Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs

By A. Naber, Daniele Valtorta. 2014

76 citations recorded by the source corpus.

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 singular set is (n–2)–dimensional; however, at this point it has not even been shown that Hn−2(C)<∞ unless the coefficients are smooth. Fundamentally, this is due to the need of an ɛ‐regularity theorem that requires higher smoothness of the coefficients as the frequency increases. We introduce new techniques for estimating the critical and singular set, which avoids the need of any such ɛ‐regularity. Consequently, we prove that if the frequency of u is bounded by Λ, then we have the estimates Hn−2(C(u))≤CΛ2 and Hn−2(S(u))≤CΛ2 , depending on whether the equation is critical or not. More importantly, we prove corresponding estimates for the effective critical and singular sets. Even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature. We also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem.© 2017 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b1259eecc17cd262ff3a92460a3a31b0832e8bca

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In Applied Mathematics

1. A. Naber
2. Wenshuai Jiang
3. Jeff Cheeger
4. G. Tian
5. Andrea Mondino
6. Nicola Gigli
7. Richard Bamler
8. Bing Wang

In Mathematical Physics

1. A. Naber
2. Jeff Cheeger
3. Wenshuai Jiang
4. Daniele Valtorta
5. Bruce Kleiner
6. Ben Andrews
7. Richard Bamler

Citations in the corpus

• Regularity of Einstein manifolds and the codimension 4 conjecture (2014, methodology): “y bounded on compact subsets. The previous two properties are standard. They follow by working in a sufficiently smooth coordinate chart and using the monotonicity of the frequency, see [HL], [CNV12], [NV14]. Note that the frequency is not monotone until one restricts to a sufficiently regular co ordi”
• L2 curvature bounds on manifolds with bounded Ricci curvature (2016, background): “53003 [23] Naber, Aaron; Valtorta, Daniele, Volume estimates on the critical sets of solutions to elliptic {PDE}s, Comm.”