In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region...

solutions to such equations is known as the Dirichlet problem. In the sciences and engineering, a Dirichlet boundary condition may also be referred to...

studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle...

boundary-value problems, and heat diffusion, and hydrodynamics. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in...

the initial conditions. Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified...

most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns...

theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with...

proven using the Dirichlet hyperbola method, and was first established by Dirichlet in 1849.: 37–38, 69  The Dirichlet divisor problem, precisely stated...

theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. Dirichlet's principle...

be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for...

mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can...

In natural language processing, latent Dirichlet allocation (LDA) is a Bayesian network (and, therefore, a generative statistical model) for modeling...

in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided...

In astrophysics, Dirichlet's ellipsoidal problem, named after Peter Gustav Lejeune Dirichlet, asks under what conditions there can exist an ellipsoidal...

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...

consider the Dirichlet problem in a connected domain (or manifold with boundary) U. Let λn be the eigenvalues for the Dirichlet problem of the Laplacian...

problem on bounding Δ k ( x ) = D k ( x ) − x P k ( log ⁡ ( x ) ) {\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))} Dirichlet's divisor problem:...

{\displaystyle x} be a point inside Ω {\displaystyle \Omega } . Consider the Dirichlet problem: { Δ u ( x ) = 0 if  x ∈ Ω u ( x ) = h ( x ) if  x ∈ Γ . {\displaystyle...

domain D in the plane. Denote by λn the Dirichlet eigenvalues for D: that is, the eigenvalues of the Dirichlet problem for the Laplacian: { Δ u + λ u = 0 u...

with respect to x {\displaystyle x} . P2 is a two-dimensional problem (Dirichlet problem) P2  : { u x x ( x , y ) + u y y ( x , y ) = f ( x , y )  in ...

a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels. (Such diffeomorphisms will be holomorphic...

theory) Dirichlet eigenvalue Dirichlet's ellipsoidal problem Dirichlet eta function (number theory) Dirichlet form Dirichlet function (topology) Dirichlet hyperbola...

commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the...

Media. ISBN 978-3-540-41160-4. Serrin, James (1969-05-08). "The problem of Dirichlet for quasilinear elliptic differential equations with many independent...

to the new coordinates and Γ denotes its Christoffel symbols. The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain...

In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes...

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there...

solutions to the Dirichlet problem are unique, provided they exist.[citation needed] Monge–Ampère equations arise naturally in several problems in Riemannian...

thorn is a type of set used for discussing solutions to the Dirichlet problem and related problems of potential theory. The Lebesgue spine was introduced in...

to solve a more general Dirichlet problem by introducing his method of the arithmetic mean. Due to his work on the Dirichlet principle of potential theory...