我们在后续提到的窄带（narrow band）方法和迹方法（trace）的分析中，对解要求：
u ~ ∈ H 2 ( γ )
\widetilde{u} \in H^{2}(\gamma)
u ∈ H 2 ( γ ) ∥ u ∥ H 2 ( N ( δ ) ) ≲ δ 1 2 ∥ u ~ ∥ H 2 ( γ )
\|u\|_{H^{2}(\mathcal{N}(\delta))} \lesssim \delta^{\frac{1}{2}}\|\widetilde{u}\|_{H^{2}(\gamma)}
∥ u ∥ H 2 ( N ( δ ) ) ≲ δ 2 1 ∥ u ∥ H 2 ( γ )
事实上，我们定义自然扩张u ( x ) = u ~ ( x − d ( x ) ∇ d ( x ) ) ∀ x ∈ N ( δ )
u(\mathrm{x})=\tilde{u}(\mathrm{x}-d(\mathrm{x}) \nabla d(\mathrm{x})) \quad \forall \mathrm{x} \in \mathcal{N}(\delta)
u ( x ) = u ~ ( x − d ( x ) ∇ d ( x ) ) ∀ x ∈ N ( δ )
问题在于，这里我们需要P d \mathbf{P}_{d} P d C 2 C^2 C 2 γ \gamma γ C 3 C^3 C 3 C 2 C^2 C 2 C 2 C^2 C 2 H 2 H^2 H 2
取ε = c δ ≤ δ 2 \varepsilon=c \delta \leq \frac{\delta}{2} ε = c δ ≤ 2 δ N ( δ + 2 ε ) ⊂ N
\mathcal{N}(\delta+2 \varepsilon) \subset \mathcal{N}
N ( δ + 2 ε ) ⊂ N d ˉ ε ( x ) : = d ⋆ ρ ε ( x ) = ∫ B ε d ( x − y ) ρ ε ( y ) d y ∀ x ∈ N ( δ )
\bar{d}_{\varepsilon}(\mathrm{x}):=d \star \rho_{\varepsilon}(\mathrm{x})=\int_{B_{\varepsilon}} d(\mathrm{x}-\mathrm{y}) \rho_{\varepsilon}(\mathrm{y}) d \mathrm{y} \quad \forall \mathrm{x} \in \mathcal{N}(\delta)
d ˉ ε ( x ) : = d ⋆ ρ ε ( x ) = ∫ B ε d ( x − y ) ρ ε ( y ) d y ∀ x ∈ N ( δ ) d d d B ε : = B ( 0 , ε ) B_{\varepsilon}:=B(0, \varepsilon) B ε : = B ( 0 , ε ) ρ ε ( x ) \rho_{\varepsilon}(\mathbf{x}) ρ ε ( x ) γ E \gamma_{E} γ E γ ε : = { x ∈ N : d ˉ ε ( x ) = 0 }
\gamma_{\varepsilon}:=\left\{\mathrm{x} \in \mathcal{N}: \quad \bar{d}_{\varepsilon}(\mathrm{x})=0\right\}
γ ε : = { x ∈ N : d ˉ ε ( x ) = 0 } d ε d_{\varepsilon} d ε γ ε \gamma_\varepsilon γ ε
首先，我们先不加证明地引用几个结论。
d H ( X , Y ) = max { sup x ∈ X i n f y ϵ Y d ( x , y ) , sup y ∈ Y i n f x ϵ X ( x , y ) }
d_{H}(X, Y)=\max \left\{\sup _{x \in X} i n f_{y \epsilon Y} d(x, y), \sup _{y \in Y} i n f_{x \epsilon X}(x, y)\right\}
d H ( X , Y ) = max { x ∈ X sup i n f y ϵ Y d ( x , y ) , y ∈ Y sup i n f x ϵ X ( x , y ) } d ˉ ε \bar{d}_{\varepsilon} d ˉ ε d d d d ˉ ε \bar{d}_{\varepsilon} d ˉ ε γ \gamma γ γ ε \gamma_\varepsilon γ ε ∣ d ∣ W ∞ 2 |d|_{W_{\infty}^{2}} ∣ d ∣ W ∞ 2 d ˉ ε \bar{d}_{\varepsilon} d ˉ ε
Lemma 26 (properties of d ε ) . The function d ε ∈ C ∞ ( N ( δ ) ) and satisfies ∥ d ε ∥ W ∞ 2 ( N ( δ ) ) + ε ∥ d ε ∥ W ∞ 3 ( N ( δ ) ) ≲ ∣ d ∣ W ∞ 2 ( N ) Moreover, the following error estimates hold ∥ ∇ ( d − d ε ) ∥ L ∞ ( N ( δ ) ) ≲ δ ∣ d ∣ W ∞ 2 ( N ) , ∥ 1 − ∇ d ⋅ ∇ d ε ∥ L ∞ ( N ( δ ) ) ≲ δ 2 ∣ d ∣ W ∞ 2 ( N )
