Relaxation and regularization of nonconvex variational problems

We are interested in variational problems of the form min ∝W(∇u) dx, withW 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 minuy=±1 ∝∝ux2+ε|uyy|dxdy with various boundary conditions. The present paper gives an expository overview of our methods and results.

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