Linear isotropic elasticity: displacement form and data

Assume Countable Choice. Let $d\in \{2,3\}$ and let $D\subset {\mathbb{R}}^d$ be nonempty, open, bounded, connected, and Lipschitz. Use the real spaces $V=H_0^1(D;{\mathbb{R}}^d)$ and $Q=L_0^2(D)$, norms $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D)}$ and $\mathrm{\parallel }q{\mathrm{\parallel }}_Q=\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(D)}$, and $b(v,q)=-{\int }_{D}q\mathrm{div}v$ as in Stokes velocity and pressure spaces and weak forms. For a real normed space $X$, write $X^{\mathrm{\prime }}$ for its continuous real dual with operator norm $\mathrm{\parallel }F{\mathrm{\parallel }}_{X^{\mathrm{\prime }}}={\sup }_{\mathrm{\parallel }x{\mathrm{\parallel }}_X\le 1}\mathrm{\mid }F(x)\mathrm{\mid }$. Let $\mu \in L^{\mathrm{\infty }}(D;\mathbb{R})$ satisfy $0<{\mu }_0\le \mu (x)\le {\mu }_1<\mathrm{\infty }$ almost everywhere, let $\lambda \ge 0$ be a scalar, and let $f\in V^{\mathrm{\prime }}$. For any $w,z\in H^1(D;{\mathbb{R}}^d)$ define $\epsilon (w)=(\mathrm{\nabla }w+\mathrm{\nabla }{w}^T)\mathrm{/}2$ and $a_{\mu }(w,z)=2{\int }_D\mu \epsilon (w):\epsilon (z),{\mathcal{E}}_{\lambda }(w,z)=a_{\mu }(w,z)+\lambda {\int }_D\mathrm{div}w\mathrm{div}z,$ and ${\sigma }_{\lambda }(w)=2\mu \epsilon (w)+\lambda (\mathrm{div}w)I_d$. The homogeneous-Dirichlet model restricts these forms to $V$, with displacement zero on the full boundary; an example may prescribe a different boundary partition but gains no well-posedness claim from this definition. This is a linear small-strain model; no nonlinear, injectivity, or contrast-independent estimate is asserted.

Countable Choice is inherited through the imported Sobolev and Stokes-space conventions; this definition makes no arbitrary-index selection.