ON MEAN VALUE PROPERTIES INVOLVING A LOGARITHM-TYPE WEIGHT.

Two new assertions characterizing analytically disks in the Euclidean plane ℝ² 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. [ABSTRACT FROM AUTHOR]