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Olivier Kneuss

Publications and source records attributed to Olivier Kneuss.

4 recordsLinked to original sources
arXiv2017

On the boundary conditions in estimating $\nabla ω$ by div $ω$ and curl $ω.$

In this paper we study under what boundary conditions the inequality $$\|\nablaω\|_{L^2(Ω)}^2\leq C\left(\|{\rm curl}ω\|_{L^2(Ω)}^2+ \|{\rm div}ω\|_{L^2(Ω)}^2+\|ω\|_{L^2(Ω)}^2\right) $$ holds true. It is known that such an estimate holds if either the tangential or normal component of $ω$ vanishes on the boundary $\partialω.$ We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions we give an interpolation result between these two classical boundary conditions.

math.AP↗
arXiv2016

Flows generated by divergence free vector fields with compact support

We are concerned with the theory of existence and uniqueness of flows generated by divergence free vector fields with compact support. Hence, assuming that the velocity vector fields are measurable, bounded, and the flows in the Euclidean space are measure preserving, we show two counterexamples of uniqueness/existence for such flows. First we consider the autonomous case in dimension 3, and then, the non autonomous one in dimension 2.

math.AP↗
arXiv2016

Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with $L^p$ Data

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\in L^p$ with $p>1$, i.e. bi-Sobolev solutions for the prescribed Jacobian inequality in the plane for right-hand sides $f\in L^p$. More precisely, for any $1<q<(p+1)/2$ we construct a solution which belongs to $W^{1,q}$ and which preserves the boundary pointwise. For bounded right-hand sides $f\in L^{\infty}$, we provide bi-Lipschitz solutions. The basic building block of our construction are Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets.

math.AP↗