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 embedded in , a -valued vector field of class decomposes as the sum of a harmonic gradient from inside , a harmonic gradient from outside , and a tangent divergence-free field, provided that , where and depend on the Lipschitz constant of the surface. We also proved that the decomposition is valid for when is -smooth (i.e. is locally the graph of Lipschitz function with derivatives in ). 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 -class on a surface. A natural endeavor is now to use this description, via balayage, to describe volumetric silent magnetizations.