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Hopf Maximum Principle
The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory. Generalizing the maximum principle for harmonic functions which was already known to Gauss in 1839, Eberhard Hopf proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of R''n'' and attains a maximum in the domain then the function is constant. The simple idea behind Hopf's proof, the comparison technique he introduced for this purpose, has led to an enormous range of important applications and generalizations. Mathematical formulation Let ''u'' = ''u''(''x''), ''x'' = (''x''1, …, ''x''''n'') be a ''C''2 function which satisfies the differential inequality : Lu = \sum_ a_(x)\frac + \sum_i b_i(x)\frac \geq 0 in an open domain (connected open subset of R''n'') Ω, where the symmetric matrix ''a''''ij'' = ''a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Sphere Property
Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior design, the trade of designing an architectural interior Places * Interior, South Dakota * Interior, Washington * Interior Township, Michigan * British Columbia Interior, commonly known as "The Interior" Government agencies * Interior ministry, sometimes called the ministry of home affairs * United States Department of the Interior Other uses * Interior (topology), mathematical concept that includes, for example, the inside of a shape * Interior FC, a football team in Gambia See also * * * List of geographic interiors * Interiors (other) * Inter (other) * Inside (other) Inside may refer to: * Insider, a member of any group of people of limited number and generally restricted access Film * ''In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monge–Ampère Equation
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of ''u'' and in the second-order partial derivatives of ''u''. The independent variables (''x'',''y'') vary over a given domain ''D'' of R2. The term also applies to analogous equations with ''n'' independent variables. The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1820. Important results in the theory of Monge–Ampère equations have been obtained by Sergei Bernstein, Aleksei Pogorelov, Charles Fefferman, and Louis Nir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvatur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Problem
In mathematics, a Dirichlet problem is the problem of finding 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 R''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 proved using the maximum principle. History The Dirichlet problem goes back to George Green, who studied the problem on general domains with general boundary condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Methoden Der Mathematischen Physik
''Methoden der mathematischen Physik'' (Methods of Mathematical Physics) is a 1924 book, in two volumes totalling around 1000 pages, published under the names of Richard Courant and David Hilbert. It was a comprehensive treatment of the "methods of mathematical physics" of the time. The second volume is devoted to the theory of partial differential equations. It contains presages of the finite element method, on which Courant would work subsequently, and which would eventually become basic to numerical analysis. The material of the book was worked up from the content of Hilbert's lectures. While Courant played the major editorial role, many at the University of Göttingen were involved in the writing-up, and in that sense it was a collective production. On its appearance in 1924 it apparently had little direct connection to the quantum theory questions at the centre of the theoretical physics of the time. That changed within two years, since the formulation of Schrödinger's eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, 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). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real analysis, mathematical physics, the calculus of variations and partial differential equations. He wrote textbooks widely used by generations of students of physics and mathematics. He is also known for founding the institute now bearing his name. Life and career Courant was born in Lublinitz, in the Prussian Province of Silesia. His parents were Siegmund Courant and Martha Courant ''née'' Freund of Oels. Edith Stein was Richard's cousin on the paternal side. During his youth his parents moved often, including to Glatz, then to Breslau and in 1905 to Berlin. He stayed in Breslau and entered the university there, then continued his studies at the University of Zürich and the University of Göttingen. He became David Hilbert's assist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word 'bounded' makes no sense in a general topological space without a corresponding metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem .... A bounded set is not necessarily a closed set and vise versa. For example, a subset ''S'' of a 2-dimensional real space R''2'' constrained by two parabolic curves ''x''2 + 1 and ''x''2 - 1 defined in a Cartesian coordinate system is a closed but is not b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Partial Differential Equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
