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 ${ℝ}^n$, a ${ℝ}^n$-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.