What is special about the function $f+\nabla\cdot g$ for $f,g\in L^1_\text{loc}(\Omega)$ in elliptic PDEs?

Question

The following is an excerpt in Gilbarg and Trudinger's Elliptic PDE of Second Order (the beginning of Chapter 8):

where and .

Motivated by (8.3), here are my questions:

• What is special about functions of the form $g+\nabla\cdot f$? Why is it of particular interest for elliptic PDE?
• If one wants to apply the Hilbert space techniques, the right hand side of the elliptic equation is usually assumed to be a $L^2$ or $H^{-1}$ functions, as I observe from Evans's Partial Differential Equations. What would be a possible overlap between such consideration and the one in (8.3)?

Answer 1

Such a function is of interest, since for $l\in H^{-1}$, it is possible to find $g$ and $f$ in $L^2$ such that $l(v)=\int_{\Omega}f^i D_i v+gv dx$ for all $v\in H_0^{1}$, this is induced by Riesz representation for the Hilbert space $H_0^{1}$ with inner product $<u,v>=\int_{\Omega} uv+Du\cdot Dv dx$.