Some asymptotic properties of solutions to triharmonic equations (Q6570652)

In this paper the author studies optimization problems for the ``triharmonic equation'', a sixth-order partial differential equation given by \N\[\N(\nabla^2)^3 U=0,\N\]\Nwhere functions \(Q\) (of the Lipschitz class \(\mathrm{Lip}\,1\)) satisfying \N\[\N|Q(x_1)-Q(x_2)|\leq |x_1-x_2|\N\]\Nhave to be minimized.\N\NThe main result is given in\N\NTheorem 1. For functions of the Lipschitz class \(\mathrm{Lip}\,1\), the following exact equality holds: \N\[\N\mathbb{E} (\mathrm{Lip}\,1; P_3(y))_C = \sup_{f\in \mathrm{Lip}\,1}\,|f(x)-P_3(x,y)_C|=\frac{4y}{3\pi},\N\]\Nwhere \N\[\NP_3(x,y)=\frac{8y^5}{3\pi}\,\int_{-\infty}^{+\infty}\,\frac{f(x+t)dt}{(t^2+y^2)^3}\N\]\Nis a triharmonic Poisson integral on the upper half-plane in Cartesian coordinates.\N\NThe layout of the paper is as follows:\N\begin{itemize}\N\item[1.] Problem statement (1/2 page)\N\N\item[2.] Some historical information (1 page)\N\N\item[3.] Approximation of functions by triharmonic integrals in the upper half-plane (3 pages)\N\NThis section contains the main theorem and its proof.\N\N\item[4.] References (43 items)\N\end{itemize}