Tangential and normal traces for extension domains with non-Lipschitz boundaries

Abstract

We generalize the classical vector-valued tangential and normal trace theory on Lipschitz domains to the setting of non-Lipschitz $H^1$-extension domains, which includes domains with fractal boundaries such as the Koch snowflake. We define and study these operators based on the surjectivity of the trace operator for elements of $H^1(\Omega)$ and the Hilbert structure of the associated trace space. The normal trace operator is defined on $\boldsymbol{H}({\rm div},\Omega)$ in $\mathbb{R}^n$. A generalized Stokes formula allows us to introduce the tangential trace operator on $\boldsymbol{H}({\bf curl},\Omega)$ and $\boldsymbol{H}^1(\Omega)$ in two and three dimensions. Following the approach of Buffa, Costabel and Sheen (2002) for Lipschitz domains, we define two abstract tangential boundary spaces as images of these trace operators, establish their Hilbert structure, and construct an abstract rotation operator linking them, in place of the geometric rotation based on the normal vector. We also extend Costabel's (1991) coercive bilinear form approach for Maxwell's equations to non-Lipschitz domains, yielding a theory that supports the treatment of the Hodge-Dirac operator and Green's formulas and enables the solution of boundary-value problems for the $\operatorname{\textbf{curl}}\operatorname{\textbf{curl}}+\textbf{I}$ operator within the $H^1$-extension domains framework.