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, 2018-2021

Papers in preparation

(2) Traces for extended divergence-measure fields, with  Hanan Ussif Gadi, in preparation.

(1) Varifolds with prescribed first variation, with Salvatore Stuvard, in preparation.

Submitted papers

(2) Divergence-measure fields: Gauss-Green formulas and Normal Traces, with Gui-Qiang Chen, submitted 2019.

(1) Traces and extensions of bounded divergence-measure fields on rough open sets, with Gui-Qiang Chen and Qinfeng Li, Submitted to Indiana University Mathematics Journal, 2019.

Published books

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

Published papers

(19) 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.

(18) 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.

(17) Morrey spaces and Generalized Cheeger sets, with Qinfeng Li, Advances in Calculus of Variations, (forthcoming volume and issue), published online on June 2017.

(16) 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.

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

(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.