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Problem of solving a partial differential equation subject to prescribed boundary values
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region
Dirichlet_problem
Type of constraint on solutions to differential equations
solutions to such equations is known as the Dirichlet problem. In the sciences and engineering, a Dirichlet boundary condition may also be referred to
Dirichlet_boundary_condition
Type of problem involving ODEs or PDEs
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_problem
German mathematician (1805–1859)
boundary-value problems, heat diffusion, and hydrodynamics. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns
Stochastic processes and boundary value problems
Stochastic_processes_and_boundary_value_problems
Property of differential equations describing physical phenomena
Jacques Hadamard in 1902. Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified
Well-posed_problem
Second-order partial differential equation
\mathbf {R} ^{n}} , the most common boundary value problems for Laplace's equation are the Dirichlet problem, in which the boundary values of the unknown function
Laplace's_equation
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
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Summatory function of the divisor-counting function
can be proven using the Dirichlet hyperbola method, and was first established by Dirichlet in 1849. The Dirichlet divisor problem, precisely stated, is
Divisor_summatory_function
American mathematician
Harvey, F. Reese; Lawson, H. Blaine Jr. (2009). "Dirichlet duality and the nonlinear Dirichlet problem". Communications on Pure and Applied Mathematics
H._Blaine_Lawson
Concept in potential theory
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
Dirichlet's_principle
Mathematical theorem
be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from the theory of Sobolev spaces for
Riemann_mapping_theorem
Problem in hydrodynamics
In astrophysics, Dirichlet's ellipsoidal problem, named after Peter Gustav Lejeune Dirichlet, asks under what conditions there can exist an ellipsoidal
Dirichlet's ellipsoidal problem
Dirichlet's_ellipsoidal_problem
Iterative method in conformal mapping
in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided
Schwarz_alternating_method
Modes of vibration in mathematics
mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can
Dirichlet_eigenvalue
theory) Dirichlet eigenvalue Dirichlet's ellipsoidal problem Dirichlet eta function (number theory) Dirichlet form Dirichlet function (topology) Dirichlet hyperbola
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Coordinate system in two dimensions
\theta }}\right)f(e^{\rho +i\theta })=0} When one wants to solve the Dirichlet problem in a domain with rotational symmetry, the usual thing to do is to
Log-polar_coordinates
Fundamental solution to the heat equation, given boundary values
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
Heat_kernel
Generative topic model
In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can
Latent_Dirichlet_allocation
Numerical method for solving physical or engineering problems
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
Finite_element_method
divisor problem on bounding Δ k ( x ) = D k ( x ) − x P k ( log ( x ) ) {\textstyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))} Dirichlet's divisor problem: the
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
When are solutions in the calculus of variations analytic
analytic, as happens for Dirichlet's problem for the potential function . Hilbert stated his nineteenth problem as a regularity problem for a class of elliptic
Hilbert's_nineteenth_problem
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
Lebesgue_spine
Mathematical technique
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
Perron_method
Mathematical algorithm
{\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
Walk-on-spheres_method
Property of functions which is weaker than continuity
prototypical example is the Dirichlet problem for the Laplace operator, which can be formulated as a minimization problem of the energy, subject to boundary
Semi-continuity
French mathematician
His fields of interest are noncommutative geometry, ergodic theory, Dirichlet problem, non-commutative residue. Katz, Mikhail G.; Leichtnam, Eric (2013)
Éric_Leichtnam
Extends the Jordan curve theorem to characterize the inner and outer regions
a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels. (Such diffeomorphisms will be holomorphic
Schoenflies_problem
Nonlinear second-order partial differential equation of special kind
of x {\displaystyle x} and y {\displaystyle y} only. Consider the Dirichlet problem to find u {\displaystyle u} so that L [ u ] = 0 , on Ω {\displaystyle
Monge–Ampère_equation
If there are more items than boxes holding them, one box must contain at least two items
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
Pigeonhole_principle
Constructing a strictly convex compact surface with specified Gaussian curvature
difficult problems as the Calabi conjecture of 1954, and a problem of Hermann Minkowski in Euclidean spaces concerning the Dirichlet problem for the real
Minkowski_problem
American mathematician (born 1946)