\begin{array}{l}{ \text { Lemma }\left.26 \text { (properties of } d_{\varepsilon}\right) . \text { The function } d_{\varepsilon} \in C^{\infty}(\mathcal{N}(\delta)) \text { and satisfies }} \\ {\qquad \begin{aligned}\left\|d_{\varepsilon}\right\|_{W_{\infty}^{2}}(\mathcal{N}(\delta))+\varepsilon\left\|d_{\varepsilon}\right\|_{W_{\infty}^{3}}(\mathcal{N}(\delta)) & \lesssim|d|_{W_{\infty}^{2}(\mathcal{N})} \\ \text { Moreover, the following error estimates hold } \\\left\|\nabla\left(d-d_{\varepsilon}\right)\right\|_{L_{\infty}(\mathcal{N}(\delta))} & \lesssim \delta|d|_{W_{\infty}^{2}(\mathcal{N})}, \quad\left\|1-\nabla d \cdot \nabla d_{\varepsilon}\right\|_{L_{\infty}(\mathcal{N}(\delta))} \lesssim \delta^{2}|d|_{W_{\infty}^{2}(\mathcal{N})} \end{aligned}}\end{array}
Lemma 2 6 (properties of d ε ) . The function d ε ∈ C ∞ ( N ( δ ) ) and satisfies ∥ d ε ∥ W ∞ 2 ( N ( δ ) ) + ε ∥ d ε ∥ W ∞ 3 ( N ( δ ) ) Moreover, the following error estimates hold ∥ ∇ ( d − d ε ) ∥ L ∞ ( N ( δ ) ) ≲ ∣ d ∣ W ∞ 2 ( N ) ≲ δ ∣ d ∣ W ∞ 2 ( N ) , ∥ 1 − ∇ d ⋅ ∇ d ε ∥ L ∞ ( N ( δ ) ) ≲ δ 2 ∣ d ∣ W ∞ 2 ( N ) γ ε \gamma_\varepsilon γ ε d ε d_\varepsilon d ε ∣ d ∣ W ∞ 2 |d|_{W_{\infty}^{2}} ∣ d ∣ W ∞ 2
Corollary 27 (property of P ε ) . The lift P ε belongs to C ∞ ( N ( δ ) ) and satisfies ∣ P ε ∣ W ∞ 2 ( N ( δ ) ) ≲ ∣ d ∣ W ∞ 2 ( N ) for suitable constants C 1 , C 2 so that C 1 δ ≤ ε ≤ δ 2 and C 2 ε ∣ d ∣ W ∞ 2 ( N ) ≤ 1
\begin{array}{l}{ \text { Corollary }\left.27 \text { (property of } P_{\varepsilon}\right) . \text { The lift } P_{\varepsilon} \text { belongs to } C^{\infty}(\mathcal{N}(\delta)) \text { and satisfies }} \\ {\qquad\left|\mathbf{P}_{\varepsilon}\right|_{W_{\infty}^{2}(\mathcal{N}(\delta))} \lesssim|d|_{W_{\infty}^{2}(\mathcal{N})}} \\ {\text { for suitable constants } C_{1}, C_{2} \text { so that } C_{1} \delta \leq \varepsilon \leq \frac{\delta}{2} \text { and } C_{2} \varepsilon|d|_{W_{\infty}^{2}(\mathcal{N})} \leq 1}\end{array}
Corollary 2 7 (property of P ε ) . The lift P ε belongs to C ∞ ( N ( δ ) ) and satisfies ∣ P ε ∣ W ∞ 2 ( N ( δ ) ) ≲ ∣ d ∣ W ∞ 2 ( N ) for suitable constants C 1 , C 2 so that C 1 δ ≤ ε ≤ 2 δ and C 2 ε ∣ d ∣ W ∞ 2 ( N ) ≤ 1 γ ε \gamma_\varepsilon γ ε P ε \mathbf{P}_\varepsilon P ε W ∞ 2 W_{\infty}^{2} W ∞ 2 d d d
Proposition 28 ( H 2 extension ) . Let ε and δ be as in Corollary 27 (property of P ε ) , and assume that ε ∣ d ∣ W ∞ 2 ( N ) ≤ c for a sufficiently small constant c . If u ~ ∈ H 2 ( γ ) , then u ∈ H 2 ( N ( δ ) ) and ∥ u ∥ H 2 ( N ( δ ) ) ≲ δ 1 2 ∣ d ∣ W ∞ 2 ( N ) ∥ u ~ ∥ H 2 ( γ ) Moreover, the trace of u on γ coincides with u ~ , that is u an H 2 extension of u ~ .
