Hodge-Maxwell systems arise from the time-harmonic Maxwell’s equation and can be viewed as a variable-coefficient version of the Hodge Laplacian. These systems feature prominently in a number of areas of mathematical physics and differential geometry.
In the first part of this talk, I shall give a brief overview of these systems and their connection to Gaffney type inequalities, Poincar´e lemmas, Hodge decomposition and Maxwell equations.
In the second part, I shall discuss some recent advances in the regularity theory for such systems with anisotropic coefficients. The crux of the matter is to circumvent the lack of representation formulas by point-wise inequalities for appropriate maximal functions, so that regularity estimates can be read off using standard results in harmonic analysis.
The results are joint works with my PhD student Rohit Mahato.
