\(k\)-explicit inf-sup stability for the truncated Cartesian PML Maxwell system

Setup.
Let \(\Omega=(-L,L)^3\subset\mathbb R^3\) be a box, partitioned into a physical
domain \(\Omega_{\mathrm{phys}}=(-L_0,L_0)^3\), with \(0<L_0<L\), and a
Cartesian PML layer \(\Omega_{\mathrm{pml}}=\Omega\setminus\Omega_{\mathrm{phys}}\)
of thickness \(\delta=L-L_0\). For \(j=1,2,3\), let
\(\sigma_j\in L^\infty(\mathbb R)\) depend only on \(x_j\), with
\(\sigma_j\ge0\) and \(\sigma_j\equiv0\) on \([-L_0,L_0]\), and set
\[
  \widetilde{x}_j(x_j)
  =
  x_j+\mathrm{i}\int_0^{x_j}\sigma_j(t)\,dt,
  \qquad
  \alpha_j:=\frac{d\widetilde{x}_j}{dx_j}
  =
  1+\mathrm{i}\sigma_j,
  \qquad
  A
  :=
  \operatorname{diag}
  \left(
  \frac{\alpha_2\alpha_3}{\alpha_1},
  \frac{\alpha_1\alpha_3}{\alpha_2},
  \frac{\alpha_1\alpha_2}{\alpha_3}
  \right).
\]

Given \(k>0\) and \(\mathbf f\in L^2(\Omega)^3\), the truncated Cartesian PML
Maxwell problem is to find \(\mathbf E\in H_0(\operatorname{curl};\Omega)\) with
\[
  \operatorname{curl}\big(A\operatorname{curl}\mathbf E\big)
  -
  k^2A^{-1}\mathbf E
  =
  A^{-1}\mathbf f
  \quad \text{in } \Omega,
  \qquad
  \mathbf n\times \mathbf E=0
  \quad \text{on } \partial\Omega .
\]

Equivalently, \(\mathbf E\) solves the variational problem
\[
  a_{\mathrm{pml}}(\mathbf E,\mathbf v)
  =
  \langle A^{-1}\mathbf f,\mathbf v\rangle,
  \qquad
  \forall \mathbf v\in H_0(\operatorname{curl};\Omega),
\]
where
\[
  a_{\mathrm{pml}}(\mathbf u,\mathbf v)
  :=
  (A\operatorname{curl}\mathbf u,\operatorname{curl}\mathbf v)_{\Omega}
  -
  k^2(A^{-1}\mathbf u,\mathbf v)_{\Omega},
\]
and
\[
  \|\mathbf u\|_{\operatorname{curl},k,\Omega}^2
  :=
  \|\operatorname{curl}\mathbf u\|_{L^2(\Omega)}^2
  +
  k^2\|\mathbf u\|_{L^2(\Omega)}^2,
\]
where \((\cdot,\cdot)_\Omega\) is the complex \(L^2\)-inner product.

Assumptions.

(A1) Uniform non-degeneracy. There is a constant \(\sigma_{\max}\ge0\) with
\(0\le\sigma_j\le\sigma_{\max}\) for all \(j\). Hence
\(A,A^{-1}\in L^\infty(\Omega;\mathbb C^{3\times3})\) and
\[
  \|A\|_{L^\infty(\Omega)}+\|A^{-1}\|_{L^\infty(\Omega)}
  \le C_A(\sigma_{\max}).
\]

(A2) Absorption parameter. The PML strength is measured by
\[
  \tau:=k\min_j\int_{L_0}^{L}\sigma_j(t)\,dt,
\]
which is allowed to depend on \(k\), through \(\delta\), \(\sigma_{\max}\), or
the profile \(\sigma_j\).

Problem.
Find an exponent \(q\ge0\) and a PML strength prescription \(\tau=\tau(k)\),
for example \(\tau\ge c_0\log k\), such that, for all sufficiently large \(k\),
the sesquilinear form \(a_{\mathrm{pml}}\) satisfies the \(k\)-explicit inf-sup
estimate
\[
  \inf_{0\ne \mathbf u\in H_0(\operatorname{curl};\Omega)}
  \sup_{0\ne \mathbf v\in H_0(\operatorname{curl};\Omega)}
  \frac{|a_{\mathrm{pml}}(\mathbf u,\mathbf v)|}
  {\|\mathbf u\|_{\operatorname{curl},k,\Omega}
   \|\mathbf v\|_{\operatorname{curl},k,\Omega}}
  \ge
  \beta_{\mathrm{pml}}(k)
  \ge
  Ck^{-q},
\]
with \(C>0\) independent of \(k\). Determine the smallest such \(q\).

Sub-problems (a proposed route).
Let \(\mathbf E_{\infty}\) denote the exact outgoing solution of the
constant-coefficient Maxwell problem on \(\mathbb R^3\), with Silver-M\"uller
radiation condition, for the same data \(\mathbf f\).

(P1) Resolvent estimate. For the nontrapping case, prove a \(k\)-explicit
polynomial resolvent bound for the exact outgoing problem,
\[
  \|\mathbf E_{\infty}\|_{\operatorname{curl},k}
  \le
  Ck^p\|\mathbf f\|_{L^2},
  \qquad
  p\ge0,
\]
where \(C\) is independent of \(k\).

(P2) Exponential PML accuracy. Prove that the truncated PML solution
approximates the exact outgoing solution exponentially in the absorption:
there exist \(C,c>0\), independent of \(k\), with
\[
  \|\mathbf E-\mathbf E_{\infty}\|_{\operatorname{curl},k,\Omega_{\mathrm{phys}}}
  \le
  Ce^{-c\tau}\|\mathbf f\|_{L^2}.
\]

Show that (P1)-(P2), together with (A1)-(A2), imply the inf-sup estimate of the
Problem, and identify the resulting exponent \(q=q(p)\) and the required
absorption \(\tau(k)\).