\L ojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE

TL;DR

This paper demonstrates that for a wide class of elliptic PDEs, the zero sets of solutions are generically smooth, and introduces an effective Lojasiewicz inequality to support this result.

Contribution

It proves generic smoothness of nodal sets for elliptic PDE solutions and establishes a uniform Lojasiewicz gradient inequality for solutions with bounded frequency.

Findings

Zero sets are smooth for generic boundary data.

Small perturbations can eliminate singularities in zero sets.

Established an effective Lojasiewicz inequality with uniform constants.

Abstract

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. Furthermore, we can take the perturbation to be "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.