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Relaxation and regularization of nonconvex variational problems

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Abstract

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 min u y=±1 ∝∝u 2x +ε|u yy|dxdy with various boundary conditions. The present paper gives an expository overview of our methods and results.

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Authors and Affiliations

  1. Courant Institute, New York

    Robert V. Kohn & Stefan Müller

  2. Universität Bonn, Bonn, Germany

    Robert V. Kohn & Stefan Müller

Authors
  1. Robert V. Kohn
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  2. Stefan Müller
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Additional information

Dedicated to prof. L. Amerio

Conferenza tenuta da Robert Kohn il 28 settembre 1992

Partially supported by NSF grant DMS-9102829, AFOSRR grant 90-0090, and ARO contract DAAL03-92-G-0011.

Partially supported by NSF grant DMS-9002679 and by NSF grant DMS-9002679 and by SFB at the University of Bonn.

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Kohn, R.V., Müller, S. Relaxation and regularization of nonconvex variational problems. Seminario Mat. e. Fis. di Milano 62, 89–113 (1992). https://doi.org/10.1007/BF02925437

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