Łojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE

Matthew Badger, Max Engelstein, Tatiana Toro

Published 2021-08-18Version 1

In this article, we prove that for a broad class of second order elliptic PDEs, including the Laplacian, the zero sets of solutions to the Dirichlet problem are smooth for "generic" $L^2$ data. When the zero set of a solution (e.g. a harmonic function) contains a singularity, this means that we can find an arbitrarily small perturbation of the boundary data so that the zero set of the perturbed solution is smooth throughout a prescribed neighborhood of the former singularity. When the PDE has no zeroth order term, we can ensure the perturbation is "mean zero" for which there are additional technical difficulties to ensure that we do not introduce new singularities in the process of eliminating the original ones. Of independent interest, in order to prove the main theorem, we establish an effective version of the \L ojasiewicz gradient inequality with uniform constants in the class of solutions with bounded frequency.