Research

My main research interest is in analysis and partial differential equations in Euclidean spaces, sub-Riemannian manifolds and general metric measure spaces. A common thread through all my work is nonsmooth analysis. Because of its general setting, nonsomooth analysis plays a fundamental role in the unification of methods that previously have been developed separately for different areas of mathematics. Nonsmooth analysis is also widely applied to image reconstruction theory, optimal transport, control theory, robotics and mathematical biology.

My research work mainly consists of two parts. The first part is focused on the study of solutions to various nonlinear partial differential equations including the p-Laplace equations, mean curvature flow equations and Hamilton-Jacobi equations. In addition, a recent approach to the viscosity solutions of various equations based on deterministic and stochastic games is also applied in my research. The second part is focused on Sobolev functions, BV functions and other functions with fine properties on metric measure spaces, which are among the main tools used in analysis on metric spaces.

Publications

• A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces, (joint with Panu Lahti and Andrea Pinamonti), arXiv.

• Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces, (joint with P. Lahti), submitted. arXiv

• Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces, (joint with P. Lahti), submitted. arXiv

• Existence and Uniqueness of Green functions for the Cheeger p-Laplacian in PI spaces, (joint with M. Bonk and L. Capogna), arXiv.

• Horizontally quasiconvex envelope in the Heisenberg group, (joint with A. Kijowski and Q. Liu), to appear in Rev. Mat. Iberoam., arXiv. 

• Absolutely continuous mappings on doubling metric measure spaces, (joint with P. Lahti), to appear in Manuscripta Mathematica, arXiv

• Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces, (with Q. Liu), accepted by Minimax Theory and its Applications.

• Equivalence of solutions of eikonal equation in metric spaces, (joint with Q. Liu and N. Shanmugalingam), to appear in J. Differential Equations. arXiv

• Functions of bounded variation on complete and connected one-dimensional metric spaces, (with P. Lahti), to appear in Int. Math. Res. Not. IMRN. arXiv

• Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations, (with Q. Liu), to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. arXiv

• Absolutely continuous functions on compact and connected one-dimensional metric spaces, Ann. Acad. Sci. Fenn. Math. 44, (2019), 281-291. PDF file

• Strong comparison principle for p-harmonic functions in Carnot-Caratheodory spaces, (with L. Capogna), Proc. Amer. Math. Soc. 146 (2018), no. 10, 4265-4274.  arXiv

• Weakly coupled systems of fully nonlinear parabolic equations in the Heisenberg group, (with Q. Liu), Nonlinear Anal. 174 (2018), 54-78. PDF file

• Sobolev functions in the critical case are uniformly continuous in s-Ahlfors regular metric spaces when s less than or equal to one, Proc. Amer. Math. Soc. 145 (2017), no. 1, 267-272. arXiv

• Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group, (with Q. Liu and J. J. Manfredi), Calc. Var. Partial Differential Equations 55 (2016), no.4, Art. 80, 25pp. arXiv

• Sobolev embedding on a sphere containing an arbitrary Cantor set in the image, (with P. Hajlasz), Geom. Dedicata. 184 (2016), 159--173. arXiv

• A game-theoretic proof of convexity preserving properties for motion by curvature, (with Q. Liu and A. Schikorra), Indiana Univ. Math. J. 65 (2016), 171--197. PDF file