A question on mixed boundary value problem

Question

I was wondering if there exists a classical solution ($C^2$) to the problem \begin{align} -Lu&=1,\text{in}~\Omega\nonumber\\ u&=0,\text{on}~\Gamma_1\nonumber\\ \frac{\partial u}{\partial n}&=0,\text{on}~\Gamma_2\nonumber \end{align} where $L$ is a linear second order differential opeartor with variable coefficients which are Lipshitz continuous, $\Gamma_1\cup\Gamma_2=\partial\Omega$, $\Omega$ is a bounded domain in $\mathbb{R}^N$.