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Keywords:
harmonic function; Helmholtz equation; modified Helmholtz equation; mean value property; logarithmic weight; characterization of balls
harmonic function; Helmholtz equation; modified Helmholtz equation; mean value property; logarithmic weight; characterization of balls
Summary:
Two new assertions characterizing analytically disks in the Euclidean plane $\mathbb {R}^2$ 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.
References:
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