The colon ("$:$" sign) - I am seeing this a lot in Chapter 8 of PDE Evans, like this (page 497 of the 2nd edition):
THEOREM 6 (Pressure as Lagrange multiplier). There exists a scalar function $p \in L^2_{\text{loc}}(U)$ such that $$\int_U D\mathbf{u} : D\mathbf{v} \, dx = \int_U p \operatorname{div}\mathbf{v}+\mathbf{f}\cdot\mathbf{v}$$ for all $\mathbf{v} \in H^1(U;\mathbb{R}^3)$ with compact support within $U$.
Here's another example that uses the colon (page 496 of the 2nd edition):
THEOREM 5 (Euler-Lagrange equation for harmonic maps). Let $\mathbf{u} \in \mathcal{A}$ satisfy $$\int_U D\mathbf{u} : D\mathbf{v} \, dx = \int_U p |D\mathbf{u}|^2\mathbf{u}\cdot\mathbf{v} \, dx$$ for eacg $\mathbf{v} \in H^1_0(U;\mathbb{R}^m) \cap L^\infty(U;\mathbb{R}^m)$.
Here is $\mathbf{u},\mathbf{v}:\mathbb R^3\supset U\rightarrow \mathbb R^3$, so that $D\mathbf{u},D\mathbf{v}$ are $3\times 3$ matrices. The $:$ here is a matrix inner product defined by $A:B\;:=\mathrm{\mathop{tr}}\,(A^\top B)$ (see also Ciarlet's Mathematical Elasticity, vol. 1, p.xxxi).