I'm trying to track the behaviour of the coefficients in Theorems 8.22 and Theorem 8.24. Particularly, I'm considering the behaviour w.r.t. to the distance from $\Omega'$ to $\partial \Omega$
I'll state an abbreviated version of theorem 8.22
Theorem 8.22 If $u\in W^{1,2}(\Omega)$ is a weak solution of a linear elliptic pde in divergence form in $\Omega$, then $u$ is locally Hölder continuous in $\Omega$ and for any Ball $B_{R_0}(y)\subset \Omega$ and $R\leq R_0$ we have
$$ \text{osc}_{B_R(y)}(u)\leq CR^\alpha(R_0^{-\alpha} \sup_{B_{R_0}(y)} |u| +k)$$
where $C=C(R_0)$ and $\alpha=\alpha(R_0)$
Theorem 8.24 gives an estimate on the Hölder norm for compact sets $\Omega'\subset\subset \Omega$ of the form
$$||u||_{C^\alpha(\Omega')}\leq C (||u||_{L^2(\Omega)}+k)\tag{1}$$
where $C$ and $\alpha$ both depend on the distance of $\Omega'$ to $\partial \Omega$ and $k$ doesn't bother me :).
What I discovered, when proving the local Hölder continuity of $u$ in Theorem 8.22 from the oscillation estimate is that the $C$ for the estimate (1) also depends on the size of $\Omega '$. Has anybody already considered this?
Why do I believe this? Let $\Omega'\subset \Omega$ and choose $R_0$ so small that $B_{R_0}(\Omega')\subset \Omega$ (Hence the dependence on the distance of the boundary).
The oscillation estimates does only hold on balls. Hence, due to compactness we find a finite covering of $\Omega'$ consisting of balls $B_{R}(x_k)$, $k=1,…n$.
To show local Hölder continuity fix $x,y\in\Omega'$
and compute (in the worst case, i.e. x,y not in the same ball, i.e. $|x-y|>R$) and $1\leq l\leq n$,
$$|u(x)-u(y)|\leq |u(x)-u(x_1)|+ … + |u(x_l)-u(y)| \leq C lR^{\alpha}\leq C n|x-y|^\alpha \tag{2}$$
(I negelected $k$
Now $n$ depends obviously on the covering, so in particular on the radius $R$ which depends on $R_0$. However, the number $n$ depends on the size of $\Omega'$ and $R$.
So the question is: Is there another way to show estimate $(2)$ without using the size of the domain? Is my covering argument wrong?
Edit: Or is there another way to show the local Hölder continuity, starting from the oscillation estimate?
Edit2: In addition, it is stated that the constants $C$ and $\alpha$ are independent of $R_0$ and $R$ in the case of the Laplace-operator and that consequently the constant $C$ in $(1)$ shall be independent of the shape $\Omega$ and $\Omega'$. Does this sound reasonable?
Best Answer
No chain of balls is needed in the case $|x-y|\ge R$. Just use the estimate $$ \frac{|u(x)-u(y)|}{|x-y|^\alpha}\le 2 R^{-\alpha} \sup_{\Omega'}|u| $$ The appearance of $R^{-\alpha} $ is not a problem, since it only depends on $\alpha$ and on the distance of $\Omega'$ to $\partial \Omega$. And $\sup_{\Omega'}|u|$ is controlled by $\|u\|_{L^2(\Omega)}$ by virtue of Theorem 8.17. (We had to control $\sup_{\Omega'}|u|$ anyway, because it's a part of $C^\alpha$ norm.)
For your Edit2: no, $C$ in (1) cannot be independent of the domain even for harmonic functions. E.g., take $u\equiv 1$ on $B(0,r)$ where $r$ is small. Then $k=0$ (homogeneous equation), and $L^2$ norm is small, but $C^\alpha$ norm is $1$. Simply put, the two sides of (1) do not scale in the same way when the domain is rescaled. So, $C$ cannot be an absolute constant.
I think you misread what the authors wrote. They actually claim that the following constants are domain-independent when $\nu=0$:
What they claim is correct. Note how (8.46) and (8.47) are set up with powers of $R$, to make both sides scale in the same way. And (8.63) is Harnack's inequality. For harmonic functions, $\alpha$ in (8.65) and (8.68) can be taken to be $1$, because we can uniformly control gradient by the size of the function.
By the way, $\nu=0$ does not mean we have Laplace's equation. The PDE has to be of the form $\operatorname{div}(A(x)\nabla u)=0$ (no lower order terms), but the coefficient matrix need not be identity. The Hölder exponent $\alpha$ can be estimated purely in terms of the ellipticity parameters, with no references to domains. The constant $C$, though, is affected by scaling as explained above.
One more thing: if you are interested in good oscillation estimates for harmonic functions specifically, it's best to deal with the Laplace equation specifically, instead of using results for general elliptic equations.