Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs

@article{Naber2014VolumeEO,
  title={Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs},
  author={A. Naber and D. Valtorta},
  journal={Communications on Pure and Applied Mathematics},
  year={2014},
  volume={70},
  pages={1835-1897}
}
In this paper we study solutions to elliptic linear equations L(u)=∂i(aij(x)∂ju)+bi(x)∂iu+c(x)u=0, either on ℝn or a Riemannian manifold, under the assumption that the coefficient functions aij are Lipschitz bounded. We focus our attention on the critical set C(u)≡{x:| ∇u |=0} and the singular set C(u)≡{x:| ∇u |=0}, and more importantly on effective versions of these. Currently, with just the Lipschitz regularity of the coefficients, the strongest results in the literature say that the… Expand
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