Corrigendum
D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev. 55, 601-667 (2013).
We learn from our mistakes
If you find another error, please communicate it directly to
denis.grebenkov@polytechnique.edu
In spite of our considerable efforts, some misprints,
imprecisions or ambiguities could infiltrate in the review.
We apologize for them and invite everybody to inform us
about such points that will be summarized in this erratum.
- As pointed out by Prof. S. Benachour, the inequalities (6.8) and (6.9) were proved by Payne and Stakgold for any dimension,
not only in two dimensions, see details in Ref. [394].
In addition, there is an error in the right-hand side of the inequality (6.9), the original inequality from Ref. [394]
(page 304) reads:
u_1(x) \leq |x - \partial\Omega| \sqrt{\lambda_1} || u_1 ||_\infty
- As pointed out by Dr. B. Helffer, there is a misprint at the first line on page 609:
For Robin boundary condition (with h > 0), \alpha_{nk} are the positive zeros of the linear combination
z J'_n(z) + h J_n(z), not of J'_n(z) + h J_n(z) (i.e., the factor 'z' in front of J'_n(z) was omitted).
This misprint does not affect the remaining discussion.
- The Payne-Weinberger inequality (4.19) is valid only for convex domains (the convexity requirement
was not explicitly stated in the review); as discussed in Ref. [395], this inequality is in general false
for non-convex domains.
- As pointed out by Dr. J. Clutterbuck, the following phrase
on page 609 is misleading:
"The Robin boundary condition and a sector of a circular annulus
can be treated similarly."
The equation (3.10) for Laplacian eigenfunctions can indeed
be easily extended to the Robin boundary condition imposed
along the "radial" boundary r = R while still keeping Dirichlet
or Neumann boudnary condition at "angular" boundary \varphi = 0
and \varphi = \beta. However, this extension does not work
when the Robin boundary condition is imposed on the whole
boundary of the sector. In fact, the normal derivative at
the "angular" boundary \varphi = 0 is \partial_n = \partial_\varphi/r
that makes the parameter of the Robin condition to depend on r.
We are not aware of the explicit form of Laplacian eigenfunctions
in this situation.
- As pointed out by Dr. S. Mayboroda, the following phrase
on pages 622-623 is misleading:
"The inequalities (6.16)-(6.17) (or their extensions) were
reported by Moler and Payne [361] (see section 6.2.2) and
were used by Filoche and Mayboroda to determine the geometrical
structure of eigenfunctions [186] (see section 6.2.6)"
We have to admit this point. As stated in the review, the
inequalities (6.16,6.17) is the special case of the inequality
(6.15) which in turn is an elementary consequence of the Holder
inequality (6.3). Moler and Payne [361] have used similar
inequalities but they did not consider the L-infinity norm
and did not write the inequalities (6.16,6.17). We have found
them in the work by Filoche and Mayboroda [186]. We apologize
for this misleading formulation. By the way, we would appreciate
any information on earlier formulation or use of these inequalities.
Your appreciation can be sent to denis.grebenkov@polytechnique.edu
Last modified 04/12/2015