| Abstract: |
| The mathematical analysis of boundary value problems for Maxwell's equations traditionally requires at least a Lipschitz-continuous boundary to properly define trace operators. Building upon the work of Creo, Lancia, Vernole, Hinz, and Teplyaev (2018) regarding fractal boundaries, we address the challenge of formulating electromagnetic problems on highly irregular geometries.
Our focus is on the large class of $H^1$-extension domains, where classical tools relying on surface measures and standard boundary regularity break down. To overcome this, we propose a boundary measure-free approach that relies entirely on the intrinsic Hilbert structure of the trace space. Within this framework, we construct the notion of normal and tangential trace operators for the spaces $\mathbf{H}(\operatorname{div}, \Omega)$ and $\mathbf{H}(\operatorname{\mathbf{curl}}, \Omega)$, providing a rigorous basis for solving electromagnetic boundary value problems in geometries where classical theory is not applicable. |
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