Operator estimates and elliptic regularity

1991 ◽  
pp. 46-66
Author(s):  
Michael E. Taylor
Keyword(s):  
Elliptic Regularity

scholarly journals Regularity and approximation of strong solutions to rate-independent systems

2017 ◽  
Vol 27 (13) ◽  
pp. 2511-2556 ◽  
Author(s):  
Filip Rindler ◽  
Sebastian Schwarzacher ◽  
Endre Süli
Keyword(s):  
Strong Solutions ◽  
Local Convexity ◽  
Finite Element Discretization ◽  
Elliptic Regularity ◽  
Element Discretization ◽  
Fully Discrete ◽  
Regularity Estimates ◽  
Mathematical Techniques ◽  
Regularity Results ◽  
Higher Regularity

Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this work, we prove the existence of Hölder-regular strong solutions for a class of rate-independent systems. We also establish additional higher regularity results that guarantee the uniqueness of strong solutions. The proof proceeds via a time-discrete Rothe approximation and careful elliptic regularity estimates depending in a quantitative way on the (local) convexity of the potential featuring in the model. In the second part of the paper, we show that our strong solutions may be approximated by a fully discrete numerical scheme based on a spatial finite element discretization, whose rate of convergence is consistent with the regularity of strong solutions whose existence and uniqueness are established.

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.

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