Elliptic PDEs in Banach space The standard textbooks discuss weak solutions and regularities in Hilbert spaces $W^{k,2}.$ I could not find a good reference on the theory based on Banach spaces $W^{k,p}.$ It would be good to point out some references to me.
My motivation is to study the following problem
$$
\nabla\cdot(a\nabla u) = \delta(x), \mbox{ in }\Omega\subset\mathbb{R}^d,
$$
with homogeneous Dirichlet boundary condition. Since the $\delta$ function is not in $H^{-1},$ one approach is to switch to seek $u$ in $W_0^{1,p}$ ($1<p<2$ such that $p'=(1-p^{-1})^{-1}$ satisfies $1-dp'^{-1}>0$) in which case $\delta$ is in the dual space $(W_0^{1,p})'$ isometric to $W_0^{1,p'}.$ This is suggested by this paper. 
Now the question is do we have regularity away from $\mathbf{x}=0$? Yes, it depends on the smoothness of $a$ (which we assumes, of course, bounded away from 0). I found a nice lecture notes dealing with some related issues in Banach spaces. But I do not know


*

*Can we improve the regularity $W^{1,p}$ away from $\mathbf{x}=0,$ for bounded $a$?

*If $a$ is Lipschitz continuous, what further we can have? 

 A: $\bullet$ For a bounded $\Omega\subset\mathbb{R}^d$ with $\partial\Omega\in C^1$, the dual space $(W_0^{1,p})'$ will be isometric to $W_0^{1,p'}$ if $a\in C(\overline{\Omega})$ with the $L_p$-theory being rather trivial, since an elliptic operator $L={\rm div}(a\nabla\cdot)$ with $a\in C(\overline{\Omega})$ in this context is equivalent to the Laplacian. When $a$ is not continuous, the $L_p$-theory becomes rather nontrivial, while the isometry generally fails outside certain neighbourhood of $p=2$. For details see "Elliptic and Parabolic Equations with Discontinuous Coefficients" by A.Maugeri, D.K.Palagachev, L.G. Softova and references therein. For instance, in the simplest case of $d=2$ every angular point on discontinuity line of $a$ might be a singular point of solution with RHS in  $C_0^{\infty}(\Omega)$, as well as every intersection point of $\partial\Omega$ with a smooth line of discontinuity of $a$. 
$\bullet$ For a bounded domain $\Omega\subset\mathbb{R}^d$ with $\partial\Omega\in C^1$, if $a$ is Lipschitz, the answer coincides with that for the Lalpacian in case $\Omega=\mathbb{R}^d$, i.e., solution $u\in W^{s,p}$ where $0<s-1<1-\frac{d}{p'}$ which is readily established using the standard PDE $L_p$-theory techniques. The problem with this techniques is that it still largely stays within Mathematical Folklore, i.e. already widely known, but not yet to be found in textbooks.
A: The following paper: "Optimal regularity for elliptic transmission problems including C1 interfaces" by Elschner, Rehberg and Schmidt, seems to cover the problem you are considering.
Another good starting point for you could be the paper "A W1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations" by Konrad Gröger. It deals with fairly general elliptic PDEs, but for the case $p>2$. However you can also obtain results for the case $p<2$ by considering the adjoint of the operator discussed in the paper. 
You will also find useful informations in the literature on optimal control of PDEs with pointwise constraints in Banach spaces. See for instance pages 343 to 345 of the book "Optimal Control of Partial Differential Equations: Theory, Methods and Applications" by Fredi Tröltzsch.
