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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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