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.

For a bipolar hydrodynamic model of semiconductors in the form of EulerPoisson equations with Dirichlet or Neumann boundary conditions, in this paper we first heuristically analyze the most probable asymptotic profile (the so-called diffusion waves) and then prove this long-time behavior rigorously. For this, we construct correction functions to show the convergence of the original solution to the diffusion wave with optimal convergence rates by the energy method. Moreover, in the case with Dirichlet boundary condition, when the initial perturbation is in some weighted L^1 space, a faster and optimal convergence rate is also given.

We show that any embedded minimal torus in$S$^{3}is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.

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