= Solution
Let $\Omega=(-1,1)^2$ and use the clamped energy space $V=H_0^2(\Omega)$, whose traces satisfy $u=\partial_nu=0$ on $\partial\Omega$. Two applications of <integration by parts> give
$$
\langle\Delta^2u,u\rangle_{L^2}
=\int_\Omega(\Delta u)^2\,dx\,dy\geq0.
$$
If equality holds, then $\Delta u=0$. The <maximum principle for harmonic functions> and the zero <Dirichlet boundary condition> imply $u=0$. The <biharmonic operator> is therefore positive definite. The same identity and conclusion hold for the simply supported conditions $u=\Delta u=0$; either standard interpretation of the paper's phrase “zero boundary conditions” gives the result.
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