I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper
At page 163, De Giorgi refers to a "Lemma by Caccioppoli and Leray" but I can't find it anywhere, the referenced book is very hard to come across.
If anyone has it ("Equazioni alle derivate parziali di tipo ellittico" by C.Miranda) and can look at what this lemma at page 153 is it would be great.
The inequality I am struggling with, is, in any case:
$$\int_{A(k)\cap B(y,\varrho_2)} (u(x)-k)^2 dx\geq (\varrho_2-\varrho_1)^2 \frac{\tau_1}{\tau_2}\sqrt{\int_{A(k)\cap \partial B(y,\varrho _1)}(u(x)-k)^2d\mu_{n-1}\cdot \int_{A(k)\cap \partial B(y,\varrho _1)}|\nabla u(x)|^2d\mu_{n-1}}$$
where $A(k)$ is the subset of the domain where the solution of the elliptic equation (with constants $\tau_1, \tau_2$) $u(x)$ is greater than $k$, $B(x,r)$ is the n-dimensional ball centred at $x$ of radius $r$ and $\partial$ indicates the boundary.
Thank you very much for any hint, reference or idea!
I was not able to find Miranda's book, however, there is some books in the literature that might be useful too you.
Take a look in Fanghua Lin and Qing Han book page 20. Also Giusti's book is a good reference for regularity theory: take a look in page 214. I think that Giusti's theorem fits more in what you want (in fact I think it is a generalization of De Giorgi theorem).