Naslov
Remarks on $\Gamma$-Convergence of Penalized Functionals of Ginzburg-Landau Type in One Dimension

Autori
Raguž, Andrija

Vrsta, podvrsta i kategorija rada
Poglavlja u knjigama, znanstveni

Knjiga
Multiscale problems in science and technology

Urednik/ci
Antonić, Nenad ; van Duijn, C. J. ; Jäger, Willi ; Mikelić, Andro

Izdavač
Springer

Grad
Berlin

Godina
2002

Raspon stranica

ISBN
3-540-43584-0

Ključne riječi
$\Gamma$-Convergence, Ginzburg-Landau functional, relaxation

Sažetak
In this note we study the Ginzburg-Landau functional $$ I^{\vep}(v):=\int_{\Omega} ({\vep}^2 v''(s)^2+W(v'(s))+a(s)(v(s)-g(s))^2)ds $$ for $v\in {\rm H}^2_{per}(\Omega)$. $\Omega\subseteq\R$ is a bounded open set, $a\in {\rm L}^{\infty}(\Omega)$, $a\geq\alpha>0$ and $g\in {\rm C}^{1}(\overline{\Omega})$, $|g'|<1$. $W$ is non-negative continuous function such that $W(\xi)=0$ iff $\xi\in\{-1, 1\}$. In view of the approach of Alberti and M\"uller in~\cite{raguzAM}, we formulate the relaxation and minimization problem related to the functional $I^{\vep}$ and we discuss the choice of relaxation and blowup procedure adjusted to capture two characteristic small scales associated to the minimizing sequences. Also, we prove $\Gamma$-convergence result for the integrands, and we highlight the idea of proof for $\Gamma$-convergence for integral functionals induced by the chosen relaxation.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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