 Victor J. Mizel
Professor
Ph.D., Massachusetts Institute of Technology
Office: Wean Hall 6128
Phone: (412) 268-2562
E-mail: vm09@andrew.cmu.edu
We deeply regret to inform you that Professor Mizel passed away on August
29, 2005. If you would like further information please send a message to
dept.head@math.cmu.edu
Research
A major research interest of mine is in the Calculus of Variations and Control Theory. It's been discovered that even regular one-dimensional variational problems can give rise to nonclassical phenomena. For instance you can have problems in which there is a nonzero gap between the actual minimum value of the variational integral and the lower limit of the values it assumes for smooth admissible functions. This "Lavrentiev phenomenon" has important implications for the numerical analysis of control and variational problems. A similar Lavrentiev phenomenon has been demonstrated to hold in stochastic control problems, as well as in isoperimetric problems and in second order problems.
I have recently been involved in the formulation and analysis of a mathematical model designed to clarify the unorthodox crystal behavior of heavily Boron-doped Silicon crystals. Experiments carried out by J. Hannon and colleagues at Sandia revealed that these crystals experience isothermal ("Brownian") oscillations which in a certain tempaerature range are higher in amplitude and frequency as the equilibrium temperature is lowered (!) This phenomenon can provide the basis for development of self-organizing materials associated with such crystals and a joint investigation in collaboration with J. Hannon and M. Marcus is well under way.
I have also been involved in the successful development, in collaboration with M. Foss and W. Hrusa, of well-posed boundary problems for a class of nonlinear elastic materials in which the Lavrentiev gap phenomenon occurs. This provides a positive resolution of a question raised in a joint work with John Ball in 1985.
Selected Publications
V. J. Mizel, "New developments concerning the Lavrentiev phenomenon" in Calculus of Variations and Differential Equations, Ioffe, Reich & Shafrir, eds. CRC Research Notes in Mathematics #410 (185–191) 2000
Victor J. Mizel, "On the ubiquity of fracture in nonlinear elasticity", J. of Elasticity 52(257–266) 1999
Victor J. Mizel & Thomas I. Seidman, "An abstract bang-bang principle and time-optimal boundary control of the heat equation" SIAM J. Control Optim. 35(1204–1216) 1997
V. J. Mizel, L. A. Peletier & W. C. Troy, "Periodic phases in second order materials, and higher order model equations", Proceedings Third European conference on Elliptic and Parabolic Equations, Pont-a-Mousson, France, June 1997.
M. Sychev and V.J. Mizel, "A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems" Trans. AMS 350 (1998) 119–133.
M.Marcus, V.J. Mizel, and Y. Pinchover, "On the best constant for Hardy's inequality in Rn", Trans. AMS 350 (1998) 3237–3255.
Buttazzo, G. and Mizel, V. J. (1995), "On a Gap Phenomenon for Isoperimetrically Constrained Variational Problems," J. Convex Anal. 2:87–101.
Cheng, C-W and Mizel, V. J. (1994), "On the Lavrentiev Phenomenon for Autonomous Second-Order Integrands," Arch. Rational Mech. Anal. 126:21–33.
Buttazzo, G. and Mizel, V. J. (1992), "Interpretation of the Lavrentiev Phenomenon by Relaxaation," J. Funct. Anal. 110:434–460.
Coleman, B.D., Marcus, M. and Mizel, V.J. (1992), "On the Thermodynamics of Periodic Phases," Archive for Rat. Mech. Anal. 117.
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