Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaces

2020 ◽  
pp. 2050014
Author(s):  
The Quan Bui ◽  
The Anh Bui ◽  
Xuan Thinh Duong
Keyword(s):  
Elliptic Equations ◽  
Global Regularity ◽  
Order Elliptic Equation ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Regularity Estimates ◽  
Sparse Operators ◽  
Second Order Elliptic Equation ◽  
Bmo Coefficients ◽  
Weighted Variable

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a [Formula: see text] domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

scholarly journals ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS FOR ELLIPTIC EQUATIONS WITH EXPONENTIAL NONLINEARITY AND SINGULAR DATA

2009 ◽  
Vol 11 (03) ◽  
pp. 395-411 ◽  
Author(s):  
LEI ZHANG
Keyword(s):  
Elliptic Equations ◽  
Type Equation ◽  
Global Solutions ◽  
Order Elliptic Equation ◽  
Liouville Type ◽  
Exponential Nonlinearity ◽  
Second Order Elliptic Equation ◽  
Singular Data ◽  
Sharp Error ◽  
Uniformly Bounded

We consider a sequence of blowup solutions of a two-dimensional, second-order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci–Chen–Lin–Tarantello, it is proved that the profile of the solutions differs from global solutions of a Liouville-type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.

MORTAR SPECTRAL ELEMENT METHODS FOR ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

2002 ◽  
Vol 12 (04) ◽  
pp. 497-524 ◽  
Author(s):  
CHRISTINE BERNARDI ◽  
NEJMEDDINE CHORFI
Keyword(s):  
Elliptic Equations ◽  
Spectral Element ◽  
Discontinuous Coefficients ◽  
Optimal Error Estimates ◽  
Order Elliptic Equation ◽  
Optimal Error ◽  
Element Discretization ◽  
Second Order Elliptic Equation ◽  
Piecewise Continuous ◽  
Dimensional Domain

We consider a second-order elliptic equation with piecewise continuous coefficients in a bounded two-dimensional domain. We propose a spectral element discretization of this problem which relies on the mortar domain decomposition technique. We prove optimal error estimates. Next, we compare several versions, conforming or not, of this discretization by means of numerical experiments.

scholarly journals Global W1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains

2018 ◽  
Vol 7 (4) ◽  
pp. 517-533 ◽  
Author(s):  
The Anh Bui
Keyword(s):  
Elliptic Equations ◽  
Variable Exponent ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Lebesgue Spaces ◽  
Quasilinear Equations ◽  
Bmo Coefficients ◽  
Reifenberg Flat ◽  
Quasilinear Elliptic

AbstractIn this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

scholarly journals Estimates for solutions of elliptic equations in a limit case

1987 ◽  
Vol 36 (3) ◽  
pp. 425-434 ◽  
Author(s):  
Ester Giarrusso ◽  
Guido Trombetti
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Limit Case ◽  
Order Elliptic Equation ◽