The term "Standard elliptic theory" loosely designates the set of results obtained mainly during 1950-1960's on linear elliptic equations. Sometimes it is mentioned in the context of nonlinear equations, but it would usually mean that the result is a straightforward consequence of the "linear standard elliptic theory". The theory has two popular versions, that respectively take place in Hölder spaces and in Sobolev spaces. In general it can be described by the following scheme.
Suppose that $\Omega$ is a bounded domain, $A$ is a linear elliptic operator on $\Omega$, and $\{B_j:j=1,\ldots,n\}$ are collection of linear operators on the boundary. Then we consider the boundary value problem $Au=f$ and $B_ju=g_j$, ($j=1,\ldots,n$).
To proceed, one has to choose:
- Scale $X^s$ of functions spaces on $\Omega$, and
- Scale $Y^s$ of functions spaces (typically of the form $Y=Y^s_1\times\cdots\times Y_n^s)$ on $\partial\Omega$, satisfying
$$
A:X^s\to X^{s-2m},\qquad\textrm{and}\qquad B_j:X^s\to Y_j^s,\qquad\textrm{are bounded.}
$$
Then one proves (for some range of $s$):
- Elliptic estimates:
$$
\|u\|_{X^s}\lesssim\|{Au}\|_{X^{s-2m}}+\|{B_1}\|_{Y_1^s}+\cdots+\|{B_mu}\|_{Y^s_n}+\|{u}\|_{X^0}.
$$
- Elliptic regularity:
$$
Au\in X^{s-2m},\ \{B_ju\}\in Y^s\iff u\in X^s.
$$
- Fredholm property:
$$
u\mapsto (Au,\{B_ju\}):X^s\to X^{s-2m}\times Y\qquad
\textrm{is Fredholm}.
$$
i.e, its kernel and cokernel are finite dimensional.
Here $2m$ is usually the order of the operator, but if you want to go beyond usual elliptic operators sometimes one has suboptimal results. Of course, the theory itself has to decide what operators $A$ it can include, and what compatibility conditions have to be imposed on the operators $A$ and $\{B_j\}$.
Also, if coefficients of $A$ and $B_j$ are not smooth, the theory will be valid for some limited range of $s$.
A prototypical example is Schauder's theory for second order elliptic equations in Hölder spaces.
Such a theory for very general elliptic systems in Hölder and Sobolev type spaces was established in 50'-60's. cf. Agmon-Douglis-Nirenberg 1959, and 1964.
There is also a version of the theory for strongly elliptic systems, that assumes a bit more but delivers stronger results. This was worked out by Garding, Nirenberg, Agmon, et al.
Finally got the answer in the book of Grisvard Elliptic Problems in Nonsmooth Domains. First we find a solution $u_1$ solving $\Delta u_1=-u_1$ on $\Omega$ and $\partial u_1/\partial n = -g(u)$ by Corollary 2.2.2.6 there. Then, we consider the equation $u_1-u$ and find that the boundary is zero. Next, we apply the difference quotient method to obtain the $H^2$ regularity. Bootstrapping results are referred to Theorem 2.5.1.1 there.
Best Answer
If you expand the divergence in $Du$, you get $$- \nabla a \cdot \nabla u - a \Delta u = Du$$ Rearranging yields $$\Delta u(x) = - \frac{1}{a(x)} \nabla a(x) \cdot \nabla u(x) - \frac{1}{a(x)} Du(x)$$ Since $a$ is smooth and bounded away from $0$, we conclude using Holder's inequality that both terms on the right are in $L^2$. In particular, their norms in $L^2$ are controlled by $$\left\lVert \frac{1}{a(x)} \nabla a(x) \cdot \nabla u(x) \right\rVert_{L^2} \leq \left\lVert \frac{1}{a} \right\rVert_{L^\infty} \lVert \nabla a \rVert_{L^\infty} \lVert u \rVert_{H^1},$$ $$\left\lVert \frac{1}{a(x)} Du(x) \right\rVert_{L^2} \leq \left\lVert \frac{1}{a} \right\rVert_{L^\infty} \lVert Du \rVert_{L^2}$$ Thus, $\Delta u \in L^2(\Omega)$, and elliptic regularity puts $u \in H^2$.