Let $B=B_1(0)\subset {\mathbb{R}}^n$ be a unit ball and let $u\in C^2(B)\cap C(\bar B)$ be a solution of $\Delta u=f$ in $B$; $u=0$ on $ \partial B$, where $f\in C^{\alpha}(B)$ with $0<\alpha<1$. Then $u\in C^{2, \alpha}(B)$ and $${||u||}_{C^{2, \alpha}(B)}\le C(n,\alpha)({|u|}_{0;B}+{|f|}_{0,\alpha;B}).$$
I am trying to prove by the following steps:(of course may not be a good idea)
(1) Let $x_0\in \partial B$. By translation and rotation we may assume $x_0=0$. Consider the Kelvin Transformation$$y=F(x):=\frac{x}{{|x|^2}}.$$ Then $F$ maps spheres to spheres but with an inverse, so denote by $B^{*}$ the image of $B$ under $F$.
(2)Define $$v(y)={|y|}^{2-n} u(\frac {y}{{|y|}^2}), y\in B^{*}.$$Then $v$ satisfies ${\Delta}_y v={|y|}^{-2-n}f(\frac{y}{{|y|}^2})$ in $B^{*}$ and $v=0$ on $\partial B^{*}$.
(3)Now I want to use the estimates on boundary for $v$ to get one for $u$. So the one I already know is:$${|v|}_{2,\alpha;B\cup T}^{*}\le C(n,\alpha)({|v|}_{0;B}+{|f|}_{0,\alpha;B\cup T}^{(2)})$$ for a boundary portion $T\subset \{x\in \mathbb{R}^n:x_n=0\}$. And I am guessing(trying to prove) that there exists a $\delta>0$ independent of $x$, such that $${|u|}_{2,\alpha;B_\delta(x)\cap B}\le C(n,\alpha)({|u|}_{0;B}+{|f|}_{0,\alpha;B}).$$
(4) I am trying for the one above because the Hörder norm is gained by summing up the seminorms. So I believe this will give an estimate for ${||u||}_{C^{2, \alpha}(B)}$.
But I am stuck at the 3rd step, so that I cannot have a global estimate for the seminorm of $u$ on boundary, thus cannot move on to the last step. So I need support about my last two steps, since maybe my guess does not hold at all. Also if you have some different idea of solving this problem, please post too. Thank you very much!