In this paper, we consider a type of cone-constrained convex program in finite-dimensional space, and are interested in characterization of the solution set of this convex program with the help of the Lagrange multiplier. We establish necessary conditions for a feasible point being an optimal solution. Moreover, some necessary conditions and sufficient conditions are established which simplifies the corresponding results in Jeyakumar et al. (J Optim Theory Appl 123(1), 83103, 2004). In particular, when the cone reduces to three specific cones, that is, the p -order cone, p cone and circular cone, we show that the obtained results can be achieved by easier ways by exploiting the special structure of those three cones.

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the <i>C</i>⁰-norm of the initial data is sufficiently small, while the <i>C</i>¹-norm of the initial data can be large. The analysis is based on <i>a priori</i> estimates, which are obtained by a generalised Lax transformation.

The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity . Under some conditions on the initial and boundary data, we show that the thickness is of the order | In |. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.

Let (Mn, g) be a complete non-compact K¨ahler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that M is holomorphically covered by a pseudoconvex domain in Cn which is homeomorphic to R2n, provided (Mn, g) has uniform linear average quadratic curvature decay.

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