Removable Singularities of Separately Harmonic Functions

Removable singularities of separately harmonic functions are considered. More precisely, we prove harmonic continuation property of a separately harmonic function u(x, y) in D \ S to the domain D, when D ⊂ Rn(x) × Rm(y), n,m > 1 and S is a closed subset of the domain D with nowhere dense projections S1 = {x ∈ Rn : (x, y) ∈ S} and S2 = {y ∈ Rm : (x, y) ∈ S}.

On Removable Singularities of Solutions of Higher-Order Differential Inequalities

We obtain sufficient conditions for solutions of the mth-order differential inequality\sum_{|\alpha|=m}\partial^{\alpha}a_{\alpha}(x,u)\geq f(x)g(|u|)\quad\text{in % }B_{1}\setminus\{0\}to have a removable singularity at zero, where {a_{\alpha}}, f, and g are some functions, and {B_{1}=\{x:|x|<1\}} is a unit ball in {{\mathbb{R}}^{n}}. We show in some examples the sharpness of these conditions.