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Dual Curvature Density Equation with Group Symmetry
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This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure of convex bodies. It is a Monge-Ampere type equation on the unit sphere. If the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation is solved in this paper for certain range of negative p using a variational method. This work generalizes recent results on the Lp dual Minkowski problem of origin-symmetric convex bodies.
Forward citations
Cited by 2 Pith papers
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Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$
For convex bodies in R^3, bounded L_p qth dual curvature with p in [0,1) and q>2+p forces a uniform diameter upper bound and volume lower bound.
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Uniqueness in the near isotropic Lp dual Minkowski problem
For -1<p<1 and q sufficiently close to n, the near-isotropic Lp dual Minkowski problem on the sphere has a unique solution, with a sharp C0 estimate; the even case covers -1<p<q<min{n,n+p}.
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