Let $ \Omega $ be a $ C^1 $ bounded domain and $ T $ a positive number. Assume that $ v $ is a weak solution for the heat equation \begin{eqnarray} \left\{\begin{matrix} \partial_tv-\Delta v=0&\text{in}&\Omega\times(0,T),\\ v=g&\text{on}&\partial\Omega\times(0,T),\\ v=0&\text{on}&\Omega\times\left\{0\right\}, \end{matrix}\right. \end{eqnarray} where $ g $ is in some function spaces to be determined. I want to get the energy-type estimates for the equation, for example, I want to obtain the boundedness for \begin{eqnarray} \int_{0}^T\int_{\Omega}|\nabla v|^2dxdt \end{eqnarray} by some norm of $ g $. In the famous book written by Evans, I know already the answer in the case $ g=0 $. However, I am unable to deal with the case that $ g $ does not vanish. My question can be stated as follows:
(1) What space should I choose for $ g $ such that the energy-type estimate exists?
(2) How to obtain the energy-type estimate above?