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In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operatorΔ  p   u  =   ∑  i    ∂   ∂   x  i|  ∇  u  |    p  −  2∂  u    ∂   x  i, that is the  p ‐Laplacian, andΔ  ̃   p   u  =   ∑  i    ∂   ∂   x  i|∂  u    ∂   x  i|    p  −  2∂  u    ∂   x  i, namely the pseudo‐ p ‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic  p ‐torsional rigidity.

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators