Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs

TL;DR

This paper develops new monotonicity and classification results for singular, degenerate, and anisotropic elliptic PDEs, advancing understanding of their solutions and singularities even in non-degenerate cases.

Contribution

It introduces novel monotonicity formulas and classification results for complex elliptic PDEs with anisotropy, singularity, and degeneracy, which were previously unexplored.

Findings

New monotonicity results for energy density

Rigidity results for solutions

Classification of singularities and degeneracies

Abstract

We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classification results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non-degenerate anisotropic equations.