Abstract In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions J(h):=(∫Rd|h| dx)a611∫Rd|63h|2 dx(∫Rd|h|m+1 dx)a+1m+1≥β0, where m > 0 for d = 1, 2, for , and . The Euler–Lagrange equation for critical points of in the non-negative radial decreasing function space is given by a free boundary problem for a generalized Lane–Emden equation, which has a unique solution (denoted by h c ) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass for the thin-film equation and the best constant of the Sz. Nagy inequality in one dimension was first noted by Witelski et al (2004 Eur. J. Appl. Math. 15 223–56). For the following critical thin film equation in multi-dimension ht+6366(h 63Δh)+6366(h 63hm)=0,x∈Rd, where m = 1 + 2/d, the critical mass is also given by . A finite time blow-up occurs for solutions with the initial mass larger than M c . On the other hand, if the initial mass is less than M c and a global non-negative entropy weak solution exists, then the second moment goes to infinity as or in for some subsequence . This shows that a part of the mass spreads to infinity.

The paper aims at analytically exhibiting for the first time the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. For doing so, a homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with re- spect to the membrane polarization. Different microstructures are shown to be distinguish- able via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell-Wagner-Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells.

The time evolution of the distribution function for the charged particles in a dilute gas is governed by the VlasovPoissonBoltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible NavierStokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].

In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.

In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.