Mean value theorems for polyharmonic functions

Abstract

The problem of characterization of polyharmonic functions by mean value expressions is analyzed. A sufficient condition for the biharmonic function is formulated and proved in the paper. Mean value theorems and their operators have various applications in function theory (approximation of functions, description of functional spaces) and in the qualitative theory of linear differential equations with partial derivatives (boundary properties, elimination of features, differential properties of solutions, etc.). The main properties of polyharmonic functions, in particular, properties with mean value are investigated in the work. The results of L. Salcman and V. Volchkov, which generalize the classical mean value theorems, are analyzed. These results also reveal deep connections between complex analysis, the theory of linear differential equations with partial derivatives, harmonic analysis, integral geometry and the theory of special functions. The new directions of research in the theory of polyharmonic functions, differential equations, as well as in computational mathematics are established with the mean value theorems (algorithms "walk on spheres"\ , application of the Monte Carlo method). Particular attention is paid to the case of biharmonic functions. It is found that when performing the equality with the mean value on the circle from the set on the complex plane, the function of class $C^2$ on the specified set satisfies the Laplace equation of the second order, i.e. the function is polyharmonic of the second order. By using a smooth function with a compact support, the required function is obtained as a weak solution of the corresponding Laplace equation. In the future, this result can be used to establish a sufficient condition of biharmonicity of a twice continuously differentiated function on a plane that satisfies the above-mentioned relation only for a pair of positive values of $r_1$, $r_2$. Thus, it is possible to formulate a new theorem on two radii.