Alumni Professor of Mathematical Sciences, Professor of Materials Science and Engineering
7208 Wean Hall
Department of Mathematical Sciences
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213
Ph.D., University of California, Berkeley
- Scuola Normale Superiore, Pisa
- Fellow of the American Mathematical Society
- SIAM Fellow
My recent activity is in applied mathematics and in analysis, in particular partial differential equations. The work in applied mathematics, joint with Shlomo Ta'asan, colleagues in the Materials Science and Engineering Dept., and our extraordinary postdocs, is directed toward understanding evolution of material microstructure. Nearly all technologically useful materials are polycrystalline microstructures composed of a myriad of small crystallites or grains separated by grain boundaries, and comprise cellular networks. A central problem in materials is to develop technologies capable of producing an arrangement, or ordering, of the grains in terms of geometry and crystallographic texture that provides desired properties for a given function. The order, if indeed it is present at all, must be conferred by the network grain boundaries or interfaces, because they are what changes during the coarsening process. Using new experimental techniques and especially developed large scale simulation, we have discovered the grain boundary character distribution (GBCD), a statistic which details texture evolution. In the simplest situation, it is a Boltzmann distribution related to the interface energy density.
Employing innovative methods in analysis, especially (Monge-Kantorovich) mass transport theory, we have, further, developed an entropy based theory that explains GBCD behavior. Thus materials arise with (non random) texture order, a new discovery. In this adventure, we have also found connections to other areas, for example, an analogue to prefix codes in information theory. The issue we face in this investigation is that stochastic or probabilistic behavior of the system is high, yet its quantitative resolution requires continuum scale methods that are very challenging to discover.
Additional work is in applications to cell biology, especially mechanisms of intracellular transport like protein motors, and models of ion transport, the Poisson-Nernst Planck Equations. These all share the features of stochastic behavior that requires continuum level upscaling for predictive modeling. These investigations are heavily invested in optimal transport theory.
Bardsley, P., Barmak, K., Eggeling, E., Epshteyn, Y., Kinderlehrer, D. and Ta’asan, S. (2017). Towards a gradient flow for microstructure. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28(4), 777–805.
Barmak, K., Eggeling, E., Emelianenko, M., Epshteyn, Y., Kinderlehrer, D., Sharp, R., and Ta’asan, S. (2011). Critical events, entropy, and the grain boundary character distribution. Phys. Rev. B, 83 (13), 134117.
Barmak, K., Eggeling, E., Kinderlehrer, D., Sharp, R., Ta’asan, S., Rollett, A.D., and Coffey, K.R. (2013). Grain growth and the puzzle of its stagnation in thin films: The curious tale of a tail and an ear. Progress in Materials Science, 58(7), 987-1055.
Barmak, K., Eggeling, E., Emelianenko, M., Epshteyn, Y., Kinderlehrer, D., Sharp, R., and Ta’asan, S. (2011). An entropy based theory of the grain boundary character distribution. Discrete Contin. Dyn. Syst., 30(2), 427–454.
Blanchet, A., Carrillo, J., Kinderlehrer, D., Kowalczyk, M., Laurencot, P. and Lisini, S. (2015). A hybrid variational principle for the Keller-Segel system in R2. ESAIM Math. Model. Numer. Anal., 49(6), 1553–1576.
Chipot, M., Hastings, S. and Kinderlehrer, D. (2004). Transport in a molecular motor system. ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS- MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 38(6), 1011–1034.
Hastings, S.,Kinderlehrer, D. and McLeod, J.B. (2007). Diffusion mediated transport in multiple state systems. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 39(4), 1208-1230.
Heath, D., Kinderlehrer, D. and Kowalczyk, M. (2002). Discrete and continuous ratchets: From coin toss to molecular motor. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 2(2), 153–167.
Jordan, R., Kinderlehrer, D. and Otto, F. (1998). The variational formulation of the Fokker-Planck equation. SIAM J. Math. Analysis, 29(1), 1–17.
Kinderlehrer, D. and Nirenberg, L. (1977). Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4(2), 373–391.
Kinderlehrer, D., Monsaingeon, L. and Xu, X. (2017). A Wasserstein gradient flow ap- proach to Poisson-Nernst-Planck equations. ESAIM Control Optim. Calc. Var., 23(1), 137–164.
James, R.D. and Kinderlehrer, D. (1994). Theory of Magnetostriction with Application To Terfenol-D. Journal of Applied Physics, 76(10, Part 2), 7012– 7014.