Inductive Eigenmodes of a resistive toroidal surface in vacuum

In this paper it has been studied the Electro-Magnetic (EM) Eigenmodes, sufficiently slow as to legitimate the pre-Maxwell approximation of Maxwell's system (or inductive Eigenmodes), of a given smooth, toroidal-unknotted, electrically resistive surface Τ with given smooth (surface) resistivity 0 < ρdegree < ∞, and lying in the (empty) R3. Within the above limitations (to be made more precise), the geometry of Τ is arbitrary. With the eigenvalue associated with the generic Eigenmode being defined as the opposite of its logarithmic time-derivative, one expects that the resulting spectrum be discrete and strictly positive. It shall be interested into the degenerate case where Τ be cut (i.e. electrically broken) along one or more of its irreducible cycles. This case will be analyzed autonomously, rather than as a limit (for ρdegree → ∞ along the cuts) of the regular case. Without cuts, the Eigenproblem under consideration is nothing but the two-dimensional (2-dim) generalization of the classical case of a smooth, unknotted, electrically conductive, simple coil in infinite vacuum. Its analysis hinges on the classical potential theory, and turns out to be a special application of the linear, integrodifferential (elliptic) equation theory on a compact, multiply connected, 2-dim manifold. The attention and approach will be confined to strong (or classical) solutions, both in Τ and C Τ = R3 / Τ. This study is divided in two parts: a General Part (Sects 1 divided 4) is devoted to the case of generic Τ and ρdegree (within the convenient smoothness requirements), whereas a Special Part (Sects 5 divided 7) deals with the (more or less formal) discussion of a couple of particular cases (Τ ≡ a canonical torus), both of which with uniform ρdegree. Some propaedeutical/supplementary information is provided in a number of Appendices