Equations and inequalities

This is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein-vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. Following [Kl-Ro] we develop new analytic methods based on Strichartz-type inequalities which result in a gain of half a derivative relative to the classical result. ...

52p noel_noel 17-01-2013 60 5   Download

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in the first paper. ...

50p noel_noel 17-01-2013 54 5   Download

ON DELAY DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS TADEUSZ JANKOWSKI Received 21 July 2004 The monotone iterative method is used to obtain sufficient conditions which guarantee that a delay differential equation with a nonlinear boundary condition has quasisolutions, extremal solutions, or a unique solution. Such results are obtained using techniques of weakly coupled lower and upper solutions or lower and upper solutions. Corresponding results are also obtained for such problems with more delayed arguments.

14p sting12 10-03-2012 39 6   Download

EIGENVALUE PROBLEMS FOR DEGENERATE NONLINEAR ELLIPTIC EQUATIONS IN ANISOTROPIC MEDIA ˘ DUMITRU MOTREANU AND VICENTIU RADULESCU ¸ Received 23 September 2004 and in revised form 26 November 2004 We study nonlinear eigenvalue problems of the type − div(a(x)∇u) = g(λ,x,u) in RN , where a(x) is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality.

21p sting12 10-03-2012 46 5   Download

ASYMPTOTIC ESTIMATES AND EXPONENTIAL STABILITY FOR HIGHER-ORDER MONOTONE DIFFERENCE EQUATIONS ´ EDUARDO LIZ AND MIHALY PITUK Received 21 May 2004 Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation.

15p sting12 10-03-2012 59 5   Download

EXPONENTIAL STABILITY OF DYNAMIC EQUATIONS ON TIME SCALES ALLAN C. PETERSON AND YOUSSEF N. RAFFOUL Received 6 July 2004 and in revised form 16 December 2004 We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by defining appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given. 1. Introduction This paper considers the exponential stability of the zero solution of the first-order vector dynamic equation x∆ = f (t,x), t ≥ 0. (1.

12p sting12 10-03-2012 44 5   Download

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

31p dauphong18 09-03-2012 40 3   Download

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Part-Metric-Related Inequality Chain and Application to the Stability Analysis of Difference Equation

9p sting10 07-03-2012 48 3   Download