Publication

Two subtle convex nonlocal approximations of the BV-norm

Hoài-Minh Nguyên
2016
Journal Articles

Abstract

Inspired by the BBM formula and by work of G. Leoni and D. Spector, we analyze the asymptotic behavior of two sequences of convex nonlocal functionals (Psi(n)(u)) and (Phi(n)(u)) which converge formally to the BV-norm of u. We show that pointwise convergence when u is not smooth can be delicate; by contrast, Gamma-convergence to the BV-norm is a robust and very useful mode of convergence. (c) 2016 Elsevier Ltd. All rights reserved.

Official source

https://infoscience.epfl.ch/entities/publication/80f65b94-82b0-4832-981d-f4f4059bce6f

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Hoài-Minh Nguyên
2016
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/80f65b94-82b0-4832-981d-f4f4059bce6f

About this result

Related concepts (12)

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as if (and only if) The function is said to be the pointwise limit function of the Sometimes, authors use the term bounded pointwise convergence when there is a constant such that .

Compact convergence

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Let be a topological space and be a metric space. A sequence of functions is said to converge compactly as to some function if, for every compact set , uniformly on as . This means that for all compact , If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.

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