\begin{array}{l}{ \text { Proposition }\left.28\left(H^{2} \text { extension }\right) . \text { Let } \varepsilon \text { and } \delta \text { be as in Corollary } 27 \text { (property of } \mathbf{P}_{\varepsilon}\right) \text { , }} \\ {\text { and assume that } \varepsilon|d|_{W_{\infty}^{2}(\mathcal{N})} \leq c \text { for a sufficiently small constant } c . \text { If } \widetilde{u} \in H^{2}(\gamma),} \\ {\text { then } u \in H^{2}(\mathcal{N}(\delta)) \text { and }} \\ {\qquad\|u\|_{H^{2}(\mathcal{N}(\delta))} \lesssim \delta^{\frac{1}{2}}|d|_{W_{\infty}^{2}(\mathcal{N})}\|\widetilde{u}\|_{H^{2}(\gamma)}} \\ {\text { Moreover, the trace of } u \text { on } \gamma \text { coincides with } \widetilde{u}, \text { that is } u \text { an } H^{2} \text { extension of } \widetilde{u} \text { . }}\end{array}
Proposition 2 8 ( H 2 extension ) . Let ε and δ be as in Corollary 2 7 (property of P ε ) , and assume that ε ∣ d ∣ W ∞ 2 ( N ) ≤ c for a sufficiently small constant c . If u ∈ H 2 ( γ ) , then u ∈ H 2 ( N ( δ ) ) and ∥ u ∥ H 2 ( N ( δ ) ) ≲ δ 2 1 ∣ d ∣ W ∞ 2 ( N ) ∥ u ∥ H 2 ( γ ) Moreover, the trace of u on γ coincides with u , that is u an H 2 extension of u . u u u u ~ \tilde u u ~ u ε u_\varepsilon u ε u ( x ) : = u ε ( x − d ε ( x ) ∇ d ε ( x ) ) ∀ x ∈ N ( δ )
u(\mathrm{x}):=u_{\varepsilon}\left(\mathrm{x}-d_{\varepsilon}(\mathrm{x}) \nabla d_{\varepsilon}(\mathrm{x})\right) \quad \forall \mathrm{x} \in \mathcal{N}(\delta)
u ( x ) : = u ε ( x − d ε ( x ) ∇ d ε ( x ) ) ∀ x ∈ N ( δ ) u ε u_\varepsilon u ε γ \gamma γ γ ε \gamma_\varepsilon γ ε u ~ \tilde u u ~ u ε = u ~ ∘ Q ε
u_{\varepsilon}=\widetilde{u} \circ \mathbf{Q}_{\varepsilon}
u ε = u ∘ Q ε Q ε = P ε − 1 : γ ε → γ
\mathbf{Q}_{\varepsilon}=\mathbf{P}_{\varepsilon}^{-1}: \gamma_{\varepsilon} \rightarrow \gamma
Q ε = P ε − 1 : γ ε → γ u u u H 2 H^2 H 2 H 2 H^2 H 2
Lemma 29 (PDE satisfied by u ε ) . If γ is closed and of class C 2 , then γ ε is also closed and of class C ∞ , and the extension u ε = u ~ ∘ Q ε satisfies on γ ε − μ ~ ε div γ ε ( 1 μ ~ ε A ~ ε ∇ γ ε u ε ) = f ~ ε where A ~ ε : = ( I − d ε D 2 d ε ) Π ( I − d ε D 2 d ε ) ∘ Q ε , Π stands for the orthogonal projection Π = ( I − ∇ d ⊗ ∇ d ) on γ and μ ~ ε : = q ε q ∘ Q ε reads μ ~ ε = det ( I − d ε D 2 d ε ) ( ∇ d ⋅ ∇ d ε ) ∘ Q ε
