Dirichlet elasticity energy bound and Galerkin estimate

Since $\mathrm{\mid }\epsilon (v){\mathrm{\mid }}_F\le \mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}_F$ and $\mathrm{\mid }\mathrm{div}v\mathrm{\mid }\le \sqrt{d}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}_F$, [F4] gives $\mathrm{\mid }{\mathcal{E}}_{\lambda }(u,v)\mathrm{\mid }\le (2{\mu }_1+d\lambda )\mathrm{\parallel }u{\mathrm{\parallel }}_V\mathrm{\parallel }v{\mathrm{\parallel }}_V$. The form is real, symmetric, and bounded.

By [F3] and $\mu \ge {\mu }_0$, ${\mathcal{E}}_{\lambda }(v,v)\ge 2{\mu }_0\mathrm{\parallel }\epsilon (v){\mathrm{\parallel }}_2^2\ge {\mu }_0\mathrm{\parallel }v{\mathrm{\parallel }}_V^2$; the volumetric term is nonnegative.

By [F1]–[F2], $V$ is complete in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_V$. Applying [F5] to the bounded functional $f$ and the form of steps 1.1–1.2 yields a unique $u\in V$.

Testing with $v=u$ gives ${\mathcal{E}}_{\lambda }(u,u)=f(u)\le \mathrm{\parallel }f{\mathrm{\parallel }}_{V^{\mathrm{\prime }}}\mathrm{\parallel }u{\mathrm{\parallel }}_V$ and, by step 1.2, ${\mu }_0\mathrm{\parallel }u{\mathrm{\parallel }}_V^2\le \mathrm{\parallel }f{\mathrm{\parallel }}_{V^{\mathrm{\prime }}}\mathrm{\parallel }u{\mathrm{\parallel }}_V$. If $u\mathrm{\ne }0$, divide by ${\mu }_0\mathrm{\parallel }u{\mathrm{\parallel }}_V$; if $u=0$, the claimed bound is immediate. Substitution into the energy identity gives the second estimate.

The form bounds in step 1.1 and the coercivity constant ${\mu }_0$ in step 1.2 satisfy the hypotheses of [F6] on every finite-dimensional conforming $V_h$. Thus its Galerkin solution is unique and the stated $M\mathrm{/}\alpha$ estimate is exactly $M=2{\mu }_1+d\lambda$, $\alpha ={\mu }_0$.