The homogeneous Dirichlet strain identity

For $v\in C_c^{\mathrm{\infty }}(D;{\mathbb{R}}^d)$, expanding the symmetric gradient gives $2\mathrm{\mid }\epsilon (v){\mathrm{\mid }}^2=\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2+{\sum }_{i,j}{\mathrm{\partial }}_j{v}_i{\mathrm{\partial }}_i{v}_j$ pointwise.

For each $i,j$, apply [F3] to $h=v_i{\mathrm{\partial }}_i{v}_j$, which is compactly supported and smooth. The product rule gives ${\int }_D{\mathrm{\partial }}_j{v}_i{\mathrm{\partial }}_i{v}_j=-{\int }_D{v}_i{\mathrm{\partial }}_j{\mathrm{\partial }}_i{v}_j$. Summing in $j$ and using equality of mixed derivatives gives $-{\sum }_i{\int }_D{v}_i{\mathrm{\partial }}_i(\mathrm{div}v)$. Apply [F3] again to $h=v_i\mathrm{div}v$ and sum in $i$ to obtain ${\sum }_i{\int }_D({\mathrm{\partial }}_i{v}_i)\mathrm{div}v=\mathrm{\parallel }\mathrm{div}v{\mathrm{\parallel }}_2^2$.

By [A1, F1], approximate any $v\in H_0^1$ by such smooth fields in $H^1$. All three squared norms in the identity converge by [F2] and continuity of the finite sums, so the identity passes to the limit. Its nonnegative divergence term gives the displayed inequality.