\begin{array}{l}{ \text { Lemma }\left.29 \text { (PDE satisfied by } u_{\varepsilon}\right) . \text { If } \gamma \text { is closed and of class } C^{2}, \text { then } \gamma_{\varepsilon} \text { is also }} \\ {\text { closed and of class } C^{\infty}, \text { and the extension } u_{\varepsilon}=\widetilde{u} \circ \mathrm{Q}_{\varepsilon} \text { satisfies on } \gamma_{\varepsilon}} \\ {\qquad-\widetilde{\mu}_{\varepsilon} \operatorname{div}_{\gamma_{\varepsilon}}\left(\frac{1}{\tilde{\mu}_{\varepsilon}} \widetilde{\mathbf{A}}_{\varepsilon} \nabla_{\gamma_{\varepsilon}} u_{\varepsilon}\right)=\tilde{f}_{\varepsilon}} \\ {\text { where } \tilde{\mathbf{A}}_{\varepsilon}:=\left(\mathbf{I}-d_{\varepsilon} D^{2} d_{\varepsilon}\right) \Pi\left(\mathbf{I}-d_{\varepsilon} D^{2} d_{\varepsilon}\right) \circ \mathbf{Q}_{\varepsilon}, \Pi \text { stands for the orthogonal projection }} \\ {\Pi=(\mathbf{I}-\nabla d \otimes \nabla d) \text { on } \gamma \text { and } \widetilde{\mu}_{\varepsilon}:=\frac{q_{\varepsilon}}{q \circ Q_{\varepsilon}} \text { reads }} \\ {\widetilde{\mu}_{\varepsilon}=\operatorname{det}\left(\mathbf{I}-d_{\varepsilon} D^{2} d_{\varepsilon}\right)\left(\nabla d \cdot \nabla d_{\varepsilon}\right) \circ \mathbf{Q}_{\varepsilon}}\end{array}
Lemma 2 9 (PDE satisfied by u ε ) . If γ is closed and of class C 2 , then γ ε is also closed and of class C ∞ , and the extension u ε = u ∘ Q ε satisfies on γ ε − μ ε d i v γ ε ( μ ~ ε 1 A ε ∇ γ ε u ε ) = f ~ ε where A ~ ε : = ( I − d ε D 2 d ε ) Π ( I − d ε D 2 d ε ) ∘ Q ε , Π stands for the orthogonal projection Π = ( I − ∇ d ⊗ ∇ d ) on γ and μ ε : = q ∘ Q ε q ε reads μ ε = d e t ( I − d ε D 2 d ε ) ( ∇ d ⋅ ∇ d ε ) ∘ Q ε u ~ \tilde u u ~ γ \gamma γ f ~ \tilde f f ~ f ~ ε \tilde f_\varepsilon f ~ ε γ ϵ \gamma_\epsilon γ ϵ f ε ~ : = f ~ ∘ Q ε
\widetilde{f_{\varepsilon}}:=\widetilde{f} \circ \mathbf{Q}_{\varepsilon}
f ε : = f ∘ Q ε γ \gamma γ γ ϵ \gamma_\epsilon γ ϵ
Proposition 30 (PDE satisfied by u ) . Let ε and δ be as in Corollary 27 (property of P ε ) . The extension u ∈ H 2 ( N ( δ ) ) of u ~ of Proposition 28 satisfies the P D E − 1 μ ε div ( μ ε B ε ∇ u ) = f ε in N ( δ ) − 1 μ ε div ( μ ε B ε ) − 1 Π ε A ε Π ε ( I − d ε D 2 d ε ) − 1 μ ε : = 1 μ ~ ε ∘ P ε det ( I − d ε D 2 d ε ) and μ ~ ε is defined in Lemma 29
