On spectral optimization for p-Laplace type operators

Abstract

Sharp uniform estimates, both lower and upper, are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. Optimal constants are computed, rigidity of the upper estimate is derived, anisotropic attainability of the lower constant is established, along with the characterization of corresponding anisotropic extremizers for certain membranes. Sharp quantitative inequalities associated with lower constants are also proved, thereby providing anisotropic stability. The optimal lower constant remains positive even when the condition of uniform strict ellipticity of the operators is removed. Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the two-dimensional Faber-Krahn isoperimetric inequality.

arXiv comment20 pages, comments are welcome