Differential Equations and Complex Analysis Research Group

David R. Adams

I am presently interested in the study of systems of variational inequalities (VI) for second-order elliptic partial differential operators, especially with regard to questions about stability of solutions--the change in the solution induced by a change in the (obstacle) data. VI's are a way to express precisely many so-called ``free boundary'' problems that arise when the systems are overdetermined--especially when one or more constraints (in the form of obstacles) are also imposed. Physical examples include applied problems that involve a liquid-liquid interface, or the liquid-solid interface at the freezing point. A problem of special interest to me here is the simultaneous bending of several "metal" plates and the possible intereaction between their shapes--and how these shapes vary as the initial data (the forces) vary. Methods used have come from Real Analysis, Potential Theory, and Partial Differential Equations (PDE).

Rcently, I have discovered that many elements of Algebra and Group Invariant Theory play an important role. For example, one question that has come up is that of finding an analogue of the Jordan normal form for a complex-valued N x N matrix subject to similarity via certain special subgroups of the usual general linear group of all invertible N x N matrices. And from this theory new "characteristic" type polynomials appear that are "invariant" with respect to these new subgroups. These polynomials are then key to showing, for example, that the system has a solution and studying its regularity or smoothness properties.

Much of my present work with systems of variational inequalities comes about from my past work in Potential Theory, especially the most modern version (since the 1970's)-Nonlinear Potential Theory. Two and one half decades of research has now been collected and presented in the book

D. R. Adams, L. I. Hedberg. Function Spaces and Potential Theory, Grundlehren der Math. Wiss. 314, Springer-Verlag, 1996
This area of Analysis still presents many new and intriguing questions for future generations.

References to some of my work on VI's:

  1. Capacity and the obstacle problem. Appl. Math. Optim. 8 (1981), 39-57.
  2. Weakly elliptic systems with obstacle constraints, I: A 2 x 2 model Problem. Partial Differential Equations with Minimal Smoothness and Applications, IMA Volumes on Mathematics and its Applications 42 (1992), 1-14, Springer-Verlag.
  3. (with H. N. Lopes). Weakly elliptic systems of variational inequalities-a 2 x 2 model with obstacles in both components. Ann. di Mat. Pura. Appl. 169 (1995), 183-201.
  4. Weakly elliptic systems with obstacle constraints, II. An N x N model problem, to appear.