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In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is proposed and solved under the asymptotic conditions. In the end, we give a solution to the Minkowski problem for σ-finite measure on the conic domain Ω C .

We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem. The proposed finite element spaces are subspaces of H H H(curl), but not of H H H(grad curl), which are different from the existing nonconforming ones [10,12,13]. The wellposedness of the discrete problem is proved and optimal error estimates in discrete H H H(grad curl) norm, H H H(curl) norm and L L L 2 norm are derived. Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.