SPECIAL QUADRATURE ERROR ESTIMATES AND THEIR APPLICATION IN THE HARDY-LITTLEWOOD MAJORANT PROBLEM
The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexam- ples for all p = 2 2N. Montgomery conjectured that there exist counterex- amples even among idempotent polynomials. This was proved recently by Mockenhaupt and Schlag with some four-term idempotents. However, Mockenhaupt conjectured that even the classical 1 + e 2ix e 2i (k+2)x three-term character sums, should work for all 2k < p < 2k + 2 and for all k 2 N. In two previous papers we proved this conjecture for k = 0;1;2;3;4, i.e. in the range 0 < p < 10, p = 2 2N. Here we demonstrate that even the k = 5 case holds true. Renements in the technical features of our approach include use of total variation and integral mean estimates in error bounds for a certain fourth order quadrature. Our estimates make good use of the special forms of functions we encounter: linear combinations of powers and powers of log- arithms of absolute value squares of trigonometric polynomials of given degree. Thus the quadrature error estimates are less general, but we can nd better constants which are of practical use for us.
HOW TO CONCENTRATE IDEMPOTENTS
Call a sum of exponentials of the form f(x) = exp (2πiN1x) + exp (2πiN2x) + · · ·+ exp (2πiNmx), where the Nk are distinct integers, an idempotent. We have L interval concentration if there is a positive constant a, depending only on p, such that for each interval I ⊂ [0, 1] there is an idempotent f so that R I |f (x)| dx R 1 0 |f (x)| dx > a. We will explain how to produce such concentration for each p > 0. The origin of this question and the history of the development of its solution will be surveyed.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...