Laplacian of the indicator
In analogy with the one-dimensional case, the following higher-dimensional generalisations of the Dirac δ-function and its derivative have been proposed:[2] Here n is the outward normal vector.
Here the Dirac δ-function is generalised to a surface delta function on the boundary of some domain D in d ≥ 1 dimensions.
It is zero except on the boundary of the domain D (where it is infinite), and it integrates to the total surface area enclosing D, as shown below.
In one dimension and by taking D equal to the positive halfline, the usual one-dimensional δ'-function can be recovered.
Although seemingly ill-defined, derivatives of the indicator function can formally be defined using the theory of distributions or generalized functions: one can obtain a well-defined prescription by postulating that the Laplacian of the indicator, for example, is defined by two integrations by parts when it appears under an integral sign.
First, for a function f in the interval (a,b), recall the fundamental theorem of calculus assuming that f is locally integrable.
In this calculation, two integrations by parts (combined with the fundamental theorem of calculus as shown above) show that the first equality holds; the boundary terms are zero when a and b are finite, or when f vanishes at infinity.
Although derivatives of the indicator do not formally exist, following the usual rules of partial integration provides the 'correct' result.
When considering a finite d-dimensional domain D, the sum over outward normal derivatives is expected to become an integral, which can be confirmed as follows: where the limit is of x approaching surface β from inside domain D, nβ is the unit vector normal to surface β, and ∇x is now the multidimensional gradient operator.
As before, the first equality follows by two integrations by parts (in higher dimensions this proceeds by Green's second identity) where the boundary terms disappear as long as the domain D is finite or if f vanishes at infinity; e.g. both 1x∈D and ∇x1x∈D are zero when evaluated at the 'boundary' of Rd when the domain D is finite.
The third equality follows by the divergence theorem and shows, again, a sum (or, in this case, an integral) of outward normal derivatives over all boundary locations.
In electrostatics, a surface dipole (or Double layer potential) can be modelled by the limiting distribution of the Laplacian of the indicator.
The foregoing analysis shows that −nx ⋅ ∇x1x∈D can be regarded as the surface generalisation of the one-dimensional Dirac delta function.
Several options are possible, but it is convenient to let the bump function be non-negative and approach the indicator from below, i.e.
This ensures that the family of bump functions is identically zero outside of D. This is convenient, since it is possible that the function f is only defined in the interior of D. For f defined in D, we thus obtain the following: where the interior coordinate α approaches the boundary coordinate β from the interior of D, and where there is no requirement for f to exist outside of D. When f is defined on both sides of the boundary, and is furthermore differentiable across the boundary of D, then it is less crucial how the bump function approaches the indicator.
[2][11] A lot more attention has been focused on the one-dimensional Dirac delta prime potential recently.
