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Meyers–Serrin Theorem
In functional analysis the Meyers–Serrin theorem, named after James Serrin and Norman George Meyers, states that smooth functions are dense in the Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ... W^(\Omega) for arbitrary domains \Omega \subseteq \R^n. Historical relevance Originally there were two spaces: W^(\Omega) defined as the set of all functions which have weak derivatives of order up to k all of which are in L^p and H^(\Omega) defined as the closure of the smooth functions with respect to the corresponding Sobolev norm (obtained by summing over the L^p norms of the functions and all derivatives). The theorem establishes the equivalence W^(\Omega)=H^(\Omega) of both definitions. It is quite surprising that, in contradistinction to many other density theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Serrin
James Burton Serrin (1 November 1926, Chicago, Illinois – 23 August 2012, Minneapolis, Minnesota) was an American mathematician, and a professor at University of Minnesota. Life He received his doctorate from Indiana University in 1951 under the supervision of David Gilbarg. From 1954 till 1995 he was on the faculty of the University of Minnesota. Work He is known for his contributions to continuum mechanics, nonlinear analysis, and partial differential equations. Awards and honors He was elected a member of the National Academy of Sciences in 1980. Selected works *. *. *. References External Links Donald G. Aronson and Hans F. Weinberger, "James B. Serrin", Biographical Memoirs of the National Academy of Sciences (2016) See also *American mathematicians American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman George Meyers
Norman or Normans may refer to: Ethnic and cultural identity * The Normans, a people partly descended from Norse Vikings who settled in the territory of Normandy in France in the 10th and 11th centuries ** People or things connected with the Norman conquest of southern Italy in the 11th and 12th centuries ** Norman dynasty, a series of monarchs in England and Normandy ** Norman architecture, romanesque architecture in England and elsewhere ** Norman language, spoken in Normandy ** People or things connected with the French region of Normandy Arts and entertainment * ''Norman'' (film), a 2010 drama film * '' Norman: The Moderate Rise and Tragic Fall of a New York Fixer'', a 2016 film * ''Norman'' (TV series), a 1970 British sitcom starring Norman Wisdom * ''The Normans'' (TV series), a documentary * "Norman" (song), a 1962 song written by John D. Loudermilk and recorded by Sue Thompson * "Norman (He's a Rebel)", a song by Mo-dettes from '' The Story So Far'', 1980 Busines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Functions
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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