Geodesic
Two-dimensional curvature functionals with superquadratic growth
Journal of the European Mathematical Society, Tome 17 (2015) no. 12, pp. 3081-3111.

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For two-dimensional, immersed closed surfaces f:Σ→Rn, we study the curvature functionals Ep(f) and Wp(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 W2,p critical points are smooth in both cases. We also prove a compactness theorem for Wp-bounded sequences. In the case of Ep this is just Langer's theorem [16], while for Wp we have to impose a bound for the Willmore energy strictly below 8π as an additional condition. Finally, we establish versions of the Palais–Smale condition for both functionals.
DOI : 10.4171/jems/580
Classification : 53-XX, 35-XX
Keywords: Curvature functionals, Palais–Smale condition 
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     author = {Ernst Kuwert and Tobias Lamm and Yuxiang Li},
     title = {Two-dimensional curvature functionals with superquadratic growth},
     journal = {Journal of the European Mathematical Society},
     pages = {3081--3111},
     publisher = {mathdoc},
     volume = {17},
     number = {12},
     year = {2015},
     doi = {10.4171/jems/580},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/580/}
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Ernst Kuwert; Tobias Lamm; Yuxiang Li. Two-dimensional curvature functionals with superquadratic growth. Journal of the European Mathematical Society, Tome 17 (2015) no. 12, pp. 3081-3111. doi : 10.4171/jems/580. http://geodesic.mathdoc.fr/articles/10.4171/jems/580/

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