Consider an elliptic PDE such as $$\mbox{tr}(A(x)D^2 u(x)) = f(x), \quad x \in B_1\subset\mathbb R^n,$$ where $A(x)$ is symmetric and $0<\lambda I \leq A(x) \leq \Lambda I$ for all $x$. Let us assume $f$ and $A$ are smooth. Many a priori estimates of the form $\|u\|_X \leq C(\lambda,\Lambda)\|f\|_Y$ or $\|u\|_X \leq C(\lambda,\Lambda)(\|f\|_Y + \|u\|_Z)$ for some norms $X$, $Y$, $Z$ can be derived.
My question: is there a general method for determining the dependence of the constant $C$ on the ellipticity constants $\lambda$ and $\Lambda$, or is it necessary to dig into the proof of the particular estimate?
In the case where $A$ is a constant matrix, the sketch would be something like this: divide the equation by $\lambda$ and find a matrix $M$ such that $M^2 = A/\lambda$. Letting $v(x) = u(Mx)$, we have $$\Delta v = \mbox{tr}(M^2 D^2u) = \mbox{tr}((A/\lambda) D^2u) = f(x) / \lambda.$$ Applying the estimate to $v$, we obtain $\|v\|_{X} \leq C\|f/\lambda\|_{Y}$, with $C$ independent of $\lambda$ and $\Lambda$. Finally, we change variables back to $u$, which gives us a factor of $\sqrt{\Lambda/\lambda}$ to some power (depending on the scaling of the norms), since $M$ satisfies $I \leq M \leq \sqrt{\Lambda/\lambda} I$. (The norms are now over $M(B_1)$ instead of $B_1$, but we can fix this by rescaling $u$, at the cost of another power of $\sqrt{\Lambda/\lambda}$ which we can compute explicitly.)
But in the variable-coefficient case, one would need a different change of variables at every point, which would lead to cross terms in the equation for $v$. Alternatively, one could fix a point $x_0$, rewrite the equation as $$\mbox{tr}(A(x_0)D^2 u) = f(x) - \mbox{tr}((A(x) - A(x_0)) D^2 u),$$ and apply the constant coefficient case, but our estimate becomes $$\|u\|_X \leq C(\lambda,\Lambda) (\|f\|_Y + \|(A(x_0) - A(x))D^2u\|_Y),$$ where we now understand how $C$ depends on $\lambda$ and $\Lambda$, but we have another term. There may be interpolation tricks to roll this term into the others, but this depends on the estimate, and would seem to be be unavailable in an $L^2$-$L^\infty$ estimate like $\|u\|_{L^\infty(B_{1/2})} \leq C(\|f\|_{L^\infty(B_1)} + \|u\|_{L^2(B_1)})$. Are there more general tricks, or do we need to redo the original proofs and be careful about our $\lambda$s and $\Lambda$s?