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Sobolev Conjugate
The Sobolev conjugate of ''p'' for 1\leq p p This is an important parameter in the Sobolev inequality, Sobolev inequalities. Motivation A question arises whether ''u'' from the Sobolev space W^(\R^n) belongs to L^q(\R^n) for some ''q'' > ''p''. More specifically, when does \, Du\, _ control \, u\, _? It is easy to check that the following inequality :\, u\, _\leq C(p,q)\, Du\, _ \qquad \qquad (*) can not be true for arbitrary ''q''. Consider u(x)\in C^\infty_c(\R^n), infinitely differentiable function with compact support. Introduce u_\lambda(x):=u(\lambda x). We have that: :\begin \, u_\lambda \, _^q &= \int_, u(\lambda x), ^qdx=\frac\int_, u(y), ^qdy=\lambda^\, u\, _^q \\ \, Du_\lambda\, _^p &= \int_, \lambda Du(\lambda x), ^pdx=\frac\int_, Du(y), ^pdy=\lambda^ \, Du \, _^p \end The inequality (*) for u_\lambda results in the following inequality for u :\, u\, _\leq \lambda^C(p,q)\, Du\, _ If 1-\frac+\frac \neq 0, then by letting \lambda going to zero or infinity we obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. Sobolev embedding theorem Let denote the Sobolev space consisting of all real-valued functions on whose first weak derivatives are functions in . Here is a non-negative integer and . The first part of the Sobolev embedding theorem states that if , and are two real numbers such that :\frac-\frac = \frac -\frac, then :W^(\mathbf^n)\subseteq W^(\mathbf^n) and the embedding is continuous. In the special case of and , Sobolev embedding gives :W^(\mathbf^n) \subseteq L^(\mathbf^n) where is the Sobolev conjugate of , given byp. (Note that 1/p^*p.) Thus, a ... [...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] |
Sergei Lvovich Sobolev
Prof Sergei Lvovich Sobolev (russian: Серге́й Льво́вич Со́болев) HFRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduced notions that are now fundamental for several areas of mathematics. Sobolev spaces can be defined by some growth conditions on the Fourier transform. They and their embedding theorems are an important subject in functional analysis. Generalized functions (later known as distributions) were first introduced by Sobolev in 1935 for weak solutions, and further developed by Laurent Schwartz. Sobolev abstracted the classical notion of differentiation, so expanding the range of application of the technique of Newton and Leibniz. The theory of distributions is considered now as the calculus of the modern epoch. Life He was born in St. Petersburg as the son of Lev Alexandrovich Sobolev, a lawyer, and his wife, Natalya Georgievna. His city was rena ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Index
In mathematics, two real numbers p, q>1 are called conjugate indices (or Hölder conjugates) if : \frac + \frac = 1. Formally, we also define q = \infty as conjugate to p=1 and vice versa References Additional references * * {{Latin phrases Lists of Latin phrases, V ca:Locució llatina#V da:Latinske ord og vendinger#V fr:Liste de locutions latines#V id:Daftar frasa Latin#V it:Locuzioni latine#V nl:Lijst van Latijns .... Conjugate indices are used in Hölder's inequality. If p, q>1 are conjugate indices, the spaces ''L''''p'' and ''L''''q'' are dual to each other (see ''L''''p'' space). See also * Beatty's theorem References * Antonevich, A. ''Linear Functional Equations'', Birkhäuser, 1999. . Functional analysis {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon (199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
