Removable Singularity of the Polyharmonic Equation

Abstract. Let x0 ∈ Ω ⊂ R, n ≥ 2, be a domain and let m ≥ 2. We will prove that a solution u of the polyharmonic equation ∆u = 0 in Ω \ {x0} has a removable singularity at x0 if and only if |∆u(x)| = o(|x − x0|) ∀k = 0, 1, 2, . . . , m − 1 as |x − x0| → 0 for n ≥ 3 and = o(log(|x−x0|)) ∀k = 0, 1, 2, . . . , m− 1 as |x−x0| → 0 for n = 2. For m ≥ 2 we will also prove that u has a removable singularity at x0 if |u(x)| = o(|x − x0|) as |x−x0| → 0 for n ≥ 3 and |u(x)| = o(|x−x0| log(|x−x0|)) as |x−x0| → 0 for n = 2.