Pierre Bousquet

Hardy-Sobolev inequalities for vector fields and canceling linear differential operators

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.

Strong density for higher order Sobolev spaces into compact manifolds

Given a compact manifold \(N^n\), an integer \(k \in \mathbb{N}_*\) and an exponent \(1 \le p < \infty\), we prove that the class \(C^\infty(\overline{Q}^m; N^n)\) of smooth maps on the cube with values into \(N^n\) is dense with respect to the strong topology in the Sobolev space \(W^{k, p}(Q^m; N^n)\) when the homotopy group \(\pi_{\lfloor kp \rfloor}(N^n)\) of order \(\lfloor kp \rfloor\) is trivial. We also prove the density of maps that are smooth except for a set of dimension \(m - \lfloor kp \rfloor - 1\), without any restriction on the homotopy group of \(N^n\).

Lipschitz regularity for local minimizers of some widely degenerate problems

We consider local minimizers of the functional \[ \sum_{i=1}^N \int (|u_{x_i}|-\delta_i)^p_+\, dx+\int f\, u\, dx, \] where

Strong approximation of fractional Sobolev maps

\delta_1,\dots,\delta_N\ge 0

Density of bounded maps in Sobolev spaces into complete manifolds

and

Weak approximation by bounded Sobolev maps with values into complete manifolds

(\,\cdot\,)_+

C1 regularity of orthotropic p-harmonic functions in the plane

stands for the positive part. Under suitable assumptions on

Density of smooth maps for fractional Sobolev spaces

f

W^{s, p}

, we prove that local minimizers are Lipschitz continuous functions if

into

N=2