Publications

My research has been supported by the following grants

  • Divergence-measure fields and nonlinear conservation laws, National Science Foundation (Analysis), DMS-0540869, 2005-2009.
  • Purdue Research Foundation, Summer Faculty Award, 2008.
  • Divergence-measure fields and the structure of solutions of systems of hyperbolic conservation laws, National Science Foundation (Analysis), DMS-090245, 2009-2014.
  • PRF International Travel Grant, Summer 2016.
  • Simons Foundation, Award Number 524190, 2017-2022.
  • Geometric measure theory, image processing and nonlinear PDE, National Science Foundation (Applied Analysis), DMS-1813695, 2018-2022

 

Publications

(24) Extended divergence measure fields, the Gauss-Green Formula and Cauchy Fluxes, with Gui-Qiang Chen and Christopher Irving, Submitted, arXiv identifier 2410.09214, 70 pp., 2024.

(23) A bifurcation phenomenon in a singularly perturbed two-phase free boundary problem on phase transition, with Fernando Charro, Alaa Haj Ali, Nurul Raihen and Peiyong Wang, Nonlinear Analysis: Real World Applications, Volume 73, 2023.

(22) Divergence-measure fields: Gauss-Green formulas and Normal Traces, with Gui-Qiang Chen, Notices of the AMS, Volumen 68, Number 8, pp. 1282-1290,2021.

(21) Traces and extensions of bounded divergence-measure fields on rough open sets, with Gui-Qiang Chen and Qinfeng Li, Indiana University Mathematics Journal, Volume 69, 229-264, 2020.

(20) Cauchy Fluxes and Gauss-Green formulas for divergence-measure fields over general open sets, with Gui-Qiang Chen and Giovanni Comi, Archive for Rational Mechanics and Analysis, Volume 233, Issue 1, 87-166, 2019.

(19) Regularity of interfaces for a Pucci type like segregation problem, with Luis Cafarelli,  Stefania Patrizi and Veronica Quitalo, Annales de l'Institut Henri Poincare /Analyse non lineaire, AN 36, 939-975, 2019.

(18) Morrey spaces and Generalized Cheeger sets, with Qinfeng Li, Advances in Calculus of Variations, 12 (2), 111-133, 2019.

(17) Occupational measures and averaged shape optimization, with Qinfeng Li and Ido Bright, ESAIM: Control, Optimization and Calculus of Variations, Volume 24, Number 3, 1141-1165, 2018.

(16) On the dual of BV, Contemporary Mathematics, Volume 709, 115-129, June 2018. 

(15) Modern Real Analysis, 2nd Edition, with W. Ziemer, Graduate Texts in Mathematics Series, Springer, Volume 278, 382 pp., 2017.

(14) One-sided approximation of sets of finite perimeter, with Giovanni Comi, Rend. Lincei Mat. Appl., 28, no. 1, 181-190, 2017.

(13) Characterization of signed measures in the dual of BV and related isometric isomorphisms, with N. C. Phuc, Annali della Scuola Normale Superiore di Pisa, (5), Vol. XVII, issue 1, 385-417, 2017. 

(12) The integral of the normal and fluxes over sets of finite perimeter, with Ido Bright, Interfaces and free boundaries, 17, 245-259, 2015.

(11) On the distributional divergence of vector fields vanishing at infinity, with Thierry DePauw, Proceedings of the Royal Society of Edinburgh, 141A, 65-76, 2011.

(10) On the structure of solutions of nonlinear hyperbolic systems of conservation laws, with G. Q. Chen, Communications on Pure and Applied Analysis, Volume 10, Number 4, 1011-1036, 2011.

(9) Existence of traveling wave solutions of a high-order nonlinear acoustic wave equation, with M. Chen and T. Walsh, Physics Letters A, Volume 373, Issue 11, 1037-1043, 2009.

(8) Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, with G.Q. Chen and W. Ziemer, Communication on Pure and Applied Mathematics, 62, No 2, 242-304, 2009.

(7) Characterizations of the existence and removable singularities of divergence-measure vector fields, with N.C. Phuc, Indiana University Mathematics Journal, Volume 57, Number 4, 1573-1598, 2008.

(6) Finite element methods for non-linear acoustics and fluids, with Timothy Walsh, Journal of Computational Acoustics, Vol. 15, No. 3, 353-375, 2007.

(5) Measure-theoretic analysis and non-linear conservation laws, with G.Q. Chen and W. Ziemer, Pure and Applied Mathematics Quarterly, 3 (3), 841-879, 2007.

(4) Divergence-measure fields, sets of finite perimeter and conservation laws, with G.Q. Chen, Arch. Rat. Mech. Anal.,175,. no.2 245-267, 2005.

(3) Level set methods to compute minimal surfaces in a medium with exclusions (voids)., with D. Chopp and T. Walsh, Interfaces Free Bound. {\bf 7} (2005), no.~2, 161-185.

(2) Plane-like minimal surfaces in periodic media with exclusions, SIAM J. Math. Anal. 36 (2004), no. 2, 523--551.

(1) Generalized Snell's law for weighted minimal surfaces in heterogeneous media, with L. Zhilin, L. Xiaobiao and H.Zhao, Methods Appl. Anal. 10 (2003), no. 2, 199-214.