Two-dimensional curvature functionals with superquadratic growth

Kuwert, Ernst, Lamm, Tobias, and Li, Yuxiang. "Two-dimensional curvature functionals with superquadratic growth." Journal of the European Mathematical Society 017.12 (2015): 3081-3111. <http://eudml.org/doc/277792>.

@article{Kuwert2015,
abstract = {For two-dimensional, immersed closed surfaces $f:\Sigma \rightarrow \mathbb \{R\}^n$, we study the curvature functionals $\mathcal \{E\}^p(f)$ and $\mathcal \{W\}^p(f)$ with integrands $(1+|A|^2)^\{p/2\}$ and $(1+|H|^2)^\{p/2\}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^\{2,p\}$ critical points are smooth in both cases. We also prove a compactness theorem for $\mathcal \{W\}^p$-bounded sequences. In the case of $\mathcal \{E\}^p$ this is just Langer’s theorem [16], while for $\mathcal \{W\}^p$ we have to impose a bound for the Willmore energy strictly below $8\pi $ as an additional condition. Finally, we establish versions of the Palais–Smale condition for both functionals.},
author = {Kuwert, Ernst, Lamm, Tobias, Li, Yuxiang},
journal = {Journal of the European Mathematical Society},
keywords = {curvature functionals; Palais–Smale condition; curvature functionals; Palais-Smale condition},
language = {eng},
number = {12},
pages = {3081-3111},
publisher = {European Mathematical Society Publishing House},
title = {Two-dimensional curvature functionals with superquadratic growth},
url = {http://eudml.org/doc/277792},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Kuwert, Ernst
AU - Lamm, Tobias
AU - Li, Yuxiang
TI - Two-dimensional curvature functionals with superquadratic growth
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 12
SP - 3081
EP - 3111
AB - For two-dimensional, immersed closed surfaces $f:\Sigma \rightarrow \mathbb {R}^n$, we study the curvature functionals $\mathcal {E}^p(f)$ and $\mathcal {W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for $\mathcal {W}^p$-bounded sequences. In the case of $\mathcal {E}^p$ this is just Langer’s theorem [16], while for $\mathcal {W}^p$ we have to impose a bound for the Willmore energy strictly below $8\pi $ as an additional condition. Finally, we establish versions of the Palais–Smale condition for both functionals.
LA - eng
KW - curvature functionals; Palais–Smale condition; curvature functionals; Palais-Smale condition
UR - http://eudml.org/doc/277792
ER -