Labeling two equations with one label in LaTeX

Question

I'm trying to format an optimization problem but I am having trouble aligning and and labeling it properly in one environment. I have two equations, each written using an \begin{aligned*} environment. The first is

\documentclass{report}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
    \begin{aligned}
        & \underset{y \in X,\ u \in Y}{\text{minimize}}
        && J(y,u) \\
        &\text{subject to}
    \end{aligned}
\end{equation*}
\end{document}

and the second is

\documentclass{report}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
  \begin{cases} 
    \begin{aligned}
        -\nabla^2 y  &= u &\text{ for }  x  \text{ in }  \Omega, \\
         y &= 0 &\text{ for } x \text{ on } \partial \Omega.
    \end{aligned}
  \end{cases}
\end{equation*}
\end{document}

I'd like to be able to either join them together into one equation but have it formatted the same as the above code, or, somehow, leave them separated as two equations but then label them jointly as one equation so I can refer the pair jointly.

Answer 1

A slight variant on 2 lines using the split environment.

\documentclass{article}

\usepackage{amsmath}


\begin{document}

\begin{equation}
  \begin{split}
    &\operatorname*{minimize}_{y \in X, u \in Y} J(y,u)\\
    &\text{subject to }
    \left\{
      \begin{aligned}
        -\nabla^2 y  &= u && \text{for $x$ in $\Omega$},\\
                   y &= 0 && \text{for $x$ on $\partial \Omega$}.
      \end{aligned}
    \right.
  \end{split}
\end{equation}

\end{document}

Answer 2

I think it might become clearer if you put it into one equation (unless I am missing something obvious).

\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator*{\Minimize}{minimize}
\begin{document}
\begin{equation}
 \Minimize_{y \in X,\ u \in Y}
 J(y,u)\quad\text{subject to}~
 \begin{cases}
 \begin{aligned}
        -\nabla^2 y  &= u &&~\text{for}~ x~\text{in}~ \Omega, \\
         y &= 0 &&~\text{for}~x~\text{on}~\partial \Omega.
 \end{aligned} 
 \end{cases}
\end{equation}  
\end{document}

Answer 3

Another variant,that I think more sober:

\documentclass{article}

\usepackage{amsmath}
\usepackage{array}

\begin{document}

\begin{equation}\label{eq:optim}
  \begin{aligned}
 & \underset{y \in X, u \in Y}{\text{minimize }} J(y,u)\\
    &\text{subject to}\quad
      \begin{array}[t]{@{\vrule width 0.6pt\,}rl}
        -\nabla^2 y = u & \text{for $x$ in $\Omega$},\\
                   y = 0 & \text{for $x$ on $\partial \Omega$}.
      \end{array}
  \end{aligned}
\end{equation}

\end{document}