Let A be the unit square, $\{u_k\}$ is the set of all L2normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\limits_V dx u_k(x)^2 > 0$ ?
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My offic mate and I believe this is true. By separation of variables the eigenfunctions are just $Csin(\pi kx)sin(\pi ly)$ for some fixed constant $C > 0$. Using the trig identity $\sin^2(x) = (1\cos(2x))/2$, we see that $\int_a^b \sin^2(kx)\ dx = (1/k)\int_{ak}^{bk} \sin^2(x)\ dx = (ba)/2  (1/2k)\int_{ak}^{bk}\cos(2x)\ dx \geq (ba)/2 1/2k$ The integral of $u_k^2$ on a small square is just $\int_a^b\int_a^b \sin^2(\pi kx)\sin^2(\pi ly)\ dxdy$ so we can apply the previous line twice. 


If we take the usual trigonometric basis, it is indeed true as has been pointed out. However, there is a harder form of the question: Some of the eigenvalues are degenerate. If we allow arbritrary eigenfunctions, does the claim remain true? 

