An inequality \`a la Szeg\H{o}-Weinberger for the $p-$Laplacian on   convex sets
  In this paper we prove a sharp inequality of Szeg\H{o}-Weinberger type for the first nontrivial eigenvalue of the $p-$Laplacian with Neumann boundary conditions. This applies to convex sets with given diameter. Some variants, extensions and limit cases are investigated as well. 
L. Brasco, C. Nitsch, C. Trombetti
math.AP math.OC