elliptic partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration
Joel_Spruck
problem is a special kind of boundary value problem which can be thought of as the steady state of an evolution problem. For example, the Dirichlet problem
Elliptic boundary value problem
Elliptic_boundary_value_problem
Canadian-American mathematician (1925–2020)
studied the Dirichlet problem for Yamabe-type equations on Euclidean spaces, following part of Thierry Aubin's work on the Yamabe problem.[BN83] With
Louis_Nirenberg
harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary
Harmonic_measure
Family of stochastic processes
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes
Dirichlet_process
Singularities of holomorphic functions extend infinitely outward
Fichera in the paper (Fichera 1957), by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept
Hartogs's_extension_theorem
Mathematical measure of a function's variability
to solving the variational problem of finding a function u that satisfies the boundary conditions and has minimal Dirichlet energy. Such a solution is
Dirichlet_energy
Mathematical problem in spectral theory
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
Hearing_the_shape_of_a_drum
23 mathematical problems stated in 1900
Media. ISBN 978-3-540-41160-4. Serrin, James (1969-05-08). "The problem of Dirichlet for quasilinear elliptic differential equations with many independent
Hilbert's_problems
Manifold of functions in the calculus of variations
Nehari (1960, 1961). It is a differentiable manifold associated to the Dirichlet problem for the semilinear elliptic partial differential equation − △ u =
Nehari_manifold
Chinese-American mathematician (born 1949)
consequence, they were able to prove the general solvability of the Dirichlet problem for the Monge–Ampère equation, which at the time had been a major
Shing-Tung_Yau
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
Klein, the first edition incorporated Hilbert's treatment of the Dirichlet problem using Hilbert space techniques; Brouwer's contributions to topology;
Uniformization_theorem
Prussian mathematician (1832–1925)
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
Carl_Neumann
{\displaystyle \Gamma _{N}} portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied ( Γ D ∩ Γ N = ∅ {\displaystyle
Variational_multiscale_method
Neumann–Dirichlet method is a domain decomposition preconditioner which involves solving Neumann boundary value problem on one subdomain and Dirichlet boundary
Neumann–Dirichlet_method
Theorem on the number of primes in arithmetic sequences
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
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
analytic problems, and offers new approaches to prove results by means of probability. For example, one can apply Brownian motion to the Dirichlet problem at
Stochastic analysis on manifolds
Stochastic_analysis_on_manifolds
Collection of results for partial differential equations
continuity to prove the existence and regularity of solutions to the Dirichlet problem for elliptic PDEs. This result says that when the coefficients of
Schauder_estimates
fundamental solutions. For example, the Dirichlet problem of the heat equation on the half-line, i.e., the problem u 0 {\displaystyle u_{0}} and g 0 {\displaystyle
Fokas_method
Green's function for Laplacian
can be used to prove existence and uniqueness of solutions to the Dirichlet problem for the Poisson equation in suitably regular domains, and for suitably
Newtonian_potential
American mathematician (1943–2024)
corresponding boundary value problem for this flow, proving an analogous result to Eells and Sampson's for the Dirichlet condition and Neumann condition
Richard_S._Hamilton
Conjecture on zeros of the zeta function
this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation ( 1 − 2 2 s ) ζ ( s ) = η ( s ) = ∑
Riemann_hypothesis
French mathematician and physicist (1781–1840)
the works of Dirichlet and Hermann Schwarz, the Poisson kernel is now typically presented in the context of solving the Dirichlet problem for harmonic
Siméon_Denis_Poisson
Functions in mathematics
a Euclidean space of the same dimension. Balayage Biharmonic map Dirichlet problem Harmonic morphism Harmonic polynomial Heat equation Laplace equation
Harmonic_function
German mathematician (1880–1975)
differential equations, including the Perron method to solve the Dirichlet problem for elliptic partial differential equations. He wrote an encyclopedic
Oskar_Perron
Mathematical model in poker
Gorstein, Evan (24 July 2016). "Solving and Computing the Discrete Dirichlet Problem" (PDF). Retrieved 9 June 2021. Harrington, Dan; Robertie, Bill (2014)
Independent_Chip_Model
Italian neoclassical economist (1886–1965)
of necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables in the paper (Amoroso
Luigi_Amoroso
Analytic function in mathematics
+ 2 + 3 + 4 + ··· Arithmetic zeta function Apéry's constant Basel problem Dirichlet eta function Generalized Riemann hypothesis Lehmer pair Particular
Riemann_zeta_function
differential equation. The original conjecture predicts that for the Dirichlet problem on a bounded two-dimensional domain, the second eigenfunction has
Nodal_line_conjecture
Vector space of functions in mathematics
