Ravi Shankar, University of Washington

Title: Finding classical solutions to elliptic PDEs with singular solutions

Abstract: 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.