On mean value properties involving a logarithm-type weight

Abstract

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.

Keywords

• harmonic function
• helmholtz equation
• modified helmholtz equation
• mean value property
• logarithmic weight
• characterization of balls