Eliminating Neumann boundary condition for elliptic PDE

Question

In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:

I am now wondering if this also works with Neumann boundary conditions. More precisely:

Let $\Omega$ be open and bounded with $C^1$ boundary. Let $L$ be in divergence form $$Lu=-\sum_{i,j=1}^n(a_{ij}(x)u_{x_i})_{x_j}+\sum_{i=1}^nb_i(x)u_{x_i}+c(x)u$$ and uniformly elliptic with $a_{ij},b_i,c\in L^{\infty}(\Omega)$, $f\in L^2(\Omega)$, $\psi\in L^2(\partial\Omega)$ and $h\in L^{\infty}(\partial\Omega)$ with $h\geq 0$.

Consider now the generalized boundary value problem \begin{cases} Lu = f & \text{in } \Omega \newline Bu - hu=\psi & \text{on } \partial \Omega \end{cases} for $Bu=-\sum_{i,j=1}^na_{ij}(x)u_{x_i}\cdot\nu_j(x)$ with $x\in\partial\Omega$.

Under what conditions on $\psi \in L^2(\partial \Omega)$ can the solvability (with solutions $u \in H^2(\Omega)$, such that both equations hold in the $L^2$ sense) be reduced to an equivalent problem of the form \begin{cases} Lv = g & \text{in } \Omega \newline Bv - hv=0 & \text{on } \partial \Omega? \end{cases}

Is it sufficient to find a solution $w$ to the homogeneous problem and then argue with $v=u-w$?

Answer 1

Yes, this works, it is a general feature of linear equations (not restricted to pdes).

First find $w$ that satisfies the boundary condition $$ Bw-hw = \psi, $$ and has sufficient regularity, then solve $$ Lv = f - Lw, \ Bv-hv=0 $$ for $v$.

Note that this decomposition approach was necessary to get a weak formulation for the Dirichlet problem, while for Neumann boundary conditions this is not necessary.