Dirichlet Problem
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. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: :Given a function ''f'' that has values everywhere on the boundary of a region in \mathbb^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u=f on the boundary? This requirement is called the Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proven using the maximum principle. History The Dirichlet problem goes back to George Green, who studied the problem on general domains with general boundary conditions in his ''Essay on the Application o ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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. Formal statement Dirichlet's principle states that, if the function u ( x ) is the solution to Poisson's equation :\Delta u + f = 0 on a domain \Omega of \mathbb^n with boundary condition :u=g on the boundary \partial\Omega, then ''u'' can be obtained as the minimizer of the Dirichlet energy :E (x)= \int_\Omega \left(\frac, \nabla v, ^2 - vf\right)\,\mathrmx amongst all twice differentiable functions v such that v=g on \partial\Omega (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet. History The name "Dirichlet's principle" is due to Bernhard Riemann, who applied it in the study of complex analytic functions. Riemann (and others such as Carl Friedrich Gauss and P ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Subharmonic Function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the ''boundary'' of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball. ''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharm ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Perron Method
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 "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble. The Dirichlet problem is to find a harmonic function in a domain, with boundary conditions given by a continuous function \varphi(x). The Perron solution is defined by taking the pointwise supremum over a family of functions S_\varphi, :u(x) = \sup_ v(x) where S_\varphi is the set of all subharmonic functions such that v(x) \leq \varphi(x) on the boundary of the domain. The Perron solution ''u(x)'' is always harmonic; however, the values it takes on the boundary may not be the same as the desired boundary values \varphi(x). A point ''y ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmon ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Poisson Integral Formula
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 disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to ''n''-dimensional problems. Two-dimensional Poisson kernels On the unit disc In the complex plane, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_^\infty r^e^ = \frac = \operatorname\left(\frac\right), \ \ \ 0 \le r < 1. This can be thought of in two ways: either as a function of ''r'' and ''θ'', or as a family of functions of ''θ'' indexed by ''r''. If is the open [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Unit Disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hölder Condition
In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that , f(x) - f(y) , \leq C\, x - y\, ^ for all and in the domain of . More generally, the condition can be formulated for functions between any two metric spaces. The number \alpha is called the ''exponent'' of the Hölder condition. A function on an interval satisfying the condition with is constant (see proof below). If , then the function satisfies a Lipschitz condition. For any , the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. If \alpha = 0, the function is simply bounded (any two values f takes are at most C apart). We have the following chain of inclusions for functions defined on a closed and bounded interval of the real line with : where . Hölder spaces Hölder spaces consisting of functions satisfying a Hölder conditio ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Fredholm Integral Equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. Equation of the first kind A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits. An inhomogeneous Fredholm equation of the first kind is written as and the problem is, given the continuous kernel function K and the function g, to find the function f. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely K(t,s)=K(t-s), and the limits of integration are ±∞, then the right hand side of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Direct Method In The Calculus Of Variations
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy. The method The calculus of variations deals with functionals J:V \to \bar, where V is some function space and \bar = \mathbb \cup \ . The main interest of the subject is to find ''minimizers'' for such functionals, that is, functions v \in V such that J(v) \leq J(u) for all u \in V . The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforeha ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |