Singularities and Smoothness in Geometric Partial Differential Equations

  • Naber, Aaron (PD/PI)

Project: Research project

Project Details

Description

Overview The goal of the proposal is to study various geometrically motivated equations and their applications. The proposal mostly centers around ideas involving nonlinear harmonic maps, Yang-Mills, Ricci curvature, and Reifenberg analysis. However, the proposal will also touch on spectral analysis, stochastic analysis and metric-measure spaces. In all there are nine projects with eight coauthors. Each project will discuss first progress expected to be made over the next year, and then goals past that. Intellectual Merit The first part of the proposal is focused on the study of nonlinear harmonic maps and their singularities. In [64] new techniques were introduced in order to study the structure of the singular set nonlinear equations, and one would like to continue to push forward and build in new ways off this structure. The first project is concerned with the energy identity and the W2,1-conjecture for nonlinear harmonic maps. Roughly, the energy identity is a conjectural picture which gives an explicit formula for the blow up behavior of sequences of nonlinear harmonic maps. The W2,1-conjecture is the easily stated conjecture that stationary harmonic maps have apriori L1 estimates for their hessians. Valtorta and I have solved these conjectures for stationary Yang-Mills, but the methods do not work for harmonic maps, and we hope to solve the problems by other means. The second project is with Yu Wang and hopes to answer some open questions by Hardt and Lin on nonlinear harmonic maps. Roughly, the goal is show one can solve for stable harmonic maps in the class of strongly H1-mappings. The second part of the proposal would address issues involving the regularity of spaces with lower and bounded Ricci curvature. Together with Wenshuai Jiang the first project of this part would study the energy identity for limits of manifolds with bounded Ricci curvature. Though similar in spirit, the problem itself is actually very different from the harmonic map case. To be
StatusFinished
Effective start/end date7/1/186/30/22

Funding

  • National Science Foundation (DMS-1809011)

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