Abstract:

In this paper, we study an inhomogeneous polyharmonic Neumann problem with $L^{p}$ boundary data on Lipschitz domains in $\mathbb{R} ^{2}$ by the method of layer potentials. Applying multi-layer ${\cal S}$-potentials, which are higher order analogues of the classical singular layer potential and defined in terms of polyharmonic fundamental solutions, the unique integral representation solution is given for the inhomogeneous polyharmonic Neumann problem on Lipschitz domains in $\mathbb{R} ^{2}$ when the boundary data are in some $L^{p}$ spaces.

Key words: Polyharmonic fundamental solutions, Multi-layer ${\cal S}$-potentials, Inhomogeneous Neumann problem, Polyharmonic equation