Reference request: Constant Hölder estimates

Question

I am reading the book "Elliptic partial differential equations of second order" by D. Gilbarg and N. S. Trudinger.

Specifically, I am interested in Hölder regularity estimates for solution of elliptic problems in divergence form with Hölder coefficients on a domain whose boundary is smooth ($C^2$ for example).

Theorem 8.33 p 210 of this book is exactly what I am looking for. However, the right-hand side of the inequality depends on a constant which depends on the Hölder norms of the coefficients in a non-explicit way. I am looking for any references (or argument) which explicit this constant. In particular, I would like to know what is the dependency on the Hölder norms of the coefficients of the elliptic problem.

For simplicity, I am restating this inequality: \begin{align} |u|_{1,\alpha}\leq C (|u|_0+|g|_0+|f|_{0,\alpha}) \end{align} where $u$ is a $C^{1,\alpha}(\overline{\Omega})$ solution of the elliptic problem \begin{align} L(u)=g+D_if^i, \end{align} with $u=0$ on the boundary, $L=D_i(a^{i,j}(x)D_ju)$, $\max |a^{i,j}|_{0,\alpha}=K<+\infty$ and $C>0$ depending on $K$. I want to know the dependency on $K$ of $C$.

Thanks in advance.

Answer 1

You can get the dependency from a scaling argument. For simplicity, I will discuss the interior (not boundary) version of the estimate. But you can think about boundary estimates in a similar way.

Let me first slightly restate this estimate in a helpful way. Let's also say that $K$ is just the $C^{0,\alpha}$ seminorm of the coefficients, not the full norm, $d$ is the dimension, and $\Lambda$ is ellipticity.

Now, the restatement is: the same estimate holds, in the unit ball (or with $B_{1/2}$ on the left and $B_1$ on the right, as usual), with $C$ depending only on $(d,\Lambda)$ and in particular not on $K$, provided that $K \leq \delta_0$ for some $\delta_0$ depending only on $(d,\Lambda)$. This restated estimate is what the proof should actually be giving you, because let's remember that Schauder estimates are just perturbations off Laplacian. (On the other hand, I cannot remember the argument in G&T, so if not, see the book of Han and Lin instead which does prove this statement.)

Now, by a simple scaling argument, one can restate this once more, without restriction on $K$, but only valid in balls which have a sufficiently small radius. The radius should be $K^{-\alpha}$ or smaller, I think. (The constant $C$ will now depend on $K$, because of the scaling of the norms.)

Now get the general result by covering the whatever domain you have with small balls of this radius, which produces (another) explicit dependence on $K$ and the geometry of the domain.