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Lions–Magenes Lemma
In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself. Statement of the lemma Let ''X''0, ''X'' and ''X''1 be three Hilbert spaces with ''X''0 ⊆ ''X'' ⊆ ''X''1. Suppose that ''X''0 is continuously embedded in ''X'' and that ''X'' is continuously embedded in ''X''1, and that ''X''1 is the dual space of ''X''0. Denote the norm on ''X'' by , , ⋅ , , ''X'', and denote the action of ''X''1 on ''X''0 by \langle\cdot,\cdot\rangle. Suppose for some T>0 that u \in L^2 (, T X_0) is such that its time derivative \dot \in L^2 (, T X_1). Then u is almost everywhere equal to a function continuous from ,T/math> into X, and moreover the following equality holds in the sense of scalar distributions on (0,T): :\frac\frac \, u\, _X^2 = \langle\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
