Research

Publications

Refereed papers

• Horizontal semiconcavity for the square of Carnot-Caratheodory distance on ideal Carnot groups and applications to Hamilton-Jacobi equations (with F. Dragoni and Y. Zhang), preprint. PDF

• A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow (with A. Kijowski, Y. Zhang and X. Zhou), submitted for publication. PDF

• A representation formula for viscosity solutions of nonlocal Hamilton-Jacobi equations and applications, (with T. Kagaya and H. Mitake), submitted for publication. PDF

• Discontinuous eikonal equations in metric measure spaces, (with N. Shanmugalingam and X. Zhou), submitted for publication. PDF

• Horizontally quasiconvex envelope in the Heisenberg group, (with A. Kijowski and X. Zhou), Rev. Mat. Iberoam. 40 (2024), 57--92. PDF

• Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces, (with X. Zhou), Minimax Theory Appl., 8 (2023), 121--138. PDF

• Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations, (with T. Kagaya and H. Mitake), NoDEA Nonlinear Diffferential Equations Appl., 30 (2023), Paper No. 13. PDF 

• Principal eigenvalue problem for infinity Laplacian in metric spaces, (with A. Mitsuishi), Adv. Nonlinear Stud., 22 (2022), 548--573. PDF 

• Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension, (with T. Kagaya), SIAM J. Math. Anal., 53 (2021), 4350--4385.  PDF  arXiv

• Equivalence of solutions of eikonal equation in metric spaces, (with N. Shanmugalingam and X. Zhou), J. Differential Equations, 272 (2021), 979--1014.  PDF  arXiv

• A game-theoretic approach to dynamic boundary problems for level-set curvature flow equations and applications, (with N. Hamamuki), Partial Differ. Equ. Appl., 2 (2021), Article 30. PDF

• Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations, (with X. Zhou), Ann. Sc. Norm. Super. Pisa Cl. Sci., 22 (2021), 2039--2076.  PDF

• Semiconvexity of viscosity solutions to fully nonlinear evolution equations via discrete games, Springer INdAM Series, 47 (2021), Geometric Properties for Parabolic and Elliptic PDE's, 205--231.  PDF

• The vanishing exponent limit for motion by a power of mean curvature, Interfaces Free Bound., 22 (2020), 51--84. 

• A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions, (with N. Hamamuki),  ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 13, 42 pp. 

• On a discrete scheme for time fractional fully nonlinear evolution equations, (with Y. Giga and H. Mitake), Asymptot. Anal., 120 (2020), 151--162.  arXiv

• Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations, (with K. Ishige and P. Salani), J. Math. Pures Appl., 141 (2020), 342--370. 

• Large exponent behavior for power-type nonlinear evolution equations and applications, J. Evol. Equ., 20 (2020), 777--810. 

• On an obstacle problem arising in large exponent asymptotics for one dimensional fully nonlinear diffusions of power type, Adv. Stud. Pure Math., 85 (2020), 281--289. 

• An obstacle problem arising in large exponent limit of power mean curvature flow equation, (with N. Yamada), Trans. Amer. Math. Soc., 372 (2019), 2103--2141. 

• Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces, (with A. Nakayasu), Discrete Contin. Dyn. Syst., 39 (2019), 157--183.

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

• Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group, (with J. J. Manfredi and X. Zhou), Calc. Var. Partial Differential Equations, 55 (2016), 55:80. Mathematica output

• A nonlinear parabolic equation with discontinuity in the highest order and applications, (with R. M. Chen),  J. Differential Equations, 260 (2016), 1200--1227.  

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

• Existence and symmetry of ground states to the Boussinesq abcd systems, (with E. S. Bao and R. M. Chen), Arch. Ration. Mech. Anal., 216 (2015), 569--591.  

• General existence of solutions to dynamic programming equations, (with A. Schikorra), Commun. Pure Appl. Anal., 14 (2015), a special issue on Emerging Trends in Nonlinear PDE., 167--184.  

• A game-tree approach to discrete infinity Laplacian with running costs, (with A. Schikorra), preprint.   arXiv

• Large-time asymptotics for Hamilton-Jacobi equations with noncoercive Hamiltonians appearing in crystal growth, (with Y. Giga and H. Mitake), Adv. Stud. Pure Math., 64 (2015), 235--242.

• Waiting time effect for motion by positive second derivatives and applications, Nonlinear Differential Equations Appl. NoDEA., 21 (2014), 589--620.  

• On the characterization of p-harmonic functions on the Heisenberg group by mean value properties, (with F. Ferrari and J. J. Manfredi), Discrete Contin. Dyn. Syst., 34 (2014), 2779--2793. 

• On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group, (with F. Ferrari and J. J. Manfredi), Commun. Contemp. Math., 16 (2014), 1350027.  

• Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations, (with Y. Giga and H. Mitake), Trans. Amer. Math. Soc., 366 (2014), 1905--1941.  

• Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians, (with Y. Giga and H. Mitake), J. Differential Equations, 252 (2012), 1263--1282. 

• Fattening and comparison principle for level-set equations of mean curvature type, SIAM J. Control Optim., 49 (2011), 2518--2541. 

• A billiard-based game interpretation of the Neumann problem for the curve shortening equation, (with Y. Giga), Adv. Differential Equations, 14 (2009), 201--240. 

• A remark on the discrete deterministic game approach for curvature flow equations, (with Y. Giga), GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), Nonlinear phenomena with energy dissipation: Mathematical analysis, modeling and simulation,  103--115. 

Other publications

• Quasiconvexity preserving property for first order nonlocal evolution equations, (with T. Kagaya and H. Mitake), RIMS Kokyuroku, Kyoto University, 2277 (2024), 57--69. 

• Large exponent asymptotics for one dimensional fully nonlinear diffusion of power type, RIMS Kokyuroku, Kyoto University, 2146 (2020), 9--20.

• On an elementary approach to optimal control with Neumann boundary condition, RIMS Kokyuroku, Kyoto University, 1695 (2010), 56--64.

• On game interpretations for the curvature flow equation and its boundary problems, RIMS Kokyuroku, Kyoto University, 1633 (2009), 138--150.

Researchmap Page: https://researchmap.jp/7000011690

Researchgate Page: https://www.researchgate.net/profile/Qing_Liu40