No-flow mixed Darcy problem and coefficient norms

Assume Countable Choice. Let $D\subset {\mathbb{R}}^d$ be a bounded connected Lipschitz polyhedral domain, $d\in \{2,3\}$, and set $V:=H_0(\mathrm{div};D)$ and $Q:=L_0^2(D)$ with the real graph and integral inner products from The real H(div) space and The real zero-mean $L^2$ space is Hilbert. A permeability tensor $K(x)$ is a measurable real symmetric $d\times d$ matrix field for which there are constants $0<{\kappa }_{\min }\le {\kappa }_{\max }<\mathrm{\infty }$ satisfying ${\kappa }_{\min }\mathrm{\mid }\xi {\mathrm{\mid }}^2\le \xi \cdot K(x)\xi \le {\kappa }_{\max }\mathrm{\mid }\xi {\mathrm{\mid }}^2$ for almost every $x\in D$ and every $\xi \in {\mathbb{R}}^d$. It is positive definite almost everywhere, so $K^{-1}$ is measurable and has eigenvalues between ${\kappa }_{\max }^{}$ and ${\kappa }_{\min }^{}$. Define $a(u,v):={\int }_D{K}^{-1}u\cdot v,b(v,q):=-{\int }_{D}q\mathrm{div}v,$ and the weighted flux norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{K^{-1}}:=a(v,v{)}^{1\mathrm{/}2}$. For data $F\in V^{\mathrm{\prime }}$ and $f\in Q$, the no-flow mixed Darcy problem is to find $(u,p)\in V\times Q$ such that $a(u,v)+b(v,p)=F(v)(v\in V),b(u,q)=-{\int }_{D}fq(q\in Q)\mathrm{.}$ The pressure space's zero-mean constraint fixes the additive constant (Neumann) gauge; $u\in H_0(\mathrm{div};D)$ encodes zero weak normal flux on the full boundary. In the standard source-only Darcy problem used for hybridization, $F=0$.