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In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

In this paper, we prove that the ODE system, $ \dot{x}=sinz+cosy$ $ \dot{y}=sinx+cosz$ $ \dot{z}=siny+cosx$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C = 1, has periodic orbits on $(2 \pi T)^3$ with rotation vectors parallel to (1, 0, 0), (0, 1, 0), and (0, 0, 1). An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.

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