Calculus of Variations and Geometric Measure Theory

W. Borrelli - R. Wu

On the nodal set of solutions to Dirac equations

created by borrelli on 14 Dec 2023
modified on 20 Dec 2023


Submitted Paper

Inserted: 14 dec 2023
Last Updated: 20 dec 2023

Pages: 34
Year: 2023

ArXiv: 2312.07918 PDF


Motivated by various geometric problems, we study the nodal set of solutions to Dirac equations on manifolds, of general form. We prove that such set has Hausdorff dimension less than or equal to $n-2$, $n$ being the ambient dimension. We extend this result, previously known only in the smooth case or in specific cases, working with locally Lipschitz coefficients. Under some additional, but still quite general, structural assumptions we provide a stratification result for the nodal set, which appears to be new already in the smooth case. This is achieved by exploiting the properties of a suitable Almgren-type frequency function, which is of independent interest.