On the representation of electromagnetic fields in discontinuously filled closed waveguides by means of continuous potentials

A closed waveguide of a constant cross section $S$ with perfectly conducting walls is considered. It is assumed that its filling is described by function $\varepsilon$ and $\mu$ invariable along the waveguide axis and piecewise continuous over the waveguide cross section. The aim of the paper is to show that, in such a system, it is possible to make a change of variables that makes it possible to work only with continuous functions. Instead of discontinuous transverse components of the electromagnetic field ${\mathbf{E}}$, it is proposed to use potentials ${{u}_{e}}$ and ${{v}_{e}}$ related to the field as ${{{\mathbf{E}}}_{ \bot }}=\nabla {{u}_{e}}+\tfrac{1}{\varepsilon}\nabla{\kern 1pt}'{{v}_{e}}$ and, instead of discontinuous transverse components of the magnetic field ${\mathbf{H}}$, to use the potentials ${{u}_{h}}$ and ${{v}_{h}}$ related to the field as ${{{\mathbf{H}}}_{ \bot }}=\nabla {{v}_{h}}+\tfrac{1}{\mu }\nabla {\kern 1pt}'{{u}_{h}}$. It is proven that any field in the waveguide admits the representation in this form if the potentials ${{u}_{e}},{{u}_{h}}$ are elements of the Sobolev space $\mathop {W_{2}^{1}}\limits^0(S)$ and ${{v}_{e}},{{v}_{h}}$ are elements of the space $W_{2}^{1}(S)$.