Abstract. We present a variant of the $p-$Laplacian operator, which arises as the first variation of a suitable Dirichlet integral. The corresponding elliptic equation is much more degenerate and singular than that for the standard $p-$Laplacian operator and regularity of the gradient of solutions appears to be a difficult issue.
We will show some regularity results (differentiability, boundedness and continuity).
The results presented are contained in some papers in collaboration with Pierre Bousquet (Toulouse), Guillaume Carlier (Paris Dauphine), Vesa Julin (Jyvaskyla), Chiara Leone (Napoli), Giovanni Pisante (Caserta) and Anna Verde (Napoli).