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Finding classical solutions to elliptic PDEs with singular solutions 

Ravi Shankar (UW)
Wednesday, November 25, 2020 - 4:00pm to 5:00pm

I will discuss some partial differential equations (PDEs) called Lagrangian mean curvature type equations (LMCs) which, like Laplace's equation, are elliptic, but as with the Monge-Ampere equation, are fully nonlinear and have singular solutions.  Derivative estimates and a Bernstein result for smooth convex solutions suggest that merely continuous convex solutions are classical.  In fact, as soon as some such convexity condition is removed, singular solutions appear.  The standard technique is to start with smooth approximations of the solution, apply such estimates to gain control of the higher order derivatives, then pass to a subsequence using Arzela-Ascoli, which now converges in the smooth topology.  For the Monge-Ampere equation, the condition removing singular solutions is preserved by approximation, but this isn't true for convex solutions of LMCs.  I will explain how with Jingyi Chen and Yu Yuan, we managed to bypass the usual a priori estimate technique by discovering a low regularity version of a transform known to improve the solvability properties of the simplest type of LMC.  With Arunima Bhattacharya, we also considered how to understand more complicated and lower regularity LMCs, culminating in an optimal classification of regularity and two new interpretations of the constant rank property for Hessian matrices.  I will end by hinting at the current outlook for the notorious sigma-2 PDE using this approach.

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