(\Omega ),} is compact. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an orthonormal basis of L 2 ( Ω
Sobolev_space
American mathematician
from Brown University with thesis Method for the Solution of the Dirichlet Problem for Certain Types of Domains. In the early 1940s, he worked as a physicist
Bernard_Epstein
Exploring properties of the integers with complex analysis
analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions
Analytic_number_theory
inverse boundary problem is the problem of finding the coefficient of a divergence form elliptic partial differential equation from its Dirichlet-to-Neumann
Poincaré–Steklov_operator
French mathematician (1875–1941)
Fourier series, Cantor-Riemann theory, the Poisson integral and the Dirichlet problem. In a 1910 paper, "Représentation trigonométrique approchée des fonctions
Henri_Lebesgue
Mathematics
boundary of the domain. It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution
Neumann_boundary_condition
Topics referred to by the same term
theory, the branch of mathematics concerned with probability Dirichlet problem, the problem of finding a function which solves a specified partial differential
Lyapunov_theorem
Concept in complex analysis
complicate) necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables is given. Cherry, W
Wirtinger_derivatives
Auxiliary functions used to probe equations, distributions, and weak formulations
the test space is often chosen to encode them. For the homogeneous Dirichlet problem for the Poisson equation, one commonly seeks u ∈ H 0 1 ( U ) {\displaystyle
Test_function
Type of calculus problem
In calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown
Initial_value_problem
a (more expensive) preconditioner consisting of the solution of a Dirichlet problem in each substructure is scalable with the number of substructures
FETI
Mathematical algorithms
isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized
Symmetrization_methods
Differential calculus on function spaces
important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle
Calculus_of_variations
American musician and mathematician
Competition and Davidson Fellowship with a mathematical project on the Dirichlet problem, whose applications include describing the flow of heat across a metal
Michael_Viscardi
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
four basic problems of classical potential theory are as follows: Interior Dirichlet problem: ∆u = 0 in Ω, u = f on ∂Ω Interior Neumann problem: ∆u = 0 in
Neumann–Poincaré_operator
Italian gay pornographic film actor
University of Rome focusing on the application of Morse theory to a Dirichlet problem traced back to Poisson equations. His doctoral advisors were Angela
Carlo_Masi
On tangency patterns of circles
discrete variant of Perron's method of constructing solutions to the Dirichlet problem. Yves Colin de Verdière proved the existence of the circle packing
Circle_packing_theorem
Concept in mathematics
also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory
Harmonic_map
Italian mathematician (1879–1943)
the main contributions of the two scientists to the Cauchy and the Dirichlet problem for holomorphic functions of several complex variables, as well as
Guido_Fubini
Formula in number theory
the coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series is essentially the
Class_number_formula
Class of problems for PDEs
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface
Cauchy_problem
Formal power series
Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require
Generating_function
Solution to a specific type of stochastic differential equation
more general. It is more suited to certain problems, for example in the solution of the Dirichlet problem. The characteristic operator A {\displaystyle
Itô_diffusion
formal terms, the degenerate metric supplies a Dirichlet boundary condition for the extension problem and, as it happens, the natural condition is for
Ambient_construction
Transformation defined on a grayscale image
walker algorithm is a segmentation algorithm solving the combinatorial Dirichlet problem, adapted to image segmentation by L. Grady in 2006. In 2011, C. Couprie
Watershed_(image_processing)
Meromorphic function on the complex plane
are, consequently, the Riemann hypothesis and its generalisations. A Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function
L-function
Boundary value problem Dirichlet problem, Dirichlet boundary condition Neumann boundary condition Stefan problem Wiener–Hopf problem Separation of variables
List of partial differential equation topics
List_of_partial_differential_equation_topics
Concept in number theory
In mathematics, the Dirichlet density (or analytic density) of a set of primes, named after Peter Gustav Lejeune Dirichlet, is a measure of the size of
Dirichlet_density
or A 22 − 1 {\displaystyle A_{22}^{-1}} involves solving decoupled Dirichlet problems on each domain, and these can be done in parallel. Consequently, we
Schur_complement_method
Distributions in probability theory
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite
Dirichlet-multinomial distribution
Dirichlet-multinomial_distribution
on subdomain do not mesh Neumann–Dirichlet method — combines Neumann problem on one subdomain with Dirichlet problem on other subdomain Neumann–Neumann
