On the equation DIV Y = f and applications to control of phases

By Jean Bourgain, Haim Brezis. 2002

182 citations recorded by the source corpus.

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Citations in the corpus

A note on div curl inequalities (2005, background): “(2) However, it should be observed that the proof of (1) is subtler than that of (2).”
The stokes equation in the Lp-setting: Well-posedness and regularity properties (2018, methodology): “For a detailed discussion of these questions, we refer here to the work of Bourgain and Brezis [39].”
New estimates for elliptic equations and Hodge type systems (2007, background): “As was pointed out in [3] this statement is equivalent via Hahn–Banach and duality to the estimate ∥∥∥∥ζ − | ∫ ζ∥∥∥∥ n/(n−1) ≤ C‖gradζ‖L1+W−1,n/(n−1) ∀ζ ∈ C (T).”
Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations (2014, background): “First, Bourgain and Brezis have observed that the construction of the solution u in Theorem 1.1—and a fortiori in the stronger theorem (Theorem 1.2)—cannot be linear [14, Proposition 2].”
New estimates for the Laplacian, the div-curl, and related Hodge systems (2004, background): “Theorem 3 resembles (and in fact implies) a result we established in [3] for the divergence equation.”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “The original result in [4] is not constructive, and it is even proved that the V ∈ L∞ s.”
Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws (2007, methodology): “We observe the recent important work by Bourgain-Brezis [BB02] and De Pauw-Pfeffer [DPP04a] for the following problem with different view of point: Find a continuous vector field to the divergence-measure equation: (3.”
Lifting, degree, and distributional Jacobian revisited (2005, background): “1) can be derived from [9, theorem 3] (announced in [7]; see also [5]), which asserts that, if g ∈ H 1/2(I ; S1), then we may write g = ei(φ1+φ2), with (0.”
The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains (2008, unknown): “Communications on Pure and Applied Mathematics, Wiley, 2008, 7 (6), pp.”
Embeddings for A-weakly differentiable functions on domains (2017, background): “More recently, it was proved by Bourgain and Brezis in [5, 7] that, for Poisson’s equation in R, n ≥ 2, ∆u = f, the surprising strong L(1)–estimate ‖Du‖L1∗ 6 c‖f‖L1 holds provided that f ∈ Cc is divergence–free.”