weak variational formulation of Poisson equation with Dirichlet boundary conditions

Question

I have given the Poisson equation with Dirichlet boundary conditions \begin{cases} -\Delta u & = f & \text{in} &\Omega \\ \quad u & = g & \text{on} & \partial\Omega \end{cases} For $g=0$ the weak variational formulation is $$\int_{\Omega} \nabla u \nabla v = \int_{\Omega} fv$$ Now I want to know: How could a weak variational formulation be defined if $g\neq0$?

Answer 1

Usually one reduces to the case of $g=0$, which is equivalent to interpreting $u=g$ on the boundary in the trace sense. For example, if you can find an $H^2$ function $w$ with $w=g$ on $\partial \Omega$ in the trace sense, then you set $v=u-w$ and find that $v$ satisfies $-\Delta v = -\Delta u + \Delta w = f + \Delta w$ and $v=0$ on $\partial \Omega$. So you obtain an equation for $v$ with zero boundary condition with a modified source term $f + \Delta w$.

You can also just keep the same weak variational form, but instead look for solutions $u\in H^1(\Omega)$ with $u=g$ in the trace sense on $\partial \Omega$ (you still take test functions $v\in H^1_0(\Omega)$). This is a linear closed subspace of $H^1$ (it is the space $w + H^1_0(\Omega)$, where $w$ is given above).