Nonlinear PDEs: regularity, rigidity, and an inverse problem

Abstract

Based on joint work with Arunima Bhattacharya, we obtain a sharp regularity result for Lagrangian mean curvature type equations with possibly H\"older continuous Lagrangian phases. Along the way, the constant rank theorem of Bian and Guan is generalized, and a different, lower regularity way to prove strict convexity is developed. Next, based on joint work with Yu Yuan, we show that smooth semiconvex solutions of the sigma-2 equation are quadratic polynomials if they are entire. Finally, a Calder\'on type inverse problem for quasilinear elliptic equations is discussed, where the author improves a recent result of Mu\~noz and Uhlmann using boundary jet linearization.