# Bhattacharya, Arunima

Bill Guthridge Fellow, Assistant Professor

Phillips 304A

arunimab@unc.edu

Website

## Research Interests

Geometric analysis, nonlinear partial differential equations

## Professional Background

B.Sc. St. Xavier’s College, Kolkata, 2012; M.Sc. Tata Institute of Fundamental Research, 2014; Ph.D. University of Oregon, 2019; Postdoc University of Washington, 2019-2022; Mathematical Sciences Research Institute, July 2022-December 2022

## Research Synopsis

My research is in geometric analysis with a focus on fully nonlinear second and fourth-order elliptic partial differential equations that arise naturally in differential geometry. I primarily study geometric variational problems applying tools from minimal surface theory, Lagrangian geometry, Kähler geometry, geometric measure theory, and the theory of elliptic equations. Area minimization problems amongst Lagrangian surfaces lead to the study of certain fourth order PDEs that I work on. Geometrically motivated variational problems for the volume functional, in the Lagrangian setting, give rise to nonlinear fourth-order elliptic equations. A key tool for the study of second-order elliptic equations is Schauder theory, which is by now well-developed with equations in divergence form occupying an important place. In fourth order, an inherent analog is an equation in linear or nonlinear form that has a double divergence structure. Equations such as bi-harmonic functions, extremal Kähler metrics, the Willmore surface, the Hamiltonian stationary equations, share this structure.

## Representative Publications

*Regularity of Hamiltonian stationary equations in symplectic manifolds.*

A. Bhattacharya, J. Chen, and M. Warren,

Advances in Mathematics, 424, Paper No. 109059, 32 pp, 2023

*Regularity for convex viscosity solutions of Lagrangian mean curvature equation*

A. Bhattacharya and R. Shankar,

Journal für die reine und angewandte Mathematik (Crelles Journal), vol. 2023, no. 803, pp. 219-232, 2023

*The Dirichlet problem for Lagrangian mean curvature equation*

A. Bhattacharya,

Analysis and PDE, to appear, 2023