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We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon. Keywords. Mean value theorem, regular polygon, harmonic polynomial., Introduction Let n [member of] N, n [greater than or equal to] 3. There is a well-known result in the theory of harmonic functions that was proved independently in works [...]

In terms of the Bessel functions, we characterize smooth solutions of some convolution equations in the complex plane and prove a two-radius theorem for solutions of homogeneous linear elliptic equations with constant coefficients whose left-hand sides are representable in the form of a product of some non-negative integer powers of the complex differentiation operators d and d. Keywords. Convolution equation, mean-value theorem, Bessel function, distribution, spherical transformation., Introduction The convolution equations generated by distributions with compact supports and the corresponding mean-value theorems were investigated by many authors (see, e.q., [1,2]). In particular, Volchkov [2, Part 3, Chapter [...]

We prove a mean value theorem that characterizes continuous weak solutions of homogeneous linear partial differential equations with constant coefficients in Euclidean domains. In this theorem the mean value of a smooth function with respect to a complex Borel measure on an ellipsoid of special form is equal to some linear combination of its partial derivatives at the center of this ellipsoid. The main result of the paper generalizes a well-known Zalcman's theorem. [ABSTRACT FROM AUTHOR]