Recent trends in harmonic and complex analysis

A limiting case for the divergence equation and related problems

4 avr. 2017, 10:45

45m

Salle de séminaire (Université d'Orléans, Mathématiques)

Orateur

M. Emmanuel Russ (Université Grenoble Alpes)

Description

Let $d\ge 2$, $\mathrm{\Omega }\subset R^d$ be a smooth bounded domain and $f\in L^d(\mathrm{\Omega })$ with ${\int }_{R}f(x)dx=0$. Bourgain and Brezis proved that there exists a vector field $X\in W^{1;d}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$ such that $divX=f$ and $||f|{|}_{W^{1;d}}+||f|{|}_{L^{\mathrm{\infty }}}\le C||f|{|}_{L^d}.$ We will discuss various extensions of this result to more general functions spaces, and present some related inequalities. This talk is based on results obtained in collaboration with P. Bousquet, P. Mironescu, Y. Wang and P. L. Yung.