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 ε|∇∇u|². But what is the behavior of the regularized solution in the limit as ε→0? Little is known in general. Our recent work [19, 20, 21] discusses a particular problem of this type, namely min _{u y}=±1 ∝∝u _x ² +ε|u _yy|dxdy with various boundary conditions. The present paper gives an expository overview of our methods and results.