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May/June 2015 Closed formulas for the polyharmonic operator under spherical symmetry
E. Jannelli
Adv. Differential Equations 20(5/6): 581-600 (May/June 2015).

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.

Citation

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E. Jannelli. "Closed formulas for the polyharmonic operator under spherical symmetry." Adv. Differential Equations 20 (5/6) 581 - 600, May/June 2015.

Information

Published: May/June 2015
First available in Project Euclid: 30 March 2015

zbMATH: 1320.31013
MathSciNet: MR3327708

Subjects:
Primary: 31B30, 33F10

Rights: Copyright © 2015 Khayyam Publishing, Inc.

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Vol.20 • No. 5/6 • May/June 2015
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