Let \(\Omega\subseteq \mathbb{R}^d\) be a bounded domain. A hypersurface \(\Sigma\subseteq \Omega\) is said to be of class \(C^2\) with constants \(r_0>0\) and \(K>0\) if the following conditions hold:
- The complement \(\Omega\setminus \Sigma\) has two connected components, denoted \(\Omega_+\) and \(\Omega_-\), and \(\Sigma\) is their common \(C^2\)-boundary relative to \(\Omega\) .
- For \(x\in \Sigma\), after a rigid transformation we may assume \(x=0\). In a neighbourhood \[ U = B_{r_0}(0) \times \Big[-\tfrac{1}{2} r_0 K_0, \tfrac{1}{2} r_0 K_0\Big] \cap \Omega, \] the coordinate \(y_d\) satisfies \[ y_d > 0 \quad \text{for } x \in U \cap \Omega_+, \qquad y_d < 0 \quad \text{for } x \in U \cap \Omega_-. \]
- The corresponding function \(h\) in the local graph representation of \(U\cap \Sigma\) satisfies \[ h(0) = 0, \qquad \nabla h(0) = 0, \qquad \|h\|_{C^2(B_{r_0}(0))} \leq K_0. \]