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?
$\bullet$ 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.