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We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L1–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems. Mathematics Subject Classification (2000) 35J25 (26D15, 35F05)

- Sagun Chanillo, Jean Van Schaftingen
- 2008

We show that divergence-free L1 vector fields on a nilpotent homogeneous group of homogeneous dimension Q are in the dual space of functions whose gradient is in LQ. This was previously obtained on Rn by Bourgain and Brezis.

We give an explicit sequence of polarizations such that for every measurable function, the sequence of iterated polarizations converge to the symmetric rearrangement of the initial function.

The function spaces Dk(R) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp = n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div–curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539–543 (see also J. Van… (More)

We obtain estimates in Besov, Triebel–Lizorkin and Lorentz spaces of differential forms on Rn in terms of their L1 norm.

- Sagun Chanillo, Jean Van Schaftingen
- 2016

Article history: Received 4 September 2015 Accepted after revision 8 October 2015 Available online 6 November 2015 Presented by Haïm Brézis As a consequence of inequalities due to Bourgain–Brézis, we obtain local-in-time wellposedness for the two-dimensional Navier–Stokes equation with velocity bounded in spacetime and initial vorticity in bounded… (More)

We study various questions related to the best constants in the following inequalities established in [1, 2, 3]; ̨̨̨Z Γ ~ φ · ~t ̨̨̨ ≤ Cn‖∇φ‖Ln |Γ| , and ̨̨̨Z Rn ~ φ · ~ μ ̨̨̨ ≤ Cn‖∇φ‖Ln‖~ μ‖ , where Γ is a closed curve in Rn, ~ φ ∈ C∞ c (Rn; Rn) and ~ μ is a bounded measure on Rn with values into Rn such that div ~ μ = 0. In 2d the answers are rather… (More)

We obtain estimates in Besov, Lizorkin-Triebel and Lorentz spaces of differential forms on Rn in terms of their L norm.

- Jean Van Schaftingen
- The American Mathematical Monthly
- 2013

- Sagun Chanillo, Jean Van Schaftingen
- 2017

M 〈f, φ〉dV ∣∣∣∣ ≤ C‖f‖L1(dV )‖∇φ‖Lm(dV ). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces. © 2017 Elsevier Inc. All rights reserved. * Corresponding author. E-mail addresses: chanillo@math.rutgers.edu (S. Chanillo), Jean.VanSchaftingen@uclouvain.be (J. Van Schaftingen), plyung@math.cuhk.edu.hk (P.-L.… (More)