Category: Harmonic functions

Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Kellogg's theorem
Kellogg's theorem is a pair of related results in the mathematical study of the regularity of harmonic functions on sufficiently smooth domains by Oliver Dimon Kellogg. In the first version, it states
Poisson kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit d
Weakly harmonic function
In mathematics, a function is weakly harmonic in a domain if for all with compact support in and continuous second derivatives, where Δ is the Laplacian. This is the same notion as a weak derivative,
Bôcher's theorem
In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher.
Schwarz reflection principle
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic
Harnack's principle
In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic f
Positive harmonic function
In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Ries
Edmund Schuster
Edmund Schuster (7 September 1851 – 5 July 1932) was a German engineer and mathematician who contributed to the field of special functions and complex analysis being a pioneer in the field of harmonic
Laplace's equation
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
Harmonic map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equa
Maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of el
Kelvin transform
The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This techniq
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, togeth
Schwarz alternating method
In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping region
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). Th
Differential forms on a Riemann surface
In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal stru
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's e
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (wh
Pluriharmonic function
In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of severa
Harnack's inequality
In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack. Harnack's inequality is used to prove Harnack's theor
Harmonic morphism
In mathematics, a harmonic morphism is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphism
Harmonic coordinates
In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in
Newtonian potential
In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough
Harmonic conjugate
In mathematics, a real-valued function defined on a connected open set is said to have a conjugate (function) if and only if they are respectively the real and imaginary parts of a holomorphic functio
Weyl's lemma (Laplace equation)
In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak s