engleski

Relaxation of Ginzburg-Landau functional with 1-Lipschitz penalizing term in one dimension by Young measures on micro-patterns

In this paper we study asymptotic behavior as $\vep\str 0$ of Ginzburg-Landau functional $$ I_{; ; \vep}; ; (v):=\int_{; ; \Omega}; ; \Big({; ; \vep}; ; ^2 v''^2(s)+W(v'(s))+a(s)(v(s)+g(s))^2\Big)ds. $$ Our consideration follows the approach introduced in the original paper~\cite{; ; AM}; ; by G.~Alberti and S.~M\"uller, where the case $g=0$ was studied. We show that their program can be modified in the case of functional $I_{; ; \vep}; ; $: we define suitable relaxation of $I_{; ; \vep}; ; $ and prove a $\Gamma$-convergence result in the topology of the so-called Young measures on micropatterns. Moreover, we identify a unique minimizing measure for the functional in the limit, which is the unique translation-invariant measure supported on the orbit of a particular periodic sawtooth function having minimal period and slope dependent on a derivative of $g$.

young measures; relaxation; gamma convergence

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