Closed formulas for the polyharmonic operator under spherical symmetry

Abstract

One of the simplest formulas of the calculus of several variables says that, when applied to spherical symmetric functions $u=u(\rho)$ in $\mathbb R^n$, the Laplace operator takes the form $\Delta u=u''(\rho)+(n-1)u'(\rho)/\rho$. In this paper, we derive the analogous explicit expression for the polyharmonic operator $\Delta^k$ in the case of spherical symmetry. Moreover, if $B$ is a ball centered at the origin and $u\in H^k_0(B)$ is spherical symmetric, then, we deduce the functional \[ J[u]= \begin{cases} \displaystyle \frac{1}{2}\int_{\Omega} (\Delta^{k/2} u(x))^2\,dx&\text{if $k$ is even}\cr \displaystyle \frac{1}{2}\int_{\Omega} |\nabla \Delta^{(k-1)/2} u(x)|^2\,dx&\text{if $k$ is odd},\cr \end{cases} \] of which $(-\Delta)^k$ is gradient as a sum of integrals in one variable. The results are based upon nontrivial combinatorial identities, which are proved by means of Zeilberger's algorithm.