Symmetry problems for the Helmholtz equation

Ramm, Alexander G.

doi:10.1016/j.aml.2019.04.008

Cited by 6 publications

(3 citation statements)

References 5 publications

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“…In conclusion, we compare Theorem 2 with the result of Ramm [15,16], concerning the so-called refined Schiffer's conjecture; the latter is similar to the celebrated Serrin's theorem [17], but involves equation (3) instead of Poisson's. Berenstein [2], p. 143, investigated this conjecture for simply connected two-dimensional domains with smooth boundary (see also [14] for another approach), whereas the original Schiffer's conjecture described in [5] is discussed in the review [4].…”

Section: Main Results and Discussionmentioning

confidence: 99%

On characterization of balls via solutions to the Helmholtz equation

Kuznetsov¹

2022

Preprint

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A new analytical characterization of balls in the Euclidean space R m is obtained. Unlike previous results of this kind, using either harmonic functions or solutions to the modified Helmholtz equation, the present one is based on solutions to the Helmholtz equation. This is achieved at the expense of a restriction imposed on the size of a domain-a feature absent in the inverse mean value properties known before.

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Section: Main Results and Discussionmentioning

confidence: 99%

On characterization of balls via solutions to the Helmholtz equation

Kuznetsov¹

2022

Preprint

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“…For a connected two-dimensional domain with smooth boundary this conjecture was investigated in [15, p. 143] (cf. also [16] for another approach). It is worth mentioning that a clear description of the original Schiffer conjecture can be found in [17] (cf., also the review [18] for its discussion).…”

Section: Proof Of Theoremmentioning

confidence: 99%

Inverse Mean Value Property of Metaharmonic Functions

Kuznetsov¹

2022

J Math Sci

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A new analytical characterization of balls in the Euclidean space R m is obtained. Previous results of this kind involved either harmonic functions or solutions to the modified Helmholtz equation (both have positive fundamental solutions), whereas solutions to the Helmholtz equation are used here. This is achieved at the expense of a restriction imposed on the size of admissible domains-a feature absent in the inverse mean value properties known previously. Bibliography: 18 titles.

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“…It is worth to mention that Karp's theorem can be generalized to the symmetric problem for the Helmholtz equation proposed by Ramm. We refer the reader to his series of work, for instance, see [8,9,10] and the references therein.…”

Section: Introductionmentioning

confidence: 99%