Limiting Sobolev inequalities for vector fields and canceling linear differential operators

By Jean Van Schaftingen. 2011

133 citations recorded by the source corpus.

The estimate \[ \norm{D^{k-1}u}_{L^{n/(n-1)}} \le \norm{A(D)u}_{L^1} \] is shown to hold if and only if \(A(D)\) is elliptic and canceling. Here \(A(D)\) is a homogeneous linear differential operator \(A(D)\) of order \(k\) on \(\R^n\) from a vector space \(V\) to a vector space \(E\). The operator \(A(D)\) is defined to be canceling if \[ \bigcap_{\xi \in \R^n \setminus \{0\}} A(\xi)[V]=\{0\}. \] This result implies in particular the classical Gagliardo--Nirenberg-Sobolev inequality, the Korn--Sobolev inequality and Hodge--Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator \(L(D)\) of order \(k\) on \(\R^n\) from a vector space \(E\) to a vector space \(F\) is introduced. It is proved that \(L(D)\) is cocanceling if and only if for every \(f \in L^1(\R^n; E)\) such that \(L(D)f=0\), one has \(f \in \dot{W}^{-1, n/(n-1)}(\R^n; E)\). The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4a7c39055763b12a93783033aaad88de8c7a1d0e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Franz Gmeineder
  2. Bogdan Raiță
  3. Jean Van Schaftingen
  4. Vitaly Moroz
  5. Hongyu Ye
  6. Jiankang Xia
  7. Minbo Yang
  8. Daniele Cassani

In Computational Theory and Mathematics

  1. Franz Gmeineder
  2. Bogdan Raiță
  3. Jean Van Schaftingen
  4. Giuseppe Mingione
  5. Jiankang Xia

Citations in the corpus