The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients

Question

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by $$Lu = \partial_i(a^{ij}\partial_ju) + b^i\partial_ku + c u.$$ For an arbitrary $f$, consider the boundary value problem posed in $\Omega$ $$ \begin{cases} Lu = f \\ \text{with boundary conditions}\\ \end{cases} $$ The well-posedness of a weak solution and its regularity is extensively studied.

For instance, in Evan's PDEs we find for Dirichlet boundary conditions

THEOREM 5 (Higher boundary regularity). Let $m$ be a nonnegative integer, and assume $ a^{i j}, b^i, c \in C^{m+1}(\bar{\Omega}) \quad(i, > j=1, \ldots, n) $, and $ \partial \Omega \text { is } C^{m+2} $, and $ f \in H^m(\Omega) . $ Suppose that $u \in H_0^1(\Omega)$ is a weak solution of the boundary-value problem. Then $$ u \in H^{m+2}(\Omega), $$ and we have the estimate $$ \|u\|_{H^{m+2}(\Omega)} \leq C\left(\|f\|_{H^m(\Omega)}+\|u\|_{L^2(\Omega)}\right), $$ the constant $C$ depending only on $m, \Omega$ and the coefficients of $L$.

In Grisvard's Elliptic Problems in Nonsmooth Domains we find Theorem 2.5.1.1 which gives the same result for more general boundary conditions but with $a^{ij} \in C^{m,1}(\overline{\Omega}), b^i \in C^{m,1}(\overline{\Omega})$ and $ \partial \Omega \text { is } C^{m+1,1}.$

My question is, what is the most relaxed conditions one can assume on the coefficients $a^{ij},b^i,c$, and $f$, and still get a similar regularity in Sobolev spaces as given in the previous theorems? The boundary can be assumed to be sufficiently smooth!

Providing references for further reading is much appreciated.

Answer 1

For $m=0$ a very detailed study is given in the same book of Grisvard in Section 5.2. Basically, the diffusion $a^{ij}$ needs enough regularity so that you can use the technique of "freezing the coefficients" (Lipschitz is sufficient), while $b$ can be in $L^\infty$. Then there are geometric conditions on the coefficient $a^{ij}$ that are explained in detail in that section of Grisvard's book. In particular, with Theorem 5.2.2 you can compute the co-dimension that potentially prevents your solution from belonging to $W^{2,p}$. It's similar to working out corner singularities.