Classification of solutions to the $Q$-flat and constant $T$-curvature equation on the half-space and ball

Abstract

For conformal boundary operators associated with the Paneitz operator, we introduce a rigorous definition of the biharmonic Poisson kernel consisting of a pair of kernel functions and derive its explicit representation formula. With this powerful tool, we establish classification theorems of nonnegative solutions to the $Q$-flat and constant $T$-curvature equations on $\mathbb{R}_+^{n+1}$ and $\mathbb{B}^{n+1}$.