Laplacian of the indicator

In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can be viewed as the surface delta prime function. It is analogous to the second derivative of the Heaviside step function in one dimension. It can be obtained by letting the Laplace operator work on the indicator function of some domain D. 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. From a mathematical viewpoint, 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. 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). However, a different generalisation is possible. 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 (i.e. the one-dimensional surface delta prime function) can be viewed as the first and second derivative of the Heaviside step function, i.