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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.
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the ...
The left-hand side of this equation is the Laplace operator, and the entire equation Δu = 0 is known as Laplace's equation. Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.
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]).
The problem has a solution; The solution is unique; The solution's behavior changes continuously with the initial conditions; Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are ...
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation : ∇ 2 f = − k 2 f , {\displaystyle \nabla ^{2}f=-k^{2}f,} where ∇ 2 is the Laplace operator, k 2 is the eigenvalue, and f is the (eigen)function.
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