Critical sets of elliptic equations

Jeff Cheeger, Aaron Naber, Daniele Valtorta

Research output: Contribution to journalArticle

Abstract

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{script}(u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set b{script}r(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+e{open})-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.

Original languageEnglish (US)
Pages (from-to)173-209
Number of pages37
JournalCommunications on Pure and Applied Mathematics
Volume68
Issue number2
DOIs
StatePublished - Feb 1 2015

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Critical Set
Elliptic Equations
Polynomials
Harmonic functions
Estimate
Stratification
Symmetry
Second Order Elliptic Equations
Singular Set
Polynomial
Hausdorff Measure
Coefficient
Harmonic Functions
Lipschitz
Regularity
Gradient
First-order

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Critical sets of elliptic equations. / Cheeger, Jeff; Naber, Aaron; Valtorta, Daniele.

In: Communications on Pure and Applied Mathematics, Vol. 68, No. 2, 01.02.2015, p. 173-209.

Research output: Contribution to journalArticle

Cheeger, Jeff ; Naber, Aaron ; Valtorta, Daniele. / Critical sets of elliptic equations. In: Communications on Pure and Applied Mathematics. 2015 ; Vol. 68, No. 2. pp. 173-209.
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