Quasilinear elliptic equations with discontinuous coefficients

Boccardo, Lucio ; Buttazzo, Giuseppe

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 1, pp. 21-28

Résumé

We prove an existence result for equations of the form $$\begin{cases} - D_{i} (a_{ij}(x,u) D_{j}u) = f \quad \text{in} \, \Omega\\ \in H_{0}^{1}(\Omega). \end{cases}$$ where the coefficients $a_{ij}(x,s)$ satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable $s$. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients $a_{ij}(x,s)$ are supposed only Borel functions