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Prescribed $L_p$ quotient curvature problem and related eigenvalue problem
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abstract
In this paper, we investigate the existence of admissible (and strictly convex) smooth solutions to the prescribed $L_p$ quotient type curvature problem with $p>1$. For cases where $p=k-l+1$ and $p> k-l+1$, we obtain an admissible solution without any additional conditions, which is strictly spherically convex under a convexity condition. Under the same convexity condition, we establish the existence of a strictly spherically convex solution for the case $p<k-l+1$, provided that the prescribed function is even, a condition known to be necessary.
Forward citations
Cited by 2 Pith papers
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The capillary $L_p$-Minkowski problem
Existence of smooth convex capillary bodies with prescribed capillary L_p-surface area measure is proved for all p>1, with a symmetry condition needed when 1<p<n+1.
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Convex capillary hypersurfaces of prescribed curvature problem
A strictly convex capillary hypersurface with prescribed k-th Weingarten curvature exists whenever the prescribed data is symmetric under horizontal reflection.
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