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Colloquim
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The Biot-Savart operator and the Hodge decomposition ONO Soichiro 16th April (Fri)
In electromagnetics, the Biot-Savart law gives magnetic flux densities induced by electric current densities. We can indentify the law with an operator that maps given electric fields to induced magnetic fields, called the Biot-Savart operator. This operator finds various applications in knot theory, fluid dynamics, plasma physics and other many fields. Investigation of this operator's mathematical properties provides us deeper understaning of these applications. In order to investigate, we have an useful tool, the Hodge decomposition, which says that the space of vector fields is decomposed into some subspaces. In this colloquium, we will introduce some results shown by Canterella, DeTurck and Gluck. The Biot-Savart operator is the right inverse operator of "rot" on the space of the divergence-free vector fields that are tangential to the boundary of a given connected compact set. The kernel is identified with the space of the gradient vector fields, the potentials of which are constant on the boundary. Futhermore, we will extend the operator to a compact self-adjoint operator, with the completion of the vector fields' space under standard $L^2$ inner product. |
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