, if \(u\) minimises, then \(\varepsilon \mapsto E[u + \varepsilon v]\) has a minimum at \(\varepsilon = 0\) for every \(v \in C^1(\overline \Omega )\) vanishing on the boundary, so its derivative there, \(\int (\nabla u\cdot \nabla v - fv) = \int (-\Delta u - f)v\), is zero for all such \(v\); hence \(-\Delta u = f\). □
The equation \(\int (\nabla u\cdot \nabla v - fv) = 0\) for all \(v\) is the Euler–Lagrange equation of \(E\) in integrated form. It makes sense when \(u\) has only first derivatives, which is the key to what follows.
Dirichlet’s principle assumes that a minimiser exists, and that is the gap. The energy is bounded below on \(\mathcal A\) (Problem 12.7.1), so it has an infimum, but a minimising sequence need not converge to anything in \(\mathcal A\). Weierstrass’s counterexample was a one-dimensional problem: minimise \(\int _{-1}^1x^2y'(x)^2\,dx\) over \(C^1\) functions with \(y(-1) = -1\), \(y(1) = 1\). The functions \(y_\varepsilon = \arctan (x/\varepsilon )/\arctan (1/\varepsilon )\) are admissible and their values tend to \(0\) (Software 12.6.3), so the infimum is \(0\); but a function with value \(0\) would have \(xy' = 0\), so \(y\) would be constant on each side of \(0\), and no continuous function can be \(-1\) at one end, \(1\) at the other, and constant on both halves. The minimising sequence tends to a step function, outside the admissible class.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.