$L^p$ polyharmonic Robin problems on Lipschitz domains

TL;DR

This paper extends the analysis of Robin boundary value problems for polyharmonic equations on Lipschitz domains within $L^p$ spaces, employing layer potential methods to generalize second-order results to higher orders.

Contribution

It introduces a layer potential approach for solving polyharmonic Robin problems on Lipschitz domains, generalizing existing second-order results to higher order cases.

Findings

Solved Robin BVPs using layer potentials for polyharmonic equations.

Generalized second-order results to higher order polyharmonic cases.

Established solvability in $L^p$ spaces on Lipschitz domains.

Abstract

In this paper, we study a class of boundary value problems (BVPs) with Robin conditions in some $L^p$ spaces for polyharmonic equation on Lipschitz domains. Utilizing polyharmonic fundamental solutions, these Robin BVPs are solved by the method of layer potentials. The crucial ingedients of our approach are the classical single layer potential and its higher order analog (which are called multi-layer $S$-potentials), and the main results generalize ones of second order (Laplacian) case to higher order (polyharmonic) case.