En 2004, J. Bourgain, H. Brezis et P. Mironescu have shown that if \(\Gamma \subset \mathbb{R}^n\) is a closed rectifiable curve with tangent vector \(t\) and \(\varphi : \mathbb{R}^n \to \mathbb{R}^n\) is a vector field, then \[ \int_{\Gamma} \varphi \cdot t \le \lvert \Gamma \rvert\, \lVert D \varphi \rVert_{L^n}. \] J. Bourgain and H. Brezis have generalized this to \[ \int_{\mathbb{R}^n} \varphi \cdot f \le \lVert f \rVert_{L^1} \, \lVert D \varphi \rVert_{L^n} \] whenever \(f\) is a divergence-free vector field. These inequalities are surprising since the quantity \(\lVert D \varphi \rVert_{L^n}\) does not control \(\lVert \varphi \rVert_{L^\infty} \). These inequalities have consequences in the theory of regularity of elliptic systems with \(L^1\) data.
In this domain,
- I have given elementary proofs of the circulation integral inequality and the inequality for divergence-free vector-fields,
- I have obtained inequalities when the divergence is replaced by a general higher-order operator,
- I have studied the relationship between functions satisfying this kind of estimates and the space of functions of bounded mean oscillation \(BMO\),
- with H. Brezis, I have studied the corresponding boundary estimates and posed some problems of optimal constants,
- with S. Chanillo, I have obtained corresponding inequalities on stratified homogeneous groups, including for example the Heisenberg group,
- with S. Chanillo and Po Lam Yung, we have given some applications to fluid dynamics and electromagnetism,
- with S. Chanillo and Po Lam Yung, we have proved similar estimates on a symmetric space of noncompact type.
Sagun Chanillo, Jean Van Schaftingen and Po-Lam Yung, Bourgain–Brezis estimates on symmetric spaces of non-compact type, J. Funct. Anal. 273 (2017), no. 4, 1504-1547.
doi:10.1016/j.jfa.2017.05.005 DIAL:186090 arXiv:1610.00503
Sagun Chanillo, Po-Lam Yung and Jean Van Schaftingen, The incompressible Navier Stokes flow in two dimensions with prescribed vorticity, in Sagun Chanillo, Bruno Franchi, Guozhen Lu, Carlos Perez and Eric T. Sawyer (eds.), Harmonic Analysis, Partial Differential Equations and Applications, Birkhäuser, Applied and Numerical Harmonic Analysis, 2017, 19–25.
doi:10.1007/978-3-319-52742-0_2 DIAL:184673
Armin Schikorra, Daniel Spector and Jean Van Schaftingen, An \(L^1\)–type estimate for Riesz potentials, Rev. Mat. Iberoam. 33 (2017), no. 1, 291–304.
doi:10.4171/rmi/937 DIAL:183697 CVGMT:2566 arXiv:1411.2318
Sagun Chanillo, Jean Van Schaftingen and Po-Lam Yung, Variations on a proof of a borderline Bourgain–Brezis Sobolev embedding theorem, Chinese Ann. Math. Ser. B 38 (2017), no. 1, 235–252.
doi:10.1007/s11401-016-1069-y SharedIt DIAL:183696 arXiv:1612.02888
Sagun Chanillo, Jean Van Schaftingen and Po-Lam Yung, Applications of Bourgain–Brezis inequalities to fluid mechanics and magnetism, C. R. Math. Acad. Sci. Paris 354 (2016), no. 1, 51–55.
doi:10.1016/j.crma.2015.10.005 DIAL:169283 arXiv:1509.01472
Jean Van Schaftingen, Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations, J. Fixed Point Theory Appl. 15 (2014), no. 2, 273–297.
doi:10.1007/s11784-014-0177-0 SharedIt DIAL:155931 arXiv:1311.6624
Pierre Bousquet and Jean Van Schaftingen, Hardy–Sobolev inequalities for vector fields and canceling linear differential operators, Indiana Univ. Math. J. 63 (2014), no. 5, 1419–1445.
doi:10.1512/iumj.2014.63.5395 DIAL:152137 arXiv:1305.4262
Jean Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 877–921.
doi:10.4171/JEMS/380 DIAL:131968 arXiv:1104.0192
Jean Van Schaftingen, Limiting fractional and Lorentz spaces estimates of differential forms, Proc. Amer. Math. Soc. 138 (2010), no. 1, 235–240.
doi:10.1090/S0002-9939-09-10005-9 pdf DIAL:34246 arXiv:0903.2182
Sagun Chanillo and Jean Van Schaftingen, Subelliptic Bourgain–Brezis estimates on groups, Math. Res. Lett. 16 (2009), no. 3, 487–501.
doi:10.4310/MRL.2009.v16.n3.a9 DIAL:35494 arXiv:0712.3730
Haïm Brezis and Jean Van Schaftingen, Circulation integrals and critical Sobolev spaces: problems of optimal constants, in Dorina Mitrea and Marius Mitrea (eds.), Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Amer. Math. Soc., Proc. Sympos. Pure Math., No. 79, 2008, 33–47.
Jean Van Schaftingen, Estimates for \(\mathrm{L}^1\) vector fields under higher-order differential conditions, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 867–882.
doi:10.4171/JEMS/133 MR:2443922 DIAL:36302 preprint
Haïm Brezis and Jean Van Schaftingen, Boundary estimates for elliptic systems with \(L^1\)–data, Calc. Var. Partial Differential Equations 30 (2007), no. 3, 369–388.
doi:10.1007/s00526-007-0094-9 SharedIt DIAL:37398 preprint
Jean Van Schaftingen, Function spaces between BMO and critical Sobolev spaces, J. Funct. Anal. 236 (2006), no. 2, 490–516.
doi:10.1016/j.jfa.2006.03.011 MR:2240172 DIAL:38381 preprint
Jean Van Schaftingen, Estimates for \(L^1\) vector fields with a second order condition, Acad. Roy. Belg. Bull. Cl. Sci. (6) 15 (2004), no. 1–6, 103–112.
MR:2146098 DIAL:69507 preprint
Jean Van Schaftingen, Estimates for \(L^1\)–vector fields, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 181–186.
doi:10.1016/j.crma.2004.05.013 MR:20708071 Zbl 1049.35069 DIAL:40031 preprint
Jean Van Schaftingen, A simple proof of an inequality of Bourgain, Brezis and Mironescu, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 23–26.
doi:10.1016/j.crma.2003.10.036 MR:2038078 DIAL:40341 preprint
