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The vector Laplace operator, also denoted by , is a differential operator defined over a vector field. [7] The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field , returning a vector quantity.
The following table gives an overview of Green's functions of frequently appearing differential operators, where = + +, = +, is the Heaviside step function, () is a Bessel function, () is a modified Bessel function of the first kind, and () is a modified Bessel function of the second kind. [2]
In the case of a boundary put at infinity with the boundary condition setting the solution to zero at infinity, then one has an infinite-extent Green's function. For the three-variable Laplace operator, one can for instance expand it in the rotationally invariant coordinate systems which allow separation of variables.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
On functions, the Laplace–de Rham operator is actually the negative of the Laplace–Beltrami operator, as the conventional normalization of the codifferential assures that the Laplace–de Rham operator is (formally) positive definite, whereas the Laplace–Beltrami operator is typically negative. The sign is merely a convention, and both ...
In mathematics, the infinity Laplace (or -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated .It is alternately defined, for a function : of the variables = (, …,), by
The weak solution of the p-Laplace equation with Dirichlet boundary conditions {= =in an open bounded set is the minimizer of the energy functional = | |among all functions in the Sobolev space, satisfying the boundary conditions in the sense that , (when has a smooth boundary, this is equivalent to require that functions coincide with the boundary datum in trace sense [1]).