Interpreting physics as boundary conditions in heat equation
The correct boundary condition at $y=a$ and $x=b$ are: $$ \frac{\partial u}{\partial x}\bigg |_{y=a}=0\\ \frac{\partial u}{\partial y}\bigg |_{x=b}=0\\ $$ which imply that there is no conduction of heat across these sides (heat-insulation). The other 2 boundary conditions are as stated.