Removable singularities in the boundary conditions
Yuri Egorov · Banach Center Publications · 1992
1. Let G be an open set in R and let F be its boundary. Let Γ be some part of F which is a smooth (n−1)-dimensional submanifold. Let A be a closed subset of Γ . Let u be a function harmonic in G satisfying the boundary condition Dvu = 0 on Γ A, where v is the outer normal to Γ . When can we say that Dvu = 0 on Γ , i.e. when the singularity of u on A is removable? It is evident that the answer depends on the structure of A and on the behaviour of u in a neighbourhood of A. For instance, if A is a single point, then the singularity is removable if |u(x)| = o(r2−n) as r → 0, where r is the distance from A, and can be nonremovable if n > 2 and |u(x)| = O(r2−n). Indeed, let f ∈ C∞ 0 (Γ ). We show that if |u(x)| = o(r2−n), then ∫