Mathematical Sciences: Nonlinear Elasticity, Exterior Problems and Stochastic Control

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The project consists of four parts: nonlinear partial differential equations, calculus of variations in nonlinear elasticity, stochastic control and stochastic functional differential equations. In the nonlinear partial differential equations part the investigators want to study the existence of radially symmetric solutions of two special Dirichlet problems for quasilinear elliptic equations, one of which contains a mean curvature operator. They would like to obtain the same results as Serrin, Peletier and Ni, using the calculus of variations rather than methods based on ordinary differential equations. In the nonlinear elasticity part the authors propose to re- examine certain issues in 3-dimensional nonlinear elasticity, which give rise to the one dimensional variational methods. They propose to use the connection between the elastic cavitation effect discovered by Ball and a higher dimensional form of the Lavrentiev's phenomenon. In the two stochastic parts the investigators want to study Bolza problems with regular or singular control, and to apply these ideas to another class of integrands for variational problems. They plan to incorporate memory effects in their analysis and use Hilbert space methods in the analysis of stochastic functional differential equations. This research effort belongs to applied analysis and calculus of variations, two branches of mathematics that come in contact with physical problems such as elasticity. Both investigators are senior experienced scientists with a lot of credit for their previous mathematical accomplishments.