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Sobolev Spaces For Planar Domains
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the 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 smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems. Sobolev spaces with boundary conditions Let be a bounded domain with smooth boundary. Since is contained in a large square in , it can be regarded as a domain in by identifying opposite sides of the square. The theory of Sobolev spaces on can be found in , an account which is followed in several later textbooks such as and . For an integer, the (restricted) Sobolev space is defined as the closure of in the standard Sobolev space . * . *Vanishing properties on boundary: For the elements of are referred to as " functions on wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Mapping Theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy, biholomorphic mapping ''f'' (i.e. a bijective function, bijective holomorphic function, holomorphic mapping whose inverse is also holomorphic) from ''U'' onto the open unit disk :D = \. This mapping is known as a Riemann mapping. Intuitively, the condition that ''U'' be simply connected means that ''U'' does not contain any “holes”. The fact that ''f'' is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map ''f'' is essentially unique: if ''z''0 is an element of ''U'' and φ is an arbitrary angle, then there exists precis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz–Fischer Theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces L^p from Lebesgue integration theory are complete. Modern forms of the theorem The most common form of the theorem states that a measurable function on \pi, \pi/math> is square integrable if and only if the corresponding Fourier series converges in the Lp space L^2. This means that if the ''N''th partial sum of the Fourier series corresponding to a square-integrable function ''f'' is given by S_N f(x) = \sum_^ F_n \, \mathrm^, where F_n, the ''n''th Fourier coefficient, is given by F_n =\frac\int_^\pi f(x)\, \mathrm^\, \mathrmx, then \lim_ \left\Vert S_N f - f \right\, _2 = 0, where \, \,\cdot\,\, _2 is the L^2-norm. Conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Form
In potential theory (the study of harmonic function) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can be defined on any measure space, without the need for mentioning partial derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds, for example, fractals. The benefit on these spaces is that one can do this without needing a gradient operator, and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form. Definition When working on \mathbb^n , the "classical" Dirichlet form is given by: \mathcal(u, v ) = \int_ \nabla u(x) \cdot \nabla v(x) \; dx where one often discusses \mathcal(u) := \mathcal(u, u) = \, \nabla u\, _2^2 which is often referred to as the "energy" of the function u(x). More generally, a Dirichlet form is a Markovian closed symmetric form on an ''L''2-space.Fukushima, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Test
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
