Anisotropic conductivity and weak Dirichlet-to-Neumann map

Assume the Axiom of Choice and Countable Choice. Let $n\ge 2$ and let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded connected open set with smooth boundary. Write $H^1(\mathrm{\Omega })=W^{1,2}(\mathrm{\Omega };\mathbb{R})$ and let $H_0^1(\mathrm{\Omega })$ be the $H^1$-closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$. Define the boundary trace-data space ${\mathcal{B}}_{\mathrm{\Omega }}:=H^1(\mathrm{\Omega })\mathrm{/}{H}_0^1(\mathrm{\Omega })$ with quotient norm. If $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, its boundary is nonempty, and the trace-quotient lemma identifies this quotient with the fixed-atlas $H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ boundary data convention of the scalar conductivity map. If $\mathrm{\Omega }=\mathrm{\varnothing }$, then $H^1(\mathrm{\Omega })=H_0^1(\mathrm{\Omega })=\{0\}$ and the fixed-atlas empty-boundary convention $H^{1\mathrm{/}2}(\mathrm{\varnothing })=\{0\}$ gives the identification directly. Let $\gamma \in C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }};{\mathrm{Sym}}_n(\mathbb{R}))$ be uniformly positive definite, so some $0<c\le C<\mathrm{\infty }$ satisfy $c\mathrm{\mid }\xi {\mathrm{\mid }}^2\le {\xi }^T\gamma (x)\xi \le C\mathrm{\mid }\xi {\mathrm{\mid }}^2$ for all $x\in \bar{\mathrm{\Omega }}$ and $\xi \in {\mathbb{R}}^n$. For $f=[U]\in {\mathcal{B}}_{\mathrm{\Omega }}$, a weak solution with boundary datum $f$ is a class $u_{\gamma ,f}\in H^1(\mathrm{\Omega })$ with $[u_{\gamma ,f}]=f$ and

${\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }{u}_{\gamma ,f}\cdot \mathrm{\nabla }\phi dx=0(\phi \in H_0^1(\mathrm{\Omega }))\mathrm{.}$

The weak anisotropic Dirichlet-to-Neumann map is the bilinear-form candidate ${\mathrm{\Lambda }}_{\gamma }:{\mathcal{B}}_{\mathrm{\Omega }}\times {\mathcal{B}}_{\mathrm{\Omega }}\to \mathbb{R}$ given, whenever the weak solution exists, by

$⟨{\mathrm{\Lambda }}_{\gamma }f,h⟩:={\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }{u}_{\gamma ,f}\cdot \mathrm{\nabla }{V}_{h}dx,$

where $V_h\in H^1(\mathrm{\Omega })$ is any representative of $h$. The local anisotropic conductivity well-posedness lemma proves existence and uniqueness for every boundary class, independence of the representative $V_h$, and boundedness on ${\mathcal{B}}_{\mathrm{\Omega }}\times {\mathcal{B}}_{\mathrm{\Omega }}$; it therefore makes this candidate a bounded bilinear map. The map is the boundary energy form for the equation $\mathrm{div}(\gamma \mathrm{\nabla }u)=0$. Under the trace-quotient isomorphism and the local boundedness result, this form gives the usual weak map into the dual $H^{-1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ with the scalar convention.

Integer-order Sobolev spaces and their norms

Zero-boundary Sobolev space as a norm closure

Euclidean trace quotient and fixed-atlas H1/2 boundary data

Fixed-atlas Sobolev spaces and the empty-boundary convention

Scalar conductivity Dirichlet-to-Neumann map

The Lp trace operator on a bounded C1 domain

The kernel of the trace is the closure of the test functions

The Axiom of Choice implies Countable Choice

The Axiom of Countable Choice

The Axiom of Choice

Mikko Salo, §1, pp. 170–171

Daudé, Kamran, and Nicoleau, §1, equation (1.6)

The matrix formulation is required for the anisotropic pushforward under changes of variables.