Closed formulas for the polyharmonic operator under spherical symmetry.

One of the simplest formulas of the calculus of several variables says that, when applied to spherical symmetric functions u=u(ρ) in Rn, the Laplace operator takes the form Δu=u′′(ρ)+(n−1)u′(ρ)/ρ.In this paper, we derive the analogous explicit expression for the polyharmonic operator Δk in the case of spherical symmetry. Moreover, if B is a ball centered at the origin and u∈Hk0(B) is spherical symmetric, then, we deduce the functionalJ[u]=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩12∫Ω(Δk/2u(x))2dxif k is even12∫Ω|∇Δ(k−1)/2u(x)|2dxif k is odd, of which (−Δ)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.