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In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R N with r representing a positive real radius and θ an element of the unit sphere S N−1, = + + where Δ S N−1 is the Laplace–Beltrami operator on the (N − 1)-sphere, known as the spherical Laplacian.
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.
The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator.
In Cartesian coordinates, the Laplacian of a function (,,) is = = = + +. The Laplacian is a measure of how much a function is changing over a small sphere centered at the point. When the Laplacian is equal to 0, the function is called a harmonic function .
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 cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: (,,) = | | (,) (,) where the () are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary ...
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]).
Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that is a (smooth) function , we can write the Laplace equation in the following form = (^) =,, where L 2 is the square of the nondimensional angular momentum operator, ^ = ().