\begin{array}{l}{ \text { Proposition 30 (PDE satisfied by }u) \text { . Let } \varepsilon \text { and } \delta \text { be as in Corollary } 27 \text { (property }} \\ { \text { of }\left.\mathbf{P}_{\varepsilon}\right) . \text { The extension } u \in H^{2}(\mathcal{N}(\delta)) \text { of } \widetilde{u} \text { of Proposition } 28 \text { satisfies the } P D E} \\ {\qquad \begin{aligned}-\frac{1}{\mu_{\varepsilon}} \operatorname{div}\left(\mu_{\varepsilon} \mathbf{B}_{\varepsilon} \nabla u\right)=f_{\varepsilon} & \text { in } \quad \mathcal{N}(\delta) \\-\frac{1}{\mu_{\varepsilon}} \operatorname{div}\left(\mu_{\varepsilon} \mathbf{B}_{\varepsilon}\right)^{-1} \Pi_{\varepsilon} \mathbf{A}_{\varepsilon} \Pi_{\varepsilon}\left(\mathbf{I}-d_{\varepsilon} D^{2} d_{\varepsilon}\right)^{-1} \\ \qquad \mu_{\varepsilon}:=\frac{1}{\widetilde{\mu}_{\varepsilon} \circ \mathbf{P}_{\varepsilon}} \operatorname{det}\left(\mathbf{I}-d_{\varepsilon} D^{2} d_{\varepsilon}\right) \\ \text {and } \widetilde{\mu}_{\varepsilon} \text { is defined in Lemma } 29 \end{aligned}}\end{array}
Proposition 30 (PDE satisfied by u ) . Let ε and δ be as in Corollary 2 7 (property of P ε ) . The extension u ∈ H 2 ( N ( δ ) ) of u of Proposition 2 8 satisfies the P D E − μ ε 1 d i v ( μ ε B ε ∇ u ) = f ε − μ ε 1 d i v ( μ ε B ε ) − 1 Π ε A ε Π ε ( I − d ε D 2 d ε ) − 1 μ ε : = μ ε ∘ P ε 1 d e t ( I − d ε D 2 d ε ) and μ ε is defined in Lemma 2 9 in N ( δ ) γ ε \gamma_\varepsilon γ ε γ \gamma γ u u u f ε f_\varepsilon f ε γ ε \gamma_\varepsilon γ ε γ \gamma γ u = u ~ ∘ Q ε ∘ P ε
u=\widetilde{u} \circ \mathbf{Q}_{\varepsilon} \circ \mathbf{P}_{\varepsilon}
u = u ∘ Q ε ∘ P ε f ε : = f ~ ε ∘ P ε = f ~ ∘ Q ε ∘ P ε
f_{\varepsilon}:=\widetilde{f}_{\varepsilon} \circ \mathbf{P}_{\varepsilon}=\widetilde{f} \circ \mathbf{Q}_{\varepsilon} \circ \mathbf{P}_{\varepsilon}
f ε : = f ε ∘ P ε = f ∘ Q ε ∘ P ε γ ϵ \gamma_\epsilon γ ϵ γ \gamma γ u u u f f f u ~ \tilde u u ~ f ~ \tilde f f ~ x ~ = x + s ∇ d ε ( x ) ⇒ u ( x ) = u ~ ( x ~ )
\widetilde{\mathbf{x}}=\mathbf{x}+s \nabla d_{\varepsilon}(\mathbf{x}) \quad \Rightarrow \quad {u}(\mathbf{x})=\tilde u(\widetilde{\mathbf{x}})
x = x + s ∇ d ε ( x ) ⇒ u ( x ) = u ~ ( x ) x ~ = x + s ∇ d ε ( x ) ⇒ f ε ( x ) = f ~ ( x ~ )
\widetilde{\mathbf{x}}=\mathbf{x}+s \nabla d_{\varepsilon}(\mathbf{x}) \quad \Rightarrow \quad f_{\varepsilon}(\mathbf{x})=\widetilde{f}(\widetilde{\mathbf{x}})
x = x + s ∇ d ε ( x ) ⇒ f ε ( x ) = f ( x )
