## Section: New Results

### The Hardy-Hodge decomposition

Participants : Laurent Baratchart, Masimba Nemaire.

In a joint work with T. Qian and P. Dang from the university of Macao,
we proved in previous years that on a compact hypersurface $\Sigma $
embedded in
${\mathbb{R}}^{n}$, a ${\mathbb{R}}^{n}$-valued vector field of ${L}^{p}$ class decomposes as the sum
of a harmonic gradient from inside $\Sigma $, a harmonic gradient from outside
$\Sigma $, and a
tangent divergence-free field, provided that $2-\epsilon <p<2+{\epsilon}^{\text{'}}$,
where $\epsilon $
and ${\epsilon}^{\text{'}}$ depend on the Lipschitz constant of the surface. We also
proved that the decomposition is valid for $1<p<\infty $ when $\Sigma $ is
$VMO$-smooth (*i.e.* $\Sigma $ is locally the graph of Lipschitz
function with derivatives in $VMO$). By projection onto the tangent space, this
gives a Helmholtz-Hodge decomposition for vector fields
on a Lipschitz hypersurface, which is apparently new since existing results deal with smooth surfaces. In fact, the Helmholtz-Hodge
decomposition holds on Lipschitz surfaces (not just hypersurfaces),
The Hardy-Hodge decomposition generalizes the classical
Plemelj formulas from complex analysis.
We pursued this year the writing of an article on this topic, and we also
found that this decomposition yields a description of silent
magnetizations distributions of ${L}^{p}$-class on a surface.
A natural endeavor is now to use this description, *via* balayage, to describe volumetric silent magnetizations.