Second order smoothness of weak solutions of degenerate linear equations

2018 ◽  
Vol 265 (2) ◽  
pp. 671-701
Author(s):  
Richard L. Wheeden
Keyword(s):  
Weak Solutions ◽  
Linear Equations ◽  
Second Order ◽  
Degenerate Linear ◽  
Order Smoothness

scholarly journals Post-Newtonian limit: second-order Jefimenko equations

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
David Pérez Carlos ◽  
Augusto Espinoza ◽  
Andrew Chubykalo
Keyword(s):  
Linear Equations ◽  
Space Time ◽  
Second Order ◽  
Field Equations ◽  
Gravitational Fields ◽  
Mass Distributions ◽  
Minkowski Space Time ◽  
Theory Of Gravitation ◽  
Gravity Probe ◽  
Gravitational Equations

Abstract The purpose of this paper is to get second-order gravitational equations, a correction made to Jefimenko’s linear gravitational equations. These linear equations were first proposed by Oliver Heaviside in [1], making an analogy between the laws of electromagnetism and gravitation. To achieve our goal, we will use perturbation methods on Einstein field equations. It should be emphasized that the resulting system of equations can also be derived from Logunov’s non-linear gravitational equations, but with different physical interpretation, for while in the former gravitation is considered as a deformation of space-time as we can see in [2–5], in the latter gravitation is considered as a physical tensor field in the Minkowski space-time (as in [6–8]). In Jefimenko’s theory of gravitation, exposed in [9, 10], there are two kinds of gravitational fields, the ordinary gravitational field, due to the presence of masses, at rest, or in motion and other field called Heaviside field due to and acts only on moving masses. The Heaviside field is known in general relativity as Lense-Thirring effect or gravitomagnetism (The Heaviside field is the gravitational analogous of the magnetic field in the electromagnetic theory, its existence was proved employing the Gravity Probe B launched by NASA (See, for example, [11, 12]). It is a type of gravitational induction), interpreted as a distortion of space-time due to the motion of mass distributions, (see, for example [13, 14]). Here, we will present our second-order Jefimenko equations for gravitation and its solutions.

scholarly journals Higher Integrability of Weak Solutions to a Class of Double Obstacle Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhenhua Hu ◽  
Shuqing Zhou
Keyword(s):  
Differential Equation ◽  
Weak Solutions ◽  
Second Order ◽  
Obstacle Problems ◽  
Elliptic Differential Equation ◽  
Higher Integrability ◽  
Global Higher Integrability ◽  
Quasilinear Elliptic

We first introduce double obstacle systems associated with the second-order quasilinear elliptic differential equationdiv(A(x,∇u))=div f(x,u), whereA(x,∇u),f(x,u)are twon×Nmatrices satisfying certain conditions presented in the context, then investigate the local and global higher integrability of weak solutions to the double obstacle systems, and finally generalize the results of the double obstacle problems to the double obstacle systems.

As-Rigid-As-Possible Stereo under Second Order Smoothness Priors

2014 ◽  
pp. 112-126 ◽  
Author(s):  
Chi Zhang ◽  
Zhiwei Li ◽  
Rui Cai ◽  
Hongyang Chao ◽  
Yong Rui
Keyword(s):  
Second Order ◽  
Order Smoothness

A Simple Practical Criterion to Guarantee Second Order Smoothness of Blend Surfaces

1989 ◽  
Author(s):  
J. Pegna ◽  
F.-E. Wolter
Keyword(s):  
Second Order ◽  
Geometric Design ◽  
Computer Aided Geometric Design ◽  
Common Boundary ◽  
First Order ◽  
Computer Aided ◽  
The Common ◽  
Blend Surfaces ◽  
Continuous Curves ◽  
Order Smoothness

Abstract Computer Aided Geometric Design of surfaces sometimes presents problems that were not envisioned by mathematicians in differential geometry. This paper presents mathematical results that pertain to the design of second order smooth blending surfaces. Second order smoothness normally requires that normal curvatures agree along all tangent directions at all points of the common boundary of two patches, called the linkage curve. The Linkage Curve Theorem proved here shows that, for the blend to be second order smooth when it is already first order smooth, it is sufficient that normal curvatures agree in one direction other than the tangent to a first order continuous linkage curve. This result is significant for it substantiates earlier works in computer aided geometric design. It also offers simple practical means of generating second order blends for it reduces the dimensionality of the problem to that of curve fairing, and is well adapted to a formulation of the blend surface using sweeps. From a theoretical viewpoint, it is remarkable that one can generate second order smooth blends with the assumption that the linkage curve is only first order smooth. This property may be helpful to the designer since linkage curves can be constructed from low order piecewise continuous curves.

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