= Signed power test for a Laplacian eigenfunction
{title2=$(\gamma-1)\int|u|^{\gamma-2}|Du|^2=\lambda\int|u|^\gamma$}
For a smooth <Dirichlet Laplacian eigenfunction>, testing $-\Delta u=\lambda u$ with $|u|^{\gamma-2}u$ for $\gamma\geq2$ gives the displayed <energy estimate>. The <Sobolev chain rule> for $v=|u|^{\gamma/2}$ then yields $\int|Dv|^2=\lambda\gamma^2/[4(\gamma-1)]\int|u|^\gamma$. The absolute value and the sign in the test function permit nodal changes. At $\gamma=2$, use the Lipschitz absolute-value chain rule and the fact that a <gradient of a Sobolev function vanishes on a level set>.
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