Manuscript Oai365-A@1 · Oai365-A.tex · section · a3840e3d

Variational definition

The same-patch elliptic boundary object › Variational definition · lines 663–695

Oai365-A@1#eq:DNform

Let $M$ be compact, connected, and smooth, with nonempty smooth boundary and dimension $n \geq 3$ . Fix a nonempty open $Γ \subset \partial M$ . Let $g$ be a smooth Riemannian metric. On $M \times C^{2}$ , let

\nabla_{A} = d + A, A^{*} = - A

be a smooth unitary connection. Hermitian pairings are conjugate linear in the first argument.

Set

H_{Γ} = C_{c}^{\infty} (Γ; C^{2})^{H^{1/2} (\partial M; C^{2})} .

Thus boundary inputs vanish off the measured patch in the trace sense.

For $f \in H_{Γ}$ , let $u_{f}$ be the harmonic extension:

\nabla_{A}^{*} \nabla_{A} u_{f} = 0, u_{f} ∣_{\partial M} = f .

Define

B_{g, A}^{Γ} (f, h) = \int_{M} ⟨ \nabla_{A} u_{f}, \nabla_{A} u_{h} ⟩_{g} d V_{g}, Q_{g, A}^{Γ} (f) = B_{g, A}^{Γ} (f, f) .

This is the zero-frequency same-patch DN energy form. Its observation variable $h$ is also supported on $Γ$ .

Displayed formulas in this passage

Oai365-A@1#eq:DNform · line 687
$B_{g, A}^{Γ} (f, h) = \int_{M} ⟨ \nabla_{A} u_{f}, \nabla_{A} u_{h} ⟩_{g} d V_{g}, Q_{g, A}^{Γ} (f) = B_{g, A}^{Γ} (f, f) .$
unlabeled · line 669
$\nabla_{A} = d + A, A^{*} = - A$
unlabeled · line 676
$H_{Γ} = C_{c}^{\infty} (Γ; C^{2})^{H^{1/2} (\partial M; C^{2})} .$
unlabeled · line 683
$\nabla_{A}^{*} \nabla_{A} u_{f} = 0, u_{f} ∣_{\partial M} = f .$

Variational definition

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