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


[De Giorgi paper](http://books.google.co.uk/books?id=80wlU42UC9IC&pg=PA149&lpg=PA149&dq=on%20the%20analyticity%20of%20extremals%20regular%20multiple%20integrals&source=bl&ots=zyv2HxcPlr&sig=WkLspzC3Ze9i1SVszK0wwT_JOvE&hl=en&sa=X&ei=zgPLUavTHYikPZXygNgE&ved=0CC0Q6AEwAA)


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!



Caccioppoli-Leray Inequality for De Giorgi's theorem proof
I made a trip to the library and scanned the relevant pages from Miranda's 1955 book:


[page 152-153](http://ilorentz.org/beenakker/MO/Miranda_1.jpg) and [page 154-155](http://ilorentz.org/beenakker/MO/Miranda_2.jpg)


the references are:


[3] J. Leray, J.Math. pures et appl. **17**, 89-104 (1938)


[8] R. Cacciopoli, Rend.Acc.Lincei **22**, 305-310 and 376-379 (1935).


[11] R. Cacciopoli, Giorn.Mat.Battaglini, **80**, 186-212 (1950-51)


