Advances in Differential Equations

Closed formulas for the polyharmonic operator under spherical symmetry

E. Jannelli

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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.

Article information

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

Dates
First available in Project Euclid: 30 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1427744017

Mathematical Reviews number (MathSciNet)
MR3327708

Zentralblatt MATH identifier
1320.31013

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

Citation

Jannelli, E. Closed formulas for the polyharmonic operator under spherical symmetry. Adv. Differential Equations 20 (2015), no. 5/6, 581--600. https://projecteuclid.org/euclid.ade/1427744017.


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