Partial Differential Equations

Description of some of my research directions Description

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.

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.

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,

42. (with X. Guo, Y. Yu) Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form,

41. (with Truong-Son Van) Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel,

40. (with W. Jing, Y. Yu) Effective fronts of polytope shapes,

39. (with Y. Kim, Son N. T. Tu) State-constraint static Hamilton-Jacobi equations in nested domains,

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

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,

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

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

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

32. (with H. Mitake) On uniqueness sets of additive eigenvalue problems and applications,

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

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

29. (with H. Ishii, P. E. Souganidis) On the Langevin equation with variable friction.

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

27. A note on nonconvex Mean Field Games,

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

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

24. (with D. Gomes, H. Mitake) The Selection problem for discounted Hamilton-Jacobi equations: some non-convex cases,

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

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.

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

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

19. (with H. Mitake, A. Siconolfi, N. Yamada) A Lagrangian Approach to Weakly Coupled Hamilton-Jacobi Systems,

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

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

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.

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

14. (with H. Mitake) Weakly coupled systems of the infinity Laplace equations,

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

12. (with S. Armstrong, Y. Yu) Stochastic homogenization of a nonconvex Hamilton-Jacobi equation.

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

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

9. (with H. Mitake) Large-time behavior for obstacle problems for degenerate viscous Hamilton--Jacobi equations,

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

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

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

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

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

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

2. (with F. Cagnetti, D. Gomes) Aubry-Mather measures in the non convex setting,

1. Adjoint methods for static Hamilton-Jacobi equations,

2. (with Duong M. Duc and Nguyen T. Khai) Critical points of non-C2 functionals,

1. (with Duong M. Duc and Nguyen T. Khai) Morse-Palais lemma for nonsmooth functionals on normed spaces,

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

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 .

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