Boundness of Laplacian eigenfunctions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:16:03Z http://mathoverflow.net/feeds/question/61031 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61031/boundness-of-laplacian-eigenfunctions Boundness of Laplacian eigenfunctions Denis Grebenkov 2011-04-08T07:42:09Z 2011-04-08T10:41:42Z <p>Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$). </p> <p>Is it true that these eigenfunctions are uniformly bounded, i.e., <code>$sup_k \|u_k\|_\infty &lt; \infty$</code>, where <code>$\|.\|_\infty$</code> is the $L^\infty$-norm (the maximum)? In other words, does there exist a constant $C_A$ such that for any $k$ and any $x\in A$, $|u_k(x)| &lt; C_A$?</p> <p>If the answer is positive, please provide a reference or a proof.</p> <p>If the answer is negative, please provide a counter-example. In that case, what are the conditions on the domain A to make this statement true?</p> http://mathoverflow.net/questions/61031/boundness-of-laplacian-eigenfunctions/61033#61033 Answer by Denis Serre for Boundness of Laplacian eigenfunctions Denis Serre 2011-04-08T08:14:16Z 2011-04-08T08:14:16Z <p>I don't have a general answer (I guess it is <em>yes, there are uniformy bounded</em>, at least when $A$ is a smooth bounded domain). At least, let me mention the case of the torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$. The eigenfunctions are the exponentials $\exp(2i\pi m\cdot x)$, where $m\in\mathbb N^d$. They are uniformly bounded and this fact is crucial in the Riesz-Thorin interpolation theorem that the Fourier series of an $L^p$-function $f$ belongs to $\ell^{p'}$ whenever $1\le p\le2$ and $p'$ is the conjugate exponent.</p> http://mathoverflow.net/questions/61031/boundness-of-laplacian-eigenfunctions/61039#61039 Answer by Anatoly Kochubei for Boundness of Laplacian eigenfunctions Anatoly Kochubei 2011-04-08T09:54:36Z 2011-04-08T09:54:36Z <p>There is a paper on this subject (containing also further references) by C. D. Sogge, Eigenfunction and Bochner-Riesz estimates on manifolds with boundary. Math. Res. Lett. 9, No.2-3, 205-216 (2002), ArXiv: math/0202032. It is stated that typically there is some growth in $L_\infty$ metric, not uniform boundedness.</p> http://mathoverflow.net/questions/61031/boundness-of-laplacian-eigenfunctions/61041#61041 Answer by Michael Renardy for Boundness of Laplacian eigenfunctions Michael Renardy 2011-04-08T10:05:49Z 2011-04-08T10:05:49Z <p>The answer is no. The following reference specifically discusses the case of the two-dimensional disk: <a href="http://www.staff.uni-oldenburg.de/daniel.grieser/wwwpapers/diss.pdf" rel="nofollow">http://www.staff.uni-oldenburg.de/daniel.grieser/wwwpapers/diss.pdf</a></p>