EXISTENCE OF POSITIVE SOLUTIONS TOA ROBIN BOUNDARY VALUE PROBLEM FORA CLASS QUASILINEAR ORDINARYDIFFERENTIAL EQUATIONS

11尬roductlonIn this paper;we study the exlstence ofnonneg时Ive solutions ofone dl－menslonal－Lapfaclan boundary value problems一(电N》’二入M;0＜x＜1,儿1)u’仰二0,*’川十口叫)二0,队2)where入＞0md a＞0 are parmetersmd f E C’[0,1],屯(。)＝I冲－‘。,p＞1二This problem ppears In the study ofnon－Newtonlan luids(see[1;2])andnon－Newtonlan iltratlon(see[3])．The quantity p Is a characteristic of*theteristicmadium．Media with p＞2 are called dllatantn"ids and those with p＜2 arecalled pseudop…     

The one phase free boundary problem for the -Laplacian with non-constant Bernoulli boundary condition

Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the -Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure on the ``free' streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function is subject to certain convexity properties. In our earlier results we have considered the case of constant . In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the -capacitary potentials in convex rings, with boundaries.

     

Hugoniót--Maslov Conditions for Vortex Singular Solutions of the Shallow Water Equations

For the phase of vortex singular solutions of the shallow water equations we justify the Hamilton--Jacobi equation corresponding to the hydrodynamical mode of perturbation propagation. We also obtain the next correction to the Cauchy--Riemann conditions describing how the singular part of the solution affects the smooth background.     

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

Strong duals of projective limits of (LB)-spaces

We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called strong P-type. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.     

Sigma-Boundary Statistics by Length and Number

This paper presents a rationale for comparative use of length fraction and number fraction statistics in grain boundary analysis from orientation maps generated by electron back-scatter diffraction (EBSD). The length and number fraction statistics for 3 ⁿ coincidence site lattice (CSL) boundaries were measured and compared. The length fraction of 3 boundaries was 0.48 whereas the number fraction was significantly less, 0.36. A simple model was generated to estimate both the length fraction and number fraction of annealing twins (a subset of 3). The model showed that the number fraction of twins is 0.68, 0.75, 0.79 and 0.82 of the length fraction for 1, 2, 3 and 4 twins-per-grain respectively. For the experimental data the number fraction was 0.76 of the length fraction, implying that there were on average two twins-per-grain. In contrast to the 3 case, the length fraction for 9 and 27 boundaries was less than the number fraction. There are more inaccuracies involved in obtaining the number fraction than in obtaining the length fraction from EBSD maps, therefore the length fraction should be recommended as the standard reporting method. However a knowledge of the distribution in the microstructure of 3 ⁿ segments is often crucial to the inquiry in addition to the length fraction.     

The Surface Energy of MgO: Multiscale Reconstruction from Thermal Groove Geometry

The surface energy of MgO is determined using experimental data collected from equilibrated thermal grooves circumscribing island grains. Local equilibrium assumptions at each groove require that the Herring equations be satisfied at each data site, thereby yielding a large and overdetermined system of equations involving the surface energy . This inverse problem is then solved using a new technique that is statistical in nature and multiscale in implementation. The resulting discrete solution represents a statistically significant representation of the surface energy of MgO as a function of surface orientation. Comparisons to results derived from a more traditional approach, along with suggested further applications, are discussed.     

Entanglement of Resonantly Coupled Field Modes in Cavities with Vibrating Boundaries

We study time dependence of various measures of entanglement (covariance entanglement coefficient, purity entanglement coefficient, normalized distance coefficient, entropy coefficients) between resonantly coupled modes of the electromagnetic field in ideal cavities with oscillating boundaries. Two types of cavities are considered — a three-dimensional cavity possessing eigenfrequencies ₃ = 3₁, whose wall oscillates at the frequency _w = 2₁, and a one-dimensional (Fabry–Perot) cavity with an equidistant spectrum _n = n₁ where the distance between perfect mirrors oscillates at the frequencies ₁ and 2₁. The behavior of entanglement measures in these cases turns out to be completely different, although all three coefficients demonstrate qualitatively similar time dependences in each case (except some specific situations where the covariance entanglement coefficient based on traces of covariance submatrices seems to be essentially more sensitive to entanglement than other measures, which are based on determinants of covariance submatrices). Different initial states of the field, namely, vacuum, squeezed vacuum, thermal, Fock, and even/odd coherent states, are considered.     

利用激光全息干涉法测量微小机械变形方向

提出用可分离双参考光两次曝光全息测量物体变形方向的方法。设计的光路系统用两个平面反射镜将一束扩展激光束分成两束参考光和一束物光,籍此记录物体变形下的可分离双参考光两次曝光全息图。该全息图再现的两个像的图层可以被分离,再现的全息干涉图上的条纹可通过改变曝光次序和微位移再现参考光虚点光源的位置而移动。物体变形方向、干涉条纹移动方向和反射镜位移方向以及曝光次序之间有确定的关系。该干涉图含有物体变形方向和位移数值两种信息。从理论上重点分析了物体变形方向信息所遵循的基本规律,并由此得出了测量物体变形方向的方法。最后给出了实验结果。实验表明提出的方法和理论是正确的,物体变形方向的检测正确率可达100%。