caa a22 4500 445843322 CHVBK 20180317145351.0 cr unu---uuuuu 170323e20110501xx s 000 0 eng 10.1007/s00526-010-0356-9 doi (NATIONALLICENCE)springer-10.1007/s00526-010-0356-9 Asymptotic analysis of a second-order singular perturbation model for phase transitions [Elektronische Daten] [Marco Cicalese, Emanuele Spadaro, Caterina Zeppieri] We study the asymptotic behavior, as $${\varepsilon}$$ tends to zero, of the functionals $${F^k_\varepsilon}$$ introduced by Coleman and Mizel in the theory of nonlinear second-order materials; i.e.,$$F^k_\varepsilon(u):=\int\limits_{I} \left(\frac{W(u)}{\varepsilon}-k\,\varepsilon\,(u')^2+\varepsilon^3(u'')^2\right)\,dx,\quad u\in W^{2,2}(I),$$where k>0 and $${W:\mathbb{R}\to[0,+\infty)}$$ is a double-well potential with two potential wells of level zero at $${a,b\in\mathbb{R}}$$. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k 0 such that, for k<k 0, and for a class of potentials W, $${(F^k_\varepsilon)}$$ Γ(L 1)-converges to$$F^k(u):={\bf m}_k \, \#(S(u)),\quad u\in BV(I;\{a,b\}),$$where m k is a constant depending on W and k. Moreover, in the special case of the classical potential $${W(s)=\frac{(s^2-1)^2}{2}}$$, we provide an upper bound on the values of k such that the minimizers of $${F_\varepsilon^k}$$ cannot develop oscillations on some fine scale and a lower bound on the values for which oscillations occur, the latter improving a previous estimate by Mizel, Peletier and Troy. Springer-Verlag, 2010 Cicalese Marco Dipartimento di Matematica e Applicazioni "R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126, Napoli, Italy aut Spadaro Emanuele Hausdorff Center for Mathematics Bonn, Endenicher Allee 60, 53115, Bonn, Germany aut Zeppieri Caterina Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany aut Calculus of Variations and Partial Differential Equations Springer-Verlag 41/1-2(2011-05-01), 127-150 0944-2669 41:1-2<127 2011 41 526 https://doi.org/10.1007/s00526-010-0356-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00526-010-0356-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Cicalese Marco Dipartimento di Matematica e Applicazioni "R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126, Napoli, Italy aut NATIONALLICENCE 700 1- Spadaro Emanuele Hausdorff Center for Mathematics Bonn, Endenicher Allee 60, 53115, Bonn, Germany aut NATIONALLICENCE 700 1- Zeppieri Caterina Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany aut NATIONALLICENCE 773 0- Calculus of Variations and Partial Differential Equations Springer-Verlag 41/1-2(2011-05-01), 127-150 0944-2669 41:1-2<127 2011 41 526 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer