Critical Sets of Elliptic Equations

By Jeff Cheeger, A. Naber, Daniele Valtorta. 2012

101 citations recorded by the source corpus.

Given a solution u to a linear, homogeneous, second‐order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set 𝒷(u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set 𝒷r(u), which roughly consists of points x such that the gradient of u is small somewhere on Br(x) compared to the nonconstancy of u. The results are new even for harmonic functions on ℝn. Given such a u, the standard first‐order stratification {lk} of u separates points x based on the degrees of symmetry of the leading‐order polynomial of u‐u(x). In this paper we give a quantitative stratification { ln,rk } of u, which separates points based on the number of almost symmetries of approximate leading‐order polynomials of u at various scales. We prove effective estimates on the volume of the tubular neighborhood of each { ln,rk } , which lead directly to (n‐2 + ɛ)‐Minkowski type estimates for the critical set of u. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to give new proofs of the uniform (n‐2)‐Hausdorff measure estimate on the critical set and singular sets of u.© 2014 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d94baf394a6a2f3dbac9b9f23c03ad3de8008ec1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs (2014, methodology): “We will follow the mild extensions given in [3], which are discussed in Section 4.”