Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ on $\Omega$. Does there exist a domain $\Omega$ that contains the origin and such that (i) the eigenvalues are all simple (i.e. multiplicity one) and (ii) for each $k\in \mathbb N$, $\phi_k(0) \neq 0$.
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2$\begingroup$ (ii) is not really asking for anything since you only have countably many eigenfunctions, so you can take a point where $\phi_k\not= 0$ and call it $0$. $\endgroup$– Christian RemlingApr 17 at 1:20
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2$\begingroup$ In other words, pretty much anything with not too much symmetry will work, for example a suitable rectangle (for $n=2$). $\endgroup$– Christian RemlingApr 17 at 1:41
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$\begingroup$ Not sure I understand your first comment about just “picking” a point and calling it 0. After all the origin is a fixed point. $\endgroup$– AliApr 17 at 12:07
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1$\begingroup$ If you have a domain $\Omega$ and some $x_0\in\Omega$ with $\phi_k(x_0)\not= 0$, you can shift it around and move $x_0$ to $0$, or you could choose a coordinate system that has $x_0$ as its origin. $\endgroup$– Christian RemlingApr 17 at 13:23
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1$\begingroup$ If we change the domain, the eigenvalues start moving around, and since there are only countably many, it seems quite a coincidence for two or more of them to have exactly the same value. One can probably prove something along these lines, but you could also just check the example I suggested (rectangle). $\endgroup$– Christian RemlingApr 17 at 18:11
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