In the mathematical field of

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...

, Boggio's formula is an explicit formula for the Green's function for the polyharmonic Dirichlet problem on the ball of radius 1. It was discovered by the Italian mathematician Tommaso Boggio. The polyharmonic problem is to find a function ''u'' satisfying :

(-\Delta)^m u(x) = f(x)

where ''m'' is a positive integer, and

(-\Delta)

represents the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...

. The Green's function is a function satisfying :

(-\Delta)^m G(x,y) = \delta(x-y)

where

\delta

represents the Dirac delta distribution, and in addition is equal to 0 up to order ''m-1'' at the boundary. Boggio found that the Green's function on the ball in ''n'' spatial dimensions is :

G_ (x,y) = C_ , x-y, ^ \int_1^ (v^2-1)^ v^ dv

The constant

C_

is given by :

C_ =\frac,

where

e_n = \frac

Sources

* * {{Citation , last1=Gazzola , first1=Filippo , author1-link= , last2=Grunau , first2=Hans-Christoph , author2-link= , last3=Sweers , first3=Guido , title=Polyharmonic Boundary Value Problems , series=Lecture Notes in Mathematics , volume=1991 , edition= , publisher=Springer , date=2010 , place=Berlin , url=http://www1.mate.polimi.it/~gazzola/book_GGS.pdf , isbn=978-3-642-12244-6 Elliptic partial differential equations Potential theory