scholarly journals Optimal metric regularity in general relativity follows from the RT-equations by elliptic regularity theory in $L^p$ -spaces

2020 ◽  
Vol 27 (3) ◽  
pp. 199-242
Author(s):  
Moritz Reintjes ◽  
Blake Temple
Keyword(s):  
General Relativity ◽  
Metric Regularity ◽  
Regularity Theory ◽  
Elliptic Regularity ◽  
Elliptic Regularity Theory

A counterexample in elliptic regularity theory

1981 ◽  
Vol 34 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Michael Struwe
Keyword(s):  
Regularity Theory ◽  
Elliptic Regularity ◽  
Elliptic Regularity Theory

METRIC REGULARITY—A SURVEY PART II. APPLICATIONS

2016 ◽  
Vol 101 (3) ◽  
pp. 376-417 ◽  
Author(s):  
A. D. IOFFE
Keyword(s):  
Variational Analysis ◽  
State Of The Art ◽  
Metric Regularity ◽  
Optimization Theory ◽  
Regularity Theory ◽  
Implicit Function ◽  
Classical Analysis ◽  
Set Valued Mappings ◽  
Principal Parts ◽  
Modern Optimization

Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

scholarly journals How to smooth a crinkled map of space–time: Uhlenbeck compactness for L ∞ connections and optimal regularity for general relativistic shock waves by the Reintjes–Temple equations

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200177
Author(s):  
Moritz Reintjes ◽  
Blake Temple
Keyword(s):  
Shock Waves ◽  
Regularity Theory ◽  
Existence Theory ◽  
Lorentzian Geometry ◽  
Riemann Curvature Tensor ◽  
Optimal Regularity ◽  
Elliptic Regularity ◽  
Relativistic Shock ◽  
Nonlinear Elliptic ◽  
General Relativistic

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem( Γ ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T M of arbitrary manifolds M , including Lorentzian manifolds of general relativity.

scholarly journals Periodic Solutions to Klein–Gordon Systems with Linear Couplings

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jianyi Chen ◽  
Zhitao Zhang ◽  
Guijuan Chang ◽  
Jing Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Periodic Solutions ◽  
Wave Equations ◽  
Regularity Theory ◽  
Careful Analysis ◽  
Dirichlet Boundary Conditions ◽  
Quantum Field Theories ◽  
Elliptic Regularity ◽  
Klein Gordon ◽  
Higher Regularity

Abstract In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u , v u,v satisfy the Dirichlet boundary conditions on spatial interval [ 0 , π ] [0,\pi] , b > 0 b>0 and f , g f,g are 2 ⁢ π 2\pi -periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) with time period 2 ⁢ π 2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε → 0 \varepsilon\to 0 , ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

METRIC REGULARITY—A SURVEY PART 1. THEORY

2016 ◽  
Vol 101 (2) ◽  
pp. 188-243 ◽  
Author(s):  
A. D. IOFFE
Keyword(s):  
Variational Analysis ◽  
State Of The Art ◽  
Metric Regularity ◽  
Optimization Theory ◽  
Regularity Theory ◽  
Implicit Function ◽  
Classical Analysis ◽  
Set Valued Mappings ◽  
Principal Parts ◽  
Modern Optimization

Metric regularity theory lies at the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. This paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

Sign in / Sign up

Export Citation Format

Share Document