  The underlying motivation of the present work lies on one of the cornerstone
problems in spectral optimization that consists of determining sharp lower and
upper uniform estimates for fundamental frequencies of a set of uniformly
elliptic operators on a given membrane. Our approach allows to give a complete
solution of the problem for the general class of anisotropic operators in
divergence form generated by arbitrary norms on R^2, including the computation
of optimal constants and the characterization of all anisotropic extremizers.
Such achievements demand that isoanisotropic type problems be addressed within
the broader environment of nonnegative, convex and 1-homogeneous anisotropies
and involve a deep and fine analysis of least energy levels associated to
anisotropies with maximum degeneracy. As a central outcome we find out a key
close relation between shapes and fundamental frequencies for rather degenerate
elliptic operators. Our findings also permit to establish that the supremum of
anisotropic fundamental frequencies over all fixed-area membranes is infinite
for any nonzero anisotropy. This particularly proves the well-known
maximization conjecture for fundamental frequencies of the p-laplacian for any
p other than 2. Our new sharp lower estimate is just the planar isoanisotropic
counterpart of the Faber-Krahn isoperimetric inequality and for the associated
optimal constant we provide optimal geometric controls through isodiametric and
isoperimetric shape optimization.
