Relaxation and Regularization of Nonconvex Variational Problems

We are interested in variational problems of the form min ∫ W (∇u) dx, with W nonconvex. The theory of relaxation allows one to calculate the minimum value, but it does not determine a well-defined “solution” since minimizing sequences are far from unique. A natural idea for determining a solution is regularization, i.e. the addition of a higher order term such as ǫ|∇∇u|2. But what is the behavior of the regularized solution in the limit as ǫ → 0? Little is known in general. Our recent work [19, 20, 21] discusses a particular problem of this type, namely minuy=±1 ∫ ∫ ux + ǫ|uyy| dxdy with various boundary conditions. The present paper gives an expository overview of our methods and results. Partially supported by NSF grant DMS-9102829, AFOSR grant 90-0090, and ARO contract DAAL0392-G-0011. Partially supported by NSF grant DMS-9002679 and by SFB 256 at the University of Bonn.