On the resolvent arising in a parameter-elliptic multi-order boundary problem (Q2920087)

Let \(N, d \in \mathbb{Z}_+\), \(N>1\), \(h\in\mathbb{N}^d\), \(t \in \mathbb{N}^N\), \(h_0=0\), \(t\) decreasing, \(t_1= \dots =t_{h_1}> \dots >t_{h_i+1}= \dots=t_{h_{i+1}}> \dots > t_{h_{d-1}+1}= \dots=t_N\); let \(\Omega\) be a regular open set in \(\mathbb{R}^n\); for \(j,k \in \{1, \dots, N\}\), let \(a_{\alpha}^{j,k} \in C^{t_j}(\overline{\Omega})\) if \(|\alpha| \leq t_j+t_k\), \(t_j>0\), while \(a_{\alpha}^{j,k} \in L^{\infty}(\Omega)\) if \(|\alpha|<t_k\), \(t_j=0\), and \(a_{\alpha}^{j,k} \in C^{0}(\overline{\Omega})\) if \(|\alpha|=t_k\), \(t_j=0\), \(A_{j,k}(x, \xi)= \sum_{|\alpha|\leq t_j+t_k} a_{\alpha}^{j,k}(x) \xi^{\alpha}\) for \(x \in \Omega\), \(\xi \in \mathbb{R}^n\) or \(\xi=\partial\); \(\dot{A}_{j,k}(x, \xi)= \sum_{|\alpha|= t_j+t_k} a_{\alpha}^{j,k}(x) \xi^{\alpha}\), \(M=(A_{j,k}: j,k \in \{1, \dots, N\})\); \(M_{1,1}=(\dot{A}_{j,k}: j,k \in \{1,\dots,h_{d-1}\})\); \(M_{1,2}= (\dot{A}_{j,k}: j \in \{1,\dots,h_{d-1}\},\;k \in \{h_{d-1}+1,\dots,N\})\); \(M_{2,1}= (\dot{A}_{j,k}: j \in \{h_{d-1}+1,\dots,N\},\;k \in \{1,\dots,h_{d-1}\})\); \(M_{2,2}=(\dot{A}_{j,k}: j,k \in \{h_{d-1}+1,\dots,N\})\); let \(I\) be the \((N-h_{d-1}) \times (N-h_{d-1})\) identity matrix; let \(L_{\theta}= \{\lambda \in \mathbb{C}: \theta \leq |\operatorname{arg} \lambda| \leq \pi \}\); let \(H^{-t, \lambda}(\Omega)\) be the Bessel potential space \(H^{-t}(\Omega) \equiv \prod _{j=1}^N H^{-t_j}(\Omega)\) with the norm \(|\cdot|\) such that \(|u|= \inf\{|F^{-1}(|\cdot|^2+|\lambda|^{1/t_j})^{-t_j/2}Fv|_{L^2(\mathbb{R}^n)}: v\in H^{-t_j}(\mathbb{R}^n) \text{ for which } u=v_{\big|\Omega}\}\) for \(u \in H^{-t}(\Omega)\), \(\lambda \in \mathbb{C} \setminus \{0\}\), and where \(F\) denotes the Fourier transformation; let \(R\) be the resolvent of \(M(\cdot,\partial)\) in \(H^{-t}(\Omega)\) and \(\Lambda (\lambda)^{(\pm t)}=\operatorname{diag}((-\Delta^{2t_1}+|\lambda|^2)^{\pm 1/4}, \dots,(-\Delta^{2t_N}+|\lambda|^2)^{\pm 1/4})\). If the problem \(M(\cdot,\partial)u-\lambda u =f \in H^{-t}(\Omega)\) with boundary conditions of Dirichlet type satisfies a convenient parameter elliptic condition in \(L_{\theta}\) and \(q \in 2\mathbb{Z}_+\) is the least even integer for which \(2qt_N>n\), then there exists a \(\lambda_0>0\) such that for \(\lambda \in L_{\theta}\) with \(|\lambda| \geq \lambda_0\) we have \(R(\lambda)^q\) is an operator of trace class on \(H^{-t, \lambda_0}(\Omega)\) and NEWLINE\[NEWLINE\begin{multlined}\operatorname{tr}\Lambda(\lambda)^{(-t)}R(\lambda)^q \Lambda(\lambda)^{(t)}=(2 \pi)^{-n} (-\lambda)^{-q+n/(2t_N)}\\ \int _{\Omega \times \mathbb{R}^n}\operatorname{tr}(M_{2,2}(x,\xi)-M_{2,1}(x,\xi)M_{1,1}(x,\xi)^{-1}M_{1,2}(x,\xi) + I)^{-q} \text{d}x\,\text{d}\xi +o(|\lambda|^{-q+n/(2t_N)})\end{multlined}NEWLINE\]NEWLINE as \(|\lambda| \rightarrow \infty\), uniformly in \(\lambda \in L_{\theta}\), where \((-\lambda)^s\) is holomorphically defined for \(s \in \mathbb{R}, \lambda \notin \overline{\mathbb{R}_+}\) and is equal to \(|\lambda|^s\) when \(\lambda \in \mathbb{R}_-\).