Research interests


Partial Differential Equations

Description of some of my research directions Description

Some talks


Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel, IPAM Workshop III: Mean Field Games and Applications, pdf, and video of the talk link.

Optimal rate of convergence in periodic homogenization of Hamilton-Jacobi equations -- Weak KAM 2018 - Rio 2018 ICM Satellite Conference pdf.
A newer version with some new results given at CNA seminar, CMU, Jan 2020 pdf, and video of the talk link.

Crystal growth (Large time behavior of level-set mean curvature flow equations with driving and source terms), AMS sectional meeting 2019 pdf.

Min-max formulas for nonconvex effective Hamiltonians and application to stochastic homogenization (long version at LMS-EPSRC Durham Research Symposia 2018 pdf, a video of my talk link), (shorter version at AMS meeting pdf).

Video of my talk at Banff on inverse problems in periodic homogenization of Hamilton-Jacobi equations (paper 20) link.
See this extended abstract (pages 29-35, 41st Sapporo Symposium on PDEs) for some results in paper 20 and further questions link, pdf.

Books/Lecture notes


2. Hamilton--Jacobi equations: theory and applications, AMS Graduate Studies in Mathematics third draft. For earlier versions, see lecture notes.

1. (with Nam Q. Le, H. Mitake) Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampere Equations, Springer Lecture Notes in Mathematics, vol. 2183, 2017, vii + 228. link

A version (12/2016) of the second part on ``Dynamical properties of Hamilton--Jacobi equations via the nonlinear adjoint method: Large time behavior and Discounted approximation" of Mitake and myself can be found here pdf.

Current Publications


47. (with M. Klibanov, Loc H. Nguyen) Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method arXiv.

46. (with T. Sprekeler) Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations arXiv.

45. Selection problems in Large Deviations in Games under the logit choice protocol, pdf.

44. (with D. Gomes, H. Mitake) The large time profile for Hamilton--Jacobi--Bellman equations, arXiv.

43. (with H. Mitake, Truong-Son Van) Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model, Comm Math Sci. arXiv.

42. (with X. Guo, Y. Yu) Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form, SN Partial Differential Equations and Applications pdf.

41. (with Truong-Son Van) Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel, Comm. Pure Appl. Math arXiv.

40. (with W. Jing, Y. Yu) Effective fronts of polytope shapes, Minimax Theory and its Applications pdf.

39. (with Y. Kim, Son N. T. Tu) State-constraint static Hamilton-Jacobi equations in nested domains, SIAM journal on Mathematical Analysis. arXiv.

38. (with Y. Giga, H. Mitake) Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms, DCDS-B arXiv.

37. (with X. Guo, J. Peterson) Quantitative homogenization in a balanced random environment arXiv.

36. (with W. Jing, H. Mitake) Generalized ergodic problems: existence and uniqueness structures of solutions, J. Differential Equations arXiv.

35. (with Y. Giga, L. Zhang) On obstacle problem for mean curvature flow with driving force, Geometric Flows pdf.

34. (with Y. Giga, H. Mitake, T. Ohtsuka) Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations, Indiana University Math Journal arXiv.

33. (with W. H. Sandholm, S. Arigapudi) Hamilton-Jacobi Equations with Semilinear Costs and State Constraints, with Applications to Large Deviations in Games, Mathematics of Operations Research pdf.

32. (with H. Mitake) On uniqueness sets of additive eigenvalue problems and applications, Proc. Amer. Math. Soc. arXiv

31. (with H. Mitake, Y. Yu) Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting, Arch. Ration. Mech. Anal. arXiv

30. (with Y. Yu) A rigidity result for effective Hamiltonians with 3-mode periodic potentials, Advances in Math. arXiv

29. (with H. Ishii, P. E. Souganidis) On the Langevin equation with variable friction. Calculus of Variations and PDE. pdf

28. (with J. Qian, Y. Yu) Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians. Mathematische Annalen. arXiv

27. A note on nonconvex Mean Field Games, Minimax Theory and its Applications pdf

26. (with H. Ishii, H. Mitake) The vanishing discount problem and viscosity Mather measures. Part 2: boundary value problems. Journal Math. Pures Appl. arXiv

25. (with W. Jing, P. E. Souganidis) Stochastic homogenization of viscous superquadratic Hamilton-Jacobi equations in dynamic random environment, Research in the Mathematical Sciences arXiv

