Elliptic differential inclusions and applications to implicit equations

This thesis is devoted to the study of different types of elliptic differential inclusions and their applications to a wide range of implicit equations. We first introduce some general definitions and properties of set-valued analysis, the basic notions of lower semicontinuous multifunction with decomposable values and selection, in particular we make use of the Kuratowski and Ryll-Nardzewski Theorem and the Bressan-Colombo-Fryszkowski Theorem. Then, we investigate some p-Laplacian differential inclusions with a right hand-side both lower semicontinuous and upper semicontinuous in order to show some applications to implicit differential equations. Moreover, we present a different approach to differential inclusions, based on variational methods and locally Lipschitz continuous functions.