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Results tagged with geometric-analysis
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user 108637
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If the first Dirichlet eigenfunction on a set $D$ is regular up to the boundary, is $D$ regu...
One cannot expect any fatness of the domain $D$: the thinner $D$ at some boundary point, the faster the decay of the eigenfunction near this point. In particular, the first eigenfunction will be Lipsc …
- 17.2k
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Is there a harmonic function with just one singular point?
Yes, this is possible. An explicit example is
$$u(x, y, z) = 1 - I_0\left(\sqrt{x^2 + y^2}\right) \, \cos z$$
when $L = \pi$, and $u\big(\frac{\pi x}{L}, \frac{\pi x}{L}, \frac{\pi x}{L}\big)$ for a g …
- 17.2k