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On mean value properties involving a logarithm-type weight

Nikolai G. Kuznecov — 2024

Mathematica Bohemica

Two new assertions characterizing analytically disks in the Euclidean plane ℝ 2 are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

Water-wave problem for a vertical shell

Nikolai G. Kuznecov;  Vladimir G. Maz'ya — 2001

Mathematica Bohemica

The uniqueness theorem is proved for the linearized problem describing radiation and scattering of time-harmonic water waves by a vertical shell having an arbitrary horizontal cross-section. The uniqueness holds for all frequencies, and various locations of the shell are possible: surface-piercing, totally immersed and bottom-standing. A version of integral equation technique is outlined for finding a solution.

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