# Caccioppoli-Leray Inequality for De Giorgi's theorem proof

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

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!

-

## 1 Answer

I made a trip to the library and scanned the relevant pages from Miranda's 1955 book:

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)

-
thank you very much! – Moritzplatz Jun 27 '13 at 12:01