Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs
| Autor: | Daniele Valtorta, Aaron Naber |
|---|---|
| Přispěvatelé: | Naber, A, Valtorta, D |
| Rok vydání: | 2017 |
| Předmět: |
Mathematics - Differential Geometry
Smoothness Pure mathematics Applied Mathematics General Mathematics 010102 general mathematics Elliptic PDEs critical set Riemannian manifold Lipschitz continuity 01 natural sciences Set (abstract data type) Mathematics - Analysis of PDEs Differential Geometry (math.DG) Bounded function 0103 physical sciences FOS: Mathematics Point (geometry) 010307 mathematical physics 0101 mathematics Eigenvalues and eigenvectors Linear equation Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Communications on Pure and Applied Mathematics. 70:1835-1897 |
| ISSN: | 0010-3640 |
| DOI: | 10.1002/cpa.21708 |
| Popis: | In this paper we study solutions to elliptic linear equations $L(u)=\partial_i(a^{ij}(x)\partial_j u) + b^i(x) \partial_i u + c(x) u=0$, either on $R^n$ or a Riemannian manifold, under the assumption of Lipschitz control on the coefficients $a^{ij}$. We focus our attention on the critical set $Cr(u)\equiv\{x:|\nabla u|=0\}$ and the singular set $S(u)\equiv\{x:u=|\nabla u|=0\}$, and more importantly on effective versions of these. Currently, under the coefficient control we have assumed, the strongest results in the literature say that the singular set is n-2-dimensional, however at this point it has not even been shown that $H^{n-2}(S) |
| Databáze: | OpenAIRE |
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