An Eigenvalue Problem for Prescribed Curvature Equations

We study an eigenvalue problem for prescribed $\sigma _{k}$-curvature equations of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of a unique eigenvalue and its associated hypersurface, which is also unique, provided that the given data is even. Moreover, we show that the...

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Bibliographic Details
Published in:International mathematics research notices 2024-05, Vol.2024 (10), p.8296-8312
Main Author: Lee, Taehun
Format: Article
Language:English
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Summary:We study an eigenvalue problem for prescribed $\sigma _{k}$-curvature equations of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of a unique eigenvalue and its associated hypersurface, which is also unique, provided that the given data is even. Moreover, we show that the hypersurface must be strictly convex. A crucial aspect of our proof involves deriving uniform estimates in $p$ for $L_{p}$-type prescribed curvature equations.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnad220