Laplace Operator / Laplacian

In mathematics and physics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function in Euclidean space, usually written as , or .

The name is in honor of the French mathematician Pierre-Simon Laplace (1749–1827). He first used an operator in mathematics in his study of celestial mechanics, which gave a constant multiple of the mass density when applied to a given gravitational potential.  The function of is called the harmonic function, now known as Laplace's equation, and represents the possible gravitational field in free space.

The Laplace operator appears in differential equations that describe many physical phenomena. For example, it is often used in mathematical models of wave equations, heat conduction equations, fluid mechanics, and the Helmholtz equation. In electrostatics, the applications of Laplace's equation and Poisson's equation can be found everywhere. In quantum mechanics, it represents the kinetic energy term in the Schrödinger equation.

The Laplace operator is the simplest elliptic operator and is at the heart of Hodge theory and a consequence of de Rham cohomology.In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob detection and edge detection.

References

【1】https://zh.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E7%AE%97%E5%AD%90