Team, Visitors, External Collaborators
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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 Σ embedded in n, a n-valued vector field of Lp class decomposes as the sum of a harmonic gradient from inside Σ, a harmonic gradient from outside Σ, and a tangent divergence-free field, provided that 2-ε<p<2+ε', where ε and ε' depend on the Lipschitz constant of the surface. We also proved that the decomposition is valid for 1<p< when Σ is VMO-smooth (i.e. Σ 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 Lp-class on a surface. A natural endeavor is now to use this description, via balayage, to describe volumetric silent magnetizations.