# Partial differential Equation uniqueness

Let $$\Omega\in\mathbb{R}^{n}$$ be a bounded connected open set. I have the following partial differential Equation; \begin{align} \nabla\cdot\left(-D(x)\nabla \psi\right)&=F\quad \text{in}\quad \Omega\times(0,\,T)\\ \psi(x,0)&=0\quad \text{in}\quad \Omega\\ \psi(0,t)&=0\quad \text{in}\quad \partial\Omega\\ \nabla\psi(\ell,t)&=0\quad \text{on}\quad \partial \Omega\times(0,\,T) \end{align} I would like to show the uniqueness of the solutions. I assume existence of two different solutions $$\psi_{1}$$ and $$\psi_{2}$$ and defined the difference of the solutions $$\omega=\psi_{2}-\psi_{1}$$ clearly $$\omega$$ satisfies the equation \begin{align} \nabla\cdot\left(-D(x)\nabla \omega\right)&=0\quad \text{in}\quad \Omega\times(0,\,T)\\ \omega(x,0)&=0\quad \text{in}\quad \Omega\\ \omega(0,t)&=0\quad \text{in}\quad \partial\Omega\\ \nabla\omega(\ell,t)&=0\quad \text{on}\quad \partial \Omega\times(0,\,T) \end{align} I considered the following integral \begin{align} J=\int\limits_{\Omega}\omega\nabla\cdot\left(-D(x)\nabla \omega\right) dV=0 \end{align} Assuming that $$D(x)>0$$, I want to show that $$J>0$$ so that I conclude that if $$J>0$$ then all the terms inside the inside the integration are zero. Thus $$\psi_{1}=\psi_{2}$$. Am looking for nay identities to help on this. Rhanks a lot