List of numerical analysis topics
List_of_numerical_analysis_topics
Italian mathematician (1879–1961)
"Risoluzione del problema generale di Dirichlet per le funzioni biarmoniche" [Solution of the general Dirichlet problem for biharmonic functions], Rendiconti
Francesco_Severi
ISBN 978-3-642-01673-8. Caffarelli, L.; Nirenberg, L.; Spruck, J. (1985), "The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the
Hessian_equation
Type of differential operator
order 2k with coefficients having 2k continuous derivatives. The Dirichlet problem for L is to find a function u, given a function f and some appropriate
Elliptic_operator
Hong Kong mathematician
problems in differential geometry in the large. Comm. Pure Appl. Math. 6 (1953), 337–394. L. Caffarelli, L. Nirenberg, and J. Spruck. The Dirichlet problem
Shiu-Yuen_Cheng
Russian mathematician (1857–1918)
value problem of the equation of Laplace. In the theory of potential, his work from 1897 On some questions connected with Dirichlet's problem clarified
Aleksandr_Lyapunov
American mathematician (born 1950)
a simple corollary of Richard Hamilton's resolution of the Dirichlet boundary-value problem. As a consequence they found some striking geometric results
Richard_Schoen
Integral of sin(x)/x from 0 to infinity
are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral
Dirichlet_integral
Function in analytic number theory
in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number
Dirichlet_eta_function
American mathematician (1926–2003)
This perspective has enabled the application of harmonic maps to many problems in geometry and topology. Eells and Sampson's work is one of the most prominent
Joseph_H._Sampson
DIRICHLET PROBLEM
DIRICHLET PROBLEM
Boy/Male
Hindu, Indian
Problem
Girl/Female
Indian, Telugu
Destroyer of Problems
Boy/Male
Muslim
Problem solver
Girl/Female
Bengali, Indian
Eternity; Problem Solver
Girl/Female
Muslim/Islamic
Away from all Problems
Boy/Male
Indian, Tamil
People with this Name are Preferably Intelligent and Very Generous; Highly Knowledgeable in Problem Solving Skills
Boy/Male
Arabic, Indian, Muslim
Problem Solver
DIRICHLET PROBLEM
DIRICHLET PROBLEM
Male
Russian
Variant spelling of Russian Lavrentiy, LAVRENTY means "of Laurentum."
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : nickname for a goldsmith or someone with golden hair, from Old French doré ‘golden’ (see Dore 3).
Girl/Female
Polish Czechoslovakian
A flower name.
Girl/Female
Arabic, Muslim
Beautiful; Sweet
Male
Welsh
Welsh form of Celtic Brennus, BRENIN means "king."
Boy/Male
Indian, Sanskrit
First Root of Creation
Boy/Male
Indian, Sanskrit
Cloud
Boy/Male
Indian, Tamil
Victorious
Girl/Female
Indian
Name of a sahabiyyah, Pure, Clear
Girl/Female
Hindu, Indian, Tamil, Traditional
One with Eyes Like Deer; Goddess Laxmi
DIRICHLET PROBLEM
DIRICHLET PROBLEM
DIRICHLET PROBLEM
DIRICHLET PROBLEM
DIRICHLET PROBLEM
v. t.
To explain; to resolve; to unfold; to clear up (what is obscure or difficult to be understood); to work out to a result or conclusion; as, to solve a doubt; to solve difficulties; to solve a problem.
n.
To cause to stick; to bring to a stand; to pose; to puzzle; as, to stick one with a hard problem.
a.
Having the nature of a problem; not shown in fact; questionable; uncertain; unsettled; doubtful.
n.
A problem to be solved, or an example to be wrought out.
n.
To begin to deal with; as, to tackle the problem.
n.
The act of solving, or the state of being solved; the disentanglement of any intricate problem or difficult question; explanation; clearing up; -- used especially in mathematics, either of the process of solving an equation or problem, or the result of the process.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
n.
The quality, condition, or degree of being soluble or solvable; as, the solubility of a salt; the solubility of a problem or intricate difficulty.
a.
Single; not complex; not infolded or entangled; uncombined; not compounded; not blended with something else; not complicated; as, a simple substance; a simple idea; a simple sound; a simple machine; a simple problem; simple tasks.
n.
A problem of more than usual difficulty added to another on an examination paper.
a.
Susceptible of being solved; as, a soluble algebraic problem; susceptible of being disentangled, unraveled, or explained; as, the mystery is perhaps soluble.
a.
Alt. of Problematical
n.
An instrument of the ancients for finding two mean proportionals between two given lines, required in solving the problem of the duplication of the cube.
a.
Questionable; equivocal; indefinite; problematical.
n.
The quality or state of being solvable; as, the solvability of a difficulty; the solvability of a problem.
v. i.
To work, as at a puzzle; as, to puzzle over a problem.
a.
Liable to question; subject to be doubted or called in question; problematical; doubtful; suspicious.
v. t.
To have just and adequate ideas of; to apprehended the meaning or intention of; to have knowledge of; to comprehend; to know; as, to understand a problem in Euclid; to understand a proposition or a declaration; the court understands the advocate or his argument; to understand the sacred oracles; to understand a nod or a wink.
v. t.
To propose problems.
n.
One who proposes problems.