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.
Uniqueness and nonuniqueness of p-harmonic Green functions on weighted Rn and metric spaces, (joint with A. Bjorn, J. Bjorn and S. Eriksson-Bique), arXiv.
Green functions in metric measure spaces, (joint with M. Bonk and L. Capogna), arXiv.
A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow (joint with A. Kijowski, Q. Liu, Y. Zhang), Discrete Contin. Dyn. Syst. 45 (2025), no. 9, 3483--3511. arXiv.
BV functions and nonlocal functionals in metric measure spaces, (joint with P. Lahti and A. Pinamonti), J. Geom. Anal. 34 (2024), no. 10, Paper No. 318, 34 pp., arXiv.
Discontinuous eikonal equations in metric measure spaces, (joint with Q. Liu and N. Shanmugalingam), Trans. Amer. Math. Soc. Volume 378, Number 1, January 2025, 695–729, arXiv.
A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces, (joint with Panu Lahti and Andrea Pinamonti), Nonlinear Anal. 241 (2024), Paper No. 113467, 14 pp, arXiv.
Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces, (joint with P. Lahti), Analysis and Geometry in Metric Spaces 12 (2024), no. 1, Paper No. 20240001, 22 pp, arXiv
Quasiconformal and Sobolev mappings in non-Ahlfors regular metric spaces, (joint with P. Lahti), Tohoku Mathematical Journal (to appear), arXiv
Horizontally quasiconvex envelope in the Heisenberg group, (joint with A. Kijowski and Q. Liu), Rev. Mat. Iberoam. 40 (2024), no. 1, 57-92.,arXiv
Absolutely continuous mappings on doubling metric measure spaces, (joint with P. Lahti), Manuscripta Mathematica 173 (2024), no. 1-2, 1-21, arXiv
Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces, (with Q. Liu), Minimax Theory and its Applications, 8 (2023), no. 1, 121–138.. PDF file.
Equivalence of solutions of eikonal equation in metric spaces, (joint with Q. Liu and N. Shanmugalingam), J. Differential Equations 272 (2021), 979–1014. arXiv
Functions of bounded variation on complete and connected one-dimensional metric spaces, (with P. Lahti), Int. Math. Res. Not. IMRN 2021, no. 20, 15412–15443. arXiv
Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations, (with Q. Liu), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 4, 2039–2076. 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.
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 files.