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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] |
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] |
Haïm Brezis
Haïm Brezis (born 1 June 1944) is a French mathematician, who mainly works in functional analysis and partial differential equations. Biography Born in Riom-ès-Montagnes, Cantal, France. Brezis is the son of a Romanian immigrant father, who came to France in the 1930s, and a Jewish mother who fled from the Netherlands. His wife, Michal Govrin, a native Israeli, works as a novelist, poet, and theater director. Brezis received his Ph.D. from the University of Paris in 1972 under the supervision of Gustave Choquet. He is currently a professor at the Pierre and Marie Curie University and a visiting distinguished professor at Rutgers University. He is a member of the Academia Europaea (1988) and a foreign associate of the United States National Academy of Sciences (2003). In 2012 he became a fellow of the American Mathematical Society. He holds honorary doctorates from several universities including National Technical University of Athens. Brezis is listed as an ISI highly cited res ... [...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] |
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] |
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] |
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