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Geometric Analysis Seminar

Generic Smoothness for the nodal sets of solutions to the Dirichlet problem for Elliptic PDE

Max Engelstein (UMN)

Location:  zoom
Date & time: Tuesday, 14 September 2021 at 2:50PM - 3:50PM

Abstract: 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” data. 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 Lojasiewicz gradient inequality with uniform constants in the class of solutions with bounded frequency. This is joint work with M. Badger (UConn) and T. Toro (U. Washington/MSRI).