Some remarks on $$W^{s,p}$$ interior elliptic regularity estimates

2021 ◽  
Author(s):  
A. Behzadan
Keyword(s):  
Elliptic Regularity ◽  
Regularity Estimates

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.

Geodesics and the Hadamard parametrix

2017 ◽  
Author(s):  
Christopher D. Sogge
Keyword(s):  
Quadratic Form ◽  
Coordinate System ◽  
Open Subset ◽  
Special Form ◽  
Local Coordinate System ◽  
Beltrami Operator ◽  
Compact Manifolds ◽  
Elliptic Regularity ◽  
Normal Coordinates ◽  
Regularity Estimates

This chapter studies the spectrum of Laplace–Beltrami operators on compact manifolds. It begins by defining a metric on an open subset Ω‎ ⊂ Rn, in order to lift their results to corresponding ones on compact manifolds. The chapter then details some elliptic regularity estimates, before embarking on a brief review of geodesics and normal coordinates. The purpose of this review is to show that, with given a particular Laplace–Beltrami operator and any point y0 in Ω‎, one can choose a natural local coordinate system y = κ‎(x) vanishing at y0 so that the quadratic form associated with the metric takes a special form. To conclude, the chapter turns to the Hadamard parametrix.

Weighted regularity estimates for a class of higher-order nonlinear parabolic and elliptic systems

2019 ◽  
Vol 180 ◽  
pp. 184-207
Author(s):  
The Anh Bui ◽  
Xuan Thinh Duong ◽  
Xuan Truong Le
Keyword(s):  
Elliptic Systems ◽  
Higher Order ◽  
Regularity Estimates ◽  
Nonlinear Parabolic

scholarly journals Sharp regularity estimates for second order fully nonlinear parabolic equations

2016 ◽  
Vol 369 (3-4) ◽  
pp. 1623-1648 ◽  
Author(s):  
João Vitor da Silva ◽  
Eduardo V. Teixeira
Keyword(s):  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Second Order ◽  
Fully Nonlinear ◽  
Regularity Estimates ◽  
Nonlinear Parabolic

Operator estimates and elliptic regularity

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

Global Attractor of the Liquid Helium-4 System in Hk Space

2013 ◽  
Vol 444-445 ◽  
pp. 731-737
Author(s):  
Zhi Bo Hou ◽  
Li Mei Li
Keyword(s):  
Existence Theorem ◽  
Liquid Helium ◽  
Global Attractor ◽  
Iteration Procedure ◽  
Helium 4 ◽  
Bounded Set ◽  
Regularity Estimates

In this paper, by using an iteration procedure, regularity estimates of the linear semi-groups and a generalized existence theorem of global attractor, we prove that the liquid helium-4 system possesses a global attractor in space for all , which attracts any bounded set of in the-norm.

scholarly journals Regularity of Global Attractor for the Reaction-Diffusion Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Hong Luo
Keyword(s):  
Sobolev Space ◽  
Diffusion Equation ◽  
Existence Theorem ◽  
Global Attractor ◽  
Reaction Diffusion ◽  
Iteration Procedure ◽  
Reaction Diffusion Equation ◽  
Bounded Subset ◽  
Regularity Estimates ◽  
Regularity Of Global Attractor

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev spaceHkfor allk>0, which attracts any bounded subset ofHk(Ω) in theHk-norm.

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