Integral inequalities and summability of solutions of some differential problems

The aim of this note is to indicate how inequalities concerning the integral of ${\displaystyle {|\nabla u|}^2}$ on the subsets where |u(x)| is greater than k (${\displaystyle k\in I{R}^{+}}$) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of ${\displaystyle {|\nabla u|}^2}$ on the subsets where |u(x)| is less than k (${\displaystyle k\in I{R}^{+}}$) or where k ≤ |u(x)| < k+1 were used in order to prove estimates in Sobolev spaces larger than ${\displaystyle W^{1,2}\_\left\{0\right\}(\Omega )}$ for solutions of Dirichlet problems with irregular data.