Optimal anisotropies for p-Laplace type operators in the plane

Sharp lower and upper uniform estimates are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. Optimal constants are exhibited, rigidity of the upper estimate is proved, anisotropic attainability of the lower estimate is derived as well as characterization of anisotropic extremizers for circular and rectangular membranes. Sharp quantitative anisotropic inequalities associated with lower constants are also established, providing as a by-product information on anisotropic stability. When the uniform ellipticity condition is relaxed, we show that the optimal lower constant remains positive, while anisotropic extremizers no longer exist. Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the Faber-Krahn isoperimetric inequality in the plane.