Assume that $L$ is an elliptic operator of the form
$$L(u) = \sum_{i,j} a_{ij} \partial_i \partial_j u + \sum_i b_i \partial_i u$$
where $a_{i,j}, b_i$ are measurable functions on a bounded domain $\Omega\in \mathbb R^3$.
We assume $L$ is elliptic of uniform module $\lambda$, that is $(a_{ij})$ is symmetric for all $x$ and $\sum_{i,j} a_{ij}\xi_i\xi_j > \lambda \sum_i x_i^2$ for every $\xi\in \mathbb R^3$.
Assume furthermore that the boundary of $\Omega$ is a smooth orientable surface (with possibly several connected components $S_k$).
I am looking for a theorem which asserts the existence of solutions to the Dirichlet problem
$$L u = f, \quad u = \varphi \quad on \quad \partial \Omega$$
with $\varphi \in C^\infty(\bar\Omega)$, but without continuity or smoothness assumptions for the coefficient $a_i$ (only measurability and possibly bounding conditions).
More specifically, I expect a theorem that ensures that if there is a $C^2$ solution
to $$L (G(x,x')) = \delta(x-x'), \quad G(x, x') = 0 \quad \forall x\in \partial \Omega,$$
where $x'$ is viewed as a constant for the action of $L$, then there is a solution to the Dirichlet problem above.
With an eye to the usual construction of solutions to the Dirichlet problem for the Laplace equation by Green functions, such a theorem may even be easy to obtain for persons experimented with elliptic equations; or maybe you can provide a reference?
Best Answer
It is possible to construct Green's functions to nondivergence form equations like this, but with caveats:
The Green's function
Let us think through what would be required to construct a Green's function for this PDE. We start with some $\{L_\epsilon\}$ whose coefficients are smooth and converge (say almost everywhere) to the coefficients of $L$ as $\epsilon \rightarrow 0$. Solving $L_\epsilon u_\epsilon = f$ for a smooth $f$ with zero boundary conditions is not a problem; we can move the coefficients inside and write this as a divergence-form equation, for example. Let us call $R_\epsilon(f)$ the linear mapping of $f$ to the corresponding solution $u_\epsilon$.
To construct a Green's function, we want to use the Riesz representation theorem: view $R_\epsilon(\cdot)(x)$ as a linear functional on (say) $L^p$, with $x$ fixed. If we know that it is bounded $$|R_\epsilon(f)(x)|\leq C \|f\|_{L^p}, \qquad \qquad (*)$$ we can write $$ R_\epsilon(f)(x) = \int G_\epsilon(x, y) f(y) dy $$ for some function $G_\epsilon$ with $G_\epsilon(x, \cdot) \in L^{\frac{p}{p - 1}}$, the dual space.
We will return to the estimate $(*)$ shortly. The next step would be to take the limit as $\epsilon \rightarrow 0$. This is immediately problematic, as we need to argue that $G_\epsilon(x, y)$ has some compactness, that $R_\epsilon(f)$ converges to a function $R(f)$ which we can make sense of as a solution, and possibly that $R(f)$ is the unique solution with the given data. We are aided somewhat by getting a better estimate, which reads:
$$|R_\epsilon(f)(x) - R_\epsilon(f)(y)|\leq C \|f\|_{L^p}|x - y|^\alpha \qquad \qquad (**)$$
for some $\alpha > 0$. This lets you pass to the limit in $R_\epsilon(f)(\cdot)$ (uniform convergence in $x$) for each fixed $f$, recovering a function $R(f)(\cdot)$ satisfying the same estimates. The Riesz representation theorem can again be used to write $R$ as $R(f)(x) = \int G(x, y) f(y) dy$. The remaining questions, then, are: is $R(f)$ a "meaningful" solution to the PDE, and is it unique?
Unfortunately, it's not clear that $R(f)$ is $C^2$, or has two bounded derivatives: indeed, if the coefficients are only assumed to be bounded, it does not need to be (i.e. there are counterexamples where $R(f)$ fails to have bounded, or $L^p$, or $L^1$, second derivatives; they do exist distributionally in $L^\delta$ for a small $\delta$ by a theorem of F. Lin). We also cannot use distributional notions of solution, as there's no way to integrate by parts on to a test function, you will always have derivatives hitting the coefficients when you try. So possible options are: (1) just stick with $R(f)$ as a solution in the approximating sense that $R_\epsilon(f) \rightarrow R(f)$, (2) talk about solutions in $W^{2, \epsilon}$ which satisfy the PDE a.e., (3) talk about viscosity solutions, which is a concept of weak solution based on the maximum principle. As for uniqueness: it's not known in this generality for any of these classes of solution.
The estimates
The estimates $(*), (**)$ are best possible in $L^\infty$-like spaces. $(*)$ is the Aleksandrov-Backel'man-Pucci estimate, $(**)$ is due to Krylov and Safonov. Both are valid with $p = n$, or with more work for $p > n - \delta$ for a small $\delta$ depending on the ellipticity constants ($n$ is the dimension).
As mentioned, the best "Sobolev space"-type estimate places $P(f) \in W^{2, \delta}$ for some small $\delta$, due to F. Lin.
There are some additional estimates in the $y$ variable on $G$; these are usually of integral type and based on studying the adjoint equation ($L^* u = \partial_{ij} (a_{ij} u)$). The ones I am familiar with are due to P. Bauman, but am not an expert on the topic.
If you assume that $a_{ij}$ are continuous (or have small oscillation), you get a lot more estimates, including ones which can ensure uniqueness.
The books of Gilbarg Trudinger and Caffarelli Cabre are both reasonable references on this topic.