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The effect of a singular perturbation on nonconvex variational problems. (English) Zbl 0647.49021
The problem studied in the paper is to minimize \[ F_{\epsilon}(u)=\int_{\Omega}(W(u)+\epsilon | \nabla u|^ 2)dx \] subject to the constraints \(\int_{\Omega}u dx=c\), where \(\Omega\) is an open and bounded set in \(R^ n\), \(u\in L^ 1(\Omega)\), \(W\geq 0\), \(W=0\) at more than one point and \(\epsilon\) is a scalar parameter. It is proved that the minimizer \(u_{\epsilon}\to u_ 0\) in \(L^ 1(\Omega)\), where \(u_ 0\) is a solution of a certain new variational problem. \(\Gamma\)-convergence techniques are utilized.
Reviewer: A.Dontchev

MSC:
49K40 Sensitivity, stability, well-posedness
35B25 Singular perturbations in context of PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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