We discuss the boundary value problem $\Delta u=f$, $\partial^ku/\partial n^k=g$ in a bounded two-dimensional domain. For a smooth simply connected region, we prove that the only solutions of the homogeneous problem are harmonic polynomials of degree $k-1$. For multiply connected domains, this is true ``generically." In domains with corners, on the other hand, there are additional solutions.
"Higher order Neumann problems for Laplace's equation in two dimensions." Differential Integral Equations 15 (10) 1273 - 1279, 2002. https://doi.org/10.57262/die/1356060755