An Eigenvalue Problem for Prescribed Curvature Equations
Abstract 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...
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| Published in: | International mathematics research notices 2023-09 |
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| Main Author: | |
| Format: | Article |
| Language: | eng |
| Online Access: | Get full text |
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| Summary: | Abstract 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. |
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| ISSN: | 1073-7928 1687-0247 |