Regularity vs. Singularity for Elliptic and Parabolic Systems

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This project investigates partial differential equations that arise in several areas of pure and applied mathematics including meteorology, elasticity, and differential and complex geometry. The proposed problems concern the regularity (smoothness properties) of solutions. Key challenges that these problems share are that ellipticity (an agent of regularity) degenerates, and that solutions can be vector-valued. Overcoming these challenges will not only require significant new mathematical ideas, but could also help us better predict weather patterns and assist in the design of cars. The results will be communicated by publication in peer-reviewed journals, and through the writing of expository notes that the PI intends to make publicly available and will be useful to researchers and students alike.The first project tackles regularity questions for the real Monge-Ampere equation, motivated by applications to meteorology and differential geometry. The PI will investigate singularity formation for the semi-geostrophic system, which models large scale atmospheric flows. Certain examples of irregular stationary solutions in a half-plane may be useful models for blowup at boundary points. The second project concerns local regularity for the complex Monge-Ampere equation, which arises in complex geometry. Difficulties include the non-convexity of solutions, and invariance under adding certain quadratic polynomials. The PI has initiated the study of a model equation that captures these difficulties. The third project concerns non-concave uniformly elliptic equations. A challenging problem is to construct singular solutions in low dimensions. An obstruction is that in 3d, linear equations with rough coefficients have no nontrivial one-homogeneous solutions; the PI will instead consider solutions with "spiraling" one-homogenous symmetry. The fourth project concerns minimizers of variational integrals with convex integrand. A conjecture is that scalar-valued minimizers have the same regularity as the Legendre transform of the integrand. The PI proposes to confirm this when the degeneracy set of the integrand satisfies certain geometric conditions, and to construct a counterexample when these conditions aren't met. The last project aims to answer classical regularity and stability questions for parabolic systems, especially in low dimensions. The PI recently constructed examples of singular solutions to linear parabolic systems in the plane, answering a long-standing question. This may open the way to tackling related problems, e.g. to identify conditions on coefficients that prevent singularities in the linear case, and to determine regularity vs. singularity for nonlinear structures of porous medium type.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.