Upper bounds for singular perturbation problems involving gradient fields

Poliakovsky, Arkady. "Upper bounds for singular perturbation problems involving gradient fields." Journal of the European Mathematical Society 009.1 (2007): 1-43. <http://eudml.org/doc/277645>.

@article{Poliakovsky2007,
abstract = {We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_\varepsilon (v)=\varepsilon \int _\Omega |\nabla ^2v|^2dx + \varepsilon ^\{−1\}\int _\Omega (1−|\nabla v|^2)^2dx$ over $v\in H^2(\Omega )$, where $\varepsilon >0$ is a small parameter. Given $v\in W^\{1,\infty \}(\Omega )$ such that $\nabla v\in BV$ and $|\nabla v|=1$ a.e., we construct a family $\lbrace v_\varepsilon \rbrace $ satisfying: $v_\varepsilon \rightarrow v$ in $W^\{1,p\}(\Omega )$ and $E_\varepsilon (v_\varepsilon )\rightarrow \frac\{1\}\{3\}\int _\{J_\{\nabla v\}\}|\nabla ^+v−\nabla ^−v|^3d\mathcal \{H\}^\{N−1\}$ as $\varepsilon $ goes to 0.},
author = {Poliakovsky, Arkady},
journal = {Journal of the European Mathematical Society},
language = {eng},
number = {1},
pages = {1-43},
publisher = {European Mathematical Society Publishing House},
title = {Upper bounds for singular perturbation problems involving gradient fields},
url = {http://eudml.org/doc/277645},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Poliakovsky, Arkady
TI - Upper bounds for singular perturbation problems involving gradient fields
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 1
SP - 1
EP - 43
AB - We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_\varepsilon (v)=\varepsilon \int _\Omega |\nabla ^2v|^2dx + \varepsilon ^{−1}\int _\Omega (1−|\nabla v|^2)^2dx$ over $v\in H^2(\Omega )$, where $\varepsilon >0$ is a small parameter. Given $v\in W^{1,\infty }(\Omega )$ such that $\nabla v\in BV$ and $|\nabla v|=1$ a.e., we construct a family $\lbrace v_\varepsilon \rbrace $ satisfying: $v_\varepsilon \rightarrow v$ in $W^{1,p}(\Omega )$ and $E_\varepsilon (v_\varepsilon )\rightarrow \frac{1}{3}\int _{J_{\nabla v}}|\nabla ^+v−\nabla ^−v|^3d\mathcal {H}^{N−1}$ as $\varepsilon $ goes to 0.
LA - eng
UR - http://eudml.org/doc/277645
ER -