Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / G / 1 / Solution

Take an admissible variation $u+ϵ\eta$ with $\eta =0$ on the boundary. Differentiating the functional at an extremum givesIntegration by parts removes the derivatives from $\eta$; the boundary term vanishes. The fundamental lemma of the calculus of variations yieldsFor the specified squared-gradient energy this is $2\Delta u=0$, so $u$ is harmonic. The affine function $\ell =ax+by+cz$ is harmonic and satisfies the boundary data.

To verify a global minimum, write $u=\ell +v$ with zero boundary trace. ThenThe middle term vanishes by integration by parts, since $\Delta \ell =0$ and $v=0$ at the boundary. Hence $I[u]\ge I[\ell ]$, andThis proves minimality directly, rather than relying only on a necessary Euler-Lagrange equation.