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The inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.
The problem of interpolating different Sobolev spaces has been solved in full generality by Haïm Brezis and Petru Mironescu in two works dated 2018 and 2019. [ 10 ] [ 11 ] Furthermore, their results do not depend on the dimension n {\displaystyle n} and allow real values of j {\displaystyle j} and m {\displaystyle m} , rather than integer.
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations , including the theory of harmonic maps .
For example, the approach based on "upper gradients" leads to Newtonian-Sobolev space of functions. Thus, it makes sense to say that a space "supports a Poincare inequality". It turns out that whether a space supports any Poincare inequality and if so, the critical exponent for which it does, is tied closely to the geometry of the space.
with Sobolev-Slobodeckij spaces , for non-integer > defined on through transformation to the planar case , (′) for ′, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator T m {\textstyle T_{m}} extends the classical normal traces in the sense that
In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a
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.
In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems , is to reduce the dimension of the system to that of the slow manifold— center manifold theory rigorously justifies the modelling.