|
Agmon's Inequality
In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,Lemma 13.2, in: Agmon, Shmuel, ''Lectures on Elliptic Boundary Value Problems'', AMS Chelsea Publishing, Providence, RI, 2010. . consist of two closely related interpolation inequalities between the Lebesgue space L^\infty and the Sobolev spaces H^s. It is useful in the study of partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...s. Let u\in H^2(\Omega)\cap H^1_0(\Omega) where \Omega\subset\mathbb^3. Then Agmon's inequalities in 3D state that there exists a constant C such that : \displaystyle \, u\, _\leq C \, u\, _^ \, u\, _^, and : \displaystyle \, u\, _\leq C \, u\, _^ \, u\, _^. In 2D, the first inequality still holds, but not the second: let u\in H^2(\Omega)\cap H^1_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shmuel Agmon
Shmuel Agmon ( he, שמואל אגמון; born 2 February 1922) is an Israeli mathematician. He is known for his work in analysis and partial differential equations. Biography Shmuel Agmon was born in Tel Aviv to writer Nathan Agmon and Chaya Gutman, and spent the first years of his life in Nazareth. A member of the HaMahanot HaOlim youth movement, Agmon studied at the Gymnasia Rehavia and joined a hakhshara program at Kibbutz Na'an after graduating from high school. He began his studies in mathematics at the Hebrew University of Jerusalem in 1940, but enlisted in the Jewish Brigade of the British Army before graduating. He served for four years in Cyprus, Italy and Belgium during World War II. After his discharge, he completed his undergraduate and master's degrees at the Hebrew University and went to France for further studies. He obtained a Ph.D. from Paris-Sorbonne University in 1949, under the supervision of Szolem Mandelbrojt. He returned to Jerusalem after working as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation Inequality
In the field of mathematical analysis, an interpolation inequality is an inequality of the form : \, u_ \, _ \leq C \, u_ \, _^ \, u_ \, _^ \dots \, u_ \, _^, \quad n \geq 2, where for 0\leq k \leq n, u_k is an element of some particular vector space X_k equipped with norm \, \cdot\, _k and \alpha_k is some real exponent, and C is some constant independent of u_0,..,u_n. The vector spaces concerned are usually function spaces, and many interpolation inequalities assume u_0 = u_1 = \cdots = u_n and so bound the norm of an element in one space with a combination norms in other spaces, such as Ladyzhenskaya's inequality and the Gagliardo-Nirenberg interpolation inequality, both given below. Nonetheless, some important interpolation inequalities involve distinct elements u_0,..,u_n, including Hölder's Inequality and Young's inequality for convolutions which are also presented below. Applications The main applications of interpolation inequalities lie in fields of study, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ... [...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] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, 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 Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ladyzhenskaya Inequality
In mathematics, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya. The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions (for smooth enough initial data). There is an analogous inequality for functions of three real variables, but the exponents are slightly different; much of the difficulty in establishing existence and uniqueness of solutions to the three-dimensional Navier–Stokes equations stems from these different exponents. Ladyzhenskaya's inequality is one member of a broad class of inequalities known as interpolation inequalities. Let \Omega be a Lipschitz domain in \mathbb R^ for n = 2 \text 3 and let u: \Omega \rightarrow \mathbb R be a weakly differentiable function that vanishes on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brezis–Gallouet Inequality
In mathematical analysis, the Brezis–Gallouët inequality, named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provides an explicit bound, which depends only logarithmically on the second derivatives. It is useful in the study of partial differential equations. Let \Omega\subset\mathbb^2 be the exterior or the interior of a bounded domain with regular boundary, or \mathbb^2 itself. Then the Brezis–Gallouët inequality states that there exists a real C only depending on \Omega such that, for all u\in H^2(\Omega) which is not a.e. equal to 0, :\displaystyle \, u\, _\leq C \, u\, _\left(1+\Bigl(\log\bigl( 1+\frac\bigr)\Bigr)^\right). Noticing that, for any v\in H^2(\mathbb^2), there holds :\int_ \bigl( (\partial^2_ v)^2 + 2(\partial^2_ v)^2 + (\partial^2_ v)^2\bigr) = \int_ \bigl(\partial^2_ v+\partial^2_ v\bigr)^2, one deduces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.PNG "Click for more on -> Theorems In Analysis")