Inverse Mean Value Property of Metaharmonic Functions

Invariant Mean Value Property and Harmonic Functions

We give conditions on the functions σ and u on R such that if u is given by the convolution of σ and u, then u is harmonic on R.

On the mean value property of superharmonic and subharmonic functions

Recall that a function u is harmonic (superharmonic, subharmonic) in an open set U ⊂ Rn (n ≥ 1) if u ∈ C2(U) and Δu = 0 (Δu ≤ 0,Δu ≥ 0) on U . Denote by H(U) the space of harmonic functions in U and SH(U) (sH(U)) the subset of C2(U) consisting of superharmonic (subharmonic) functions in U . If A ⊂ Rn is Lebesgue measurable, L1(A) denotes the space of Lebesgue integrable functions on A and |A| d...

A Mean Value Property of Harmonic Functions on the Interior of a Hyperbola

We establish a mean value property for harmonic functions on the interior of a hyperbola. This property connects their boundary values with the interior ones on the axis of the hyperbola from the focus to infinity.

A Mean Value Property of Harmonic Functions on Prolate Ellipsoids of Revolution

We establish a mean value property for harmonic functions on the interior of a prolate ellipsoid of revolution. This property connects their boundary values with those on the interfocal segment.

Function Theoretic Methods in the Theory of Boundary Value Problems for Generalized Metaharmonic Functions