Two-dimensional curvature functionals with superquadratic growth

For two-dimensional, immersed closed surfaces $f:\mathrm{\Sigma }\to {\mathbb{R}}^n$, we study the curvature functionals ${\mathcal{E}}^p(f)$ and ${\mathcal{W}}^p(f)$ with integrands $(1+\mathrm{\mid }A{\mathrm{\mid }}^2{)}^{p\mathrm{/}2}$ and $(1+\mathrm{\mid }H{\mathrm{\mid }}^2{)}^{p\mathrm{/}2}$, 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 \cite{langer85}, 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.