Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with $rm{VMO}_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.     

Measurement of the neutron beam polarization of BL05/NOP beamline at J-PARC

We measured the neutron beam polarization of the BL05/NOP (Neutron Optics and Physics) beamline at J-PARC with an accuracy of less than 10⁻³ using polarized ³He gas as a neutron spin analyzer. Precise polarimetry of the neutron beam is necessary to understand the beamline optics as well as for the asymmetry measurements of the neutron beta decay, which are planned in this beamline.     

外磁场中单畴反铁磁颗粒的宏观量子效应

用瞬子方法研究了外加磁场对单畴双轴反铁磁颗粒宏观量子效应的影响. 当外磁场沿易磁化方向时,简并基态中的一个能量抬高,变为亚稳态,其隧穿衰变率随外磁场增大;当外磁场沿中间磁化方向时,能级的隧穿劈裂随外磁场的变化而振荡.从而提出一种观察单畴反铁磁颗粒中宏观量子相干和隧穿现象的实验方法.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.     

Abstract estimates of the rate of convergence for optimal control problems

A method for solving optimal control problems with general elliptic operators is presented and analyzed. Especially, estimates of the rate of convergence for the control problems with the proposed approach are derived independently of the underlying approximation method. Some numerical experiments with the proposed method are included.     

Electromagnetically induced transparency and evolution of a two-level system under chaotic field of arbitrary intensity

The electromagnetically induced transparency (EIT) with a (near-)resonant chaotic (amplitude-phase fluctuating, Gaussian-Markovian) coupling field is studied theoretically. The Fourier transform of the steady-state EIT spectrum, which determines a nonstationary probe absorption, is also considered. This quantity equals the average diagonal element of the (reduced) evolution operator of the coupled transition (the evolution function). The exact solution in the form of a continued fraction is obtained and used to perform numerical calculations. Moreover, a number of approximate analytical results are obtained, which, together with the results of previous publications, describe the EIT and the evolution function in all possible regimes. In particular, in the constructive-interference case the EIT increases with the coupling-field bandwidth ν at sufficiently small ν. For a strong field, the maximum of the transparency as a function of ν is less than that for a monochromatic field of the same average intensity. In contrast, for a weak field, there is a range of ν values, where the field fluctuations do not affect the EIT. The latter result is shown to hold for a broad class of stochastic fields. Received 31 December 2000 and Received in final form 14 May 2001     

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.     

Galerkin method for a Stefan-type problem in one space dimension

Based on a Landau-type transformation, both continuous and discrete in time L²-Galerkin methods are applied to a single-phase Stefan-type problem in one space dimension. Optimal rates of convergence in L^ℵ, L^∞, and H¹-norms are derived and computational results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 393–416, 1997     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997