Relaxation and regularization of nonconvex variational problems

Robert V. Kohn, Stefan Müller

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)89-113
Number of pages25
JournalRendiconti del Seminario Matematico e Fisico di Milano
Volume62
Issue number1
DOIs
StatePublished - Dec 1992

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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