24. (with D. Gomes, H. Mitake) The Selection problem for discounted Hamilton-Jacobi equations: some non-convex cases, Journal of the Mathematical Society of Japan (JMSJ) arXiv

23. (with H. Ishii, H. Mitake) The vanishing discount problem and viscosity Mather measures. Part 1: the problem on a torus. Journal Math. Pures Appl. arXiv

22. (with W. Jing, Y. Yu) Inverse problems, non-roundedness and flat pieces of the effective burning velocity from an inviscid quadratic Hamilton-Jacobi model. Nonlinearity arXiv

21. (with Y. Giga, H. Mitake) On asymptotic speed of solutions to level-set mean curvature flow equations with driving and source terms, SIAM journal on Mathematical Analysis. arXiv

20. (with S. Luo, Y. Yu) Some inverse problems in periodic homogenization of Hamilton-Jacobi equations, Arch. Ration. Mech. Anal. arXiv

19. (with H. Mitake, A. Siconolfi, N. Yamada) A Lagrangian Approach to Weakly Coupled Hamilton-Jacobi Systems, SIAM journal on Mathematical Analysis. arXiv

18. (with S. Armstrong, Y. Yu) Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension, J. Differential Equations. arXiv

17. (with H. Mitake) Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, Advances in Math. arXiv

16. (with W. Jing, P. E. Souganidis) Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. DCDS-S. arXiv

15. (with A. Ciomaga, P. E. Souganidis) Stochastic homogenization of interfaces moving with changing sign velocity, J. Differential Equations. arXiv

14. (with H. Mitake) Weakly coupled systems of the infinity Laplace equations, Trans. Amer. Math. Soc. arXiv

13. (with L. C. Evans, O. Kneuss) Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models, Trans. Amer. Math. Soc. arXiv

12. (with S. Armstrong, Y. Yu) Stochastic homogenization of a nonconvex Hamilton-Jacobi equation. Calculus of Variations and PDE. arXiv

11. (with S. Armstrong) Viscosity solutions of general viscous Hamilton-Jacobi equations, Mathematische Annalen. arXiv

10. (with S. Armstrong) Stochastic homogenization of viscous Hamilton-Jacobi equations and applications, Analysis and PDE. arXiv

9. (with H. Mitake) Large-time behavior for obstacle problems for degenerate viscous Hamilton--Jacobi equations, Calculus of Variations and PDE. arXiv

8. (with F. Cagnetti, D. Gomes, H. Mitake) A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincare Anal. Non Lineaire. arXiv

7. (with H. Mitake) A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton--Jacobi equations, Journal Math. Pures Appl. arXiv

6. (with H. Mitake) Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates, Arch. Ration. Mech. Anal. arXiv

5. (with F. Cagnetti, D. Gomes) Convergence of a Numerical Scheme for the Hamilton-Jacobi Equation: a New Approach with the Adjoint Method, Applied Numerical Mathematics. arXiv

4. (with H. Mitake) Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations, Asymptotic Analysis. arXiv

3. (with F. Cagnetti, D. Gomes) Adjoint methods for obstacle problems and weakly coupled systems of PDE, ESAIM: COCV. arXiv

2. (with F. Cagnetti, D. Gomes) Aubry-Mather measures in the non convex setting, SIAM journal on Mathematical Analysis. arXiv

1. Adjoint methods for static Hamilton-Jacobi equations, Calculus of Variations and PDE. Open access


Undergraduate publications


2. (with Duong M. Duc and Nguyen T. Khai) Critical points of non-C2 functionals, Topol. Methods Nonlinear Anal. 29 (2007), no. 1, 35-68.

1. (with Duong M. Duc and Nguyen T. Khai) Morse-Palais lemma for nonsmooth functionals on normed spaces, Proc. Amer. Math. Soc. 135 (2007), no. 3, 921-927.


Unpublished works


1. (with W. Jing, P. E. Souganidis) Homogenization of interfaces moving in spatially random temporally periodic environment. pdf.

Grants


Supported in part by NSF CAREER grant DMS-1843320 Link.

Supported in part by NSF grant DMS-1664424 Link.

Supported in part by NSF grant DMS-1361236 Link, which is transferred to Madison as DMS-1615944 Link .

Thesis


Some new methods for Hamilton--Jacobi type nonlinear partial differential equations. Link