In potential theory (a branch of mathematics), the Laplacian of the indicator is obtained by letting the Laplace operator work on the indicator function of some domain D. It is a generalisation of the derivative (or "prime function") of the Dirac delta function to higher dimensions; it is non-zero only on the surface of D. It can be viewed as a surface delta prime function, the derivative of a surface delta function (a generalization of the Dirac delta). The Laplacian of the indicator is also analogous to the second derivative of the Heaviside step function in one dimension. The Laplacian of the indicator can be thought of as having infinitely positive and negative values when evaluated very near the boundary of the domain D. Therefore, it is not strictly a function but a generalized function or measure. Similarly to the derivative of the Dirac delta function in one dimension, the Laplacian of the indicator only makes sense as a mathematical object when it appears under an integral sign; i.e. it is a distribution function. Just as in the formulation of distribution theory, it is in practice regarded as a limit of a sequence of smooth functions; one may meaningfully take the Laplacian of a bump function, which is smooth by definition, and let the bump function approach the indicator in the limit.
History Paul Dirac introduced the Dirac δ-function, as it has become known, as early as 1930. The one-dimensional Dirac δ-function is non-zero only at a single point. Likewise, the multidimensional generalisation, as it is usually made, is non-zero only at a single point. In Cartesian coordinates, the d-dimensional Dirac δ-function is a product of d one-dimensional δ-functions; one for each Cartesian coordinate (see e.g. generalizations of the Dirac delta function).
Surface delta function A generalisation of the Dirac delta is possible beyond a single point. The point zero, in one dimension, can be considered as the boundary of the positive halfline. The function 1x>0 equals 1 on the positive halfline and zero otherwise, and is also known as the Heaviside step function. Formally, the Dirac δ-function and its derivative can be viewed as the first and second derivative of the Heaviside step function, i.e. ∂x1x>0 and ∂ x 2 1 x > 0 {\displaystyle \partial _{x}^{2}\mathbf {1} _{x>0}} . The analogue of the step function in higher dimensions is the indicator function, which can be written as 1x∈D, where D is some domain. The indicator function is also known as the characteristic function. In analogy with the one-dimensional case, the following higher-dimensional generalisations of the Dirac δ-function and its derivative have been proposed:
δ ( x ) → − n x ⋅ ∇ x 1 x ∈ D , δ ′ ( x ) → ∇ x 2 1 x ∈ D . {\displaystyle {\begin{aligned}\delta (x)&\to -n_{x}\cdot \nabla _{x}\mathbf {1} _{x\in D},\\\delta '(x)&\to \nabla _{x}^{2}\mathbf {1} _{x\in D}.\end{aligned}}}
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. This definition gives the usual one-dimensional case, when the domain is taken to be the positive halfline. 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.
Surface delta prime function The one-dimensional Dirac delta prime function is generalised to a multidimensional surface delta prime function on the boundary of some domain D in d ≥ 1 dimensions. In one dimension and by taking D equal to the positive halfline, the usual one-dimensional δ'-function can be recovered. Both the normal derivative of the indicator and the Laplacian of the indicator are supported by surfaces rather than points. The generalisation is useful in e.g. quantum mechanics, as surface interactions can lead to boundary conditions in d > 1, while point interactions cannot. Naturally, point and surface interactions coincide for d=1. Both surface and point interactions have a long history in quantum mechanics, and there exists a sizeable literature on so-called surface delta potentials or delta-sphere interactions. Surface delta functions use the one-dimensional Dirac δ-function, but as a function of the radial coordinate r, e.g. δ(r−R) where R is the radius of the sphere. 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. Alternatively, the indicator (and its derivatives) can be approximated using a bump function (and its derivatives). The limit, where the (smooth) bump function approaches the indicator function, must then be put outside of the integral.
Proofs
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