On Payne-Schaefer's Conjecture about an Overdetermined Boundary Problem of Sixth Order

TL;DR

This paper proves the Payne-Schaefer conjecture for sixth-order overdetermined boundary problems in 2D and higher dimensions under certain conditions, introduces integral identities for fourth-order problems, and establishes symmetry results for second-order cases.

Contribution

It provides the first proof of Payne-Schaefer's conjecture for sixth-order problems and develops new integral identities for overdetermined boundary problems.

Findings

Proof of Payne-Schaefer conjecture in 2D

Integral identity for fourth-order problems

Symmetry results for second-order problems

Abstract

This paper considers overdetermined boundary problems. Firstly, we give a proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two dimensional case and under an additional condition for the case of dimension no less than three. Secondly, we prove an integral identity for an overdetermined problem of fourth order which can be used to deduce Bennett's symmetry theorem. Finally, we prove a symmetry result for an overdetermined problem of second order by integral identities.