= Solution
Multiply the <Dirichlet Laplacian eigenfunction> equation by $u$ and use <integration by parts>. The zero <Dirichlet boundary condition> removes the boundary term and gives
$$
\lambda\int_\Omega u^2=\int_\Omega|Du|^2.
$$
The denominator is positive because the <eigenfunction> is nonzero. The numerator is also positive: if it were zero, the given <Sobolev inequality> for $u\in W_0^{1,2}(\Omega)$ would force $u=0$. Therefore the <Dirichlet Laplacian eigenvalue> has the positive <Rayleigh quotient>
$$
\boxed{\lambda=\frac{\int_\Omega|Du|^2}{\int_\Omega u^2}>0.}
$$
\b[Every nonzero Dirichlet eigenfunction in this problem has a positive eigenvalue.]
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