Case Condition for two lines

The main trick is using aligned; however, there are other improvements to your code:

1. “div” is an operator and it's much better to define a macro for it; since \div is preempted, I used the name \divop;

2. if an operator immediately follows & in align or aligned, it should be preceded by \! or a spurious thin space would be added; a little nuisance, but it's much better than \mathrm{div}\,;

3. \boldsymbol works, but \bm is better; remember to load bm after amsmath; you might keep \boldsymbol, but \bm is easier;

4. \bold is not predefined; probably you meant \mathbf; I recommend using \mathbf{u} and \bm{\sigma} rather than \mathbf u and \bm \sigma: it's semantically sounder;

5. \overline is less appealing than \bar for a single letter;

6. \tfrac is better in the context, because \frac in aligned would print a big fraction;

7. \left and \right in that context are useless: they add unwanted space and produce oversized parentheses.

\documentclass{article}
\usepackage{amsmath,bm}

\DeclareMathOperator{\divop}{div}

\begin{document}

\begin{align*}
\begin{cases}
\begin{aligned}
&\bm{\varepsilon}(\mathbf{u}) = \tfrac{1}{2}(\nabla \mathbf{u} + \nabla \mathbf{u}^T ) \\
&\!\divop\bm{\sigma} + \mathbf{f} = \mathbf{0}
\end{aligned}&\text{in } \bm{\Omega}
\\
\mathbf{u} = \bar{\mathbf{u}} & \text{on } \partial \bm{\Omega}_D \\
\bm{\sigma} \mathbf{n} = \bar{\mathbf{t}} & \text{on } \partial \bm{\Omega}_N
\end{cases}
\end{align*}

\end{document}

Like this? B.t.w., \bold is not defined. I replaced it with \bm.

\documentclass{article}
\usepackage[utf8]{inputenc}%
\usepackage{mathtools, bm}
\newcommand{\Leg}[3][]{\left(\frac{#2}{#3}\right)_{#1}}
\DeclareMathOperator{\divg}{div}

\begin{document}

\begin{align*}
\begin{cases}
\boldsymbol\varepsilon(\bm{u}) = \frac{1}{2}\left(\nabla \bm{u} + \nabla \bm{u}^T \right) & \smash{\raisebox{-1.6ex}{in $\boldsymbol\Omega$}} \\
\divg\boldsymbol \sigma + \bm{f} = \mathbf{0} \\
\bm{u} = \overline{\bm{u}} & \text{on } \partial \boldsymbol\Omega_D \\
\boldsymbol \sigma \bm{n}= \overline{\bm{t}} & \text{on } \partial \boldsymbol\Omega_N
\end{cases}
\end{align*}
     \[ \Leg{\pi}{θ}\quad \Leg{\pi}{θ} \]%

\end{document}