Advances in Differential Equations

Closed formulas for the polyharmonic operator under spherical symmetry

E. Jannelli

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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.

Article information

Adv. Differential Equations, Volume 20, Number 5/6 (2015), 581-600.

First available in Project Euclid: 30 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31B30: Biharmonic and polyharmonic equations and functions 33F10: Symbolic computation (Gosper and Zeilberger algorithms, etc.) [See also 68W30]


Jannelli, E. Closed formulas for the polyharmonic operator under spherical symmetry. Adv. Differential Equations 20 (2015), no. 5/6, 581--600.

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