Functions of Bounded Variation and Free Discontinuity Problems

By Luigi Ambrosio, D. Pallara, N. Fusco. 2000

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Citations in the corpus

Safety Verification of Deep Neural Networks (2016, methodology): “…following formula, which expresses the constraints of Deﬁnition 9: It is noted that s p and m p for p ∈ dims k ( η k ( α x , k )) are employed when expressing Adversarial examples shown in Figures 8, 9, and 10 were found using single-path search and automatic selection of regions and manipulations.”
An approximation result for special functions with bounded deformation (2004, methodology): “We assume that the corresponding properties for BV and SBV functions are known to the reader, we refer to [5] for a good monograph on the topic.”
Symmetrization and second-order Sobolev inequalities (2004, methodology): “An updated thorough treatment of this topic is contained in [ 3 ].”
Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise (2012, unknown): “It turns out (see [1]) that η ∈ ∂φ(u) iff there is z ∈ L∞(O;RN) such that η = −div z, |z|∞ ≤ 1, and ∫ O ηudξ = φ(u).”
Mean-field optimal control as Gamma-limit of finite agent controls (2018, background): “3 Convex functionals on measures We are concerned with the main properties of functionals defined on measures, for a detailed treatment of this subject, we refer to [4].”
Penalized triograms: total variation regularization for bivariate smoothing (2004, methodology): “However, eventually the approach based on the Vitali formula interpreted in the language of Schwartzian distribution prevailed; Ambrosio et al. (2000) have given a recent account of the theory developed in the context of geometric measure theory and variational calculus, tracing its origins back to…”
Epsilon‐stable quasi‐static brittle fracture evolution (2009, methodology): “We also use the usual notation, found in [ 2 ], for the jump set Su for a BV function u, the normal to the jump set, etc.”
Optimal control of continuity equations (2015, unknown): “The function θ ∗ ηε is known to be smooth [3].”
On the behaviour of the solutions to p-Laplacian equations as p goes to 1 (2008, background): “Indeed, the space BV (Ω) is the dual of a separable space which will be denoted by Γ(Ω); its elements can be written as f − div F , with (f, F ) ∈ C0(Ω;R ) (see [14], and also [20] and [3]).”
Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View (2007, background): “28) where H0 is the multiplicity function, see [1] for the formula and the precise definition of H0.”
A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up (2018, background): “For a modern exposition of this vast subject and a detailed list of references, see [2, 8, 11].”
The Dirichlet Problem for the Total Variation Flow (2001, background): “in [1]), and the Dunford integral of the de nition of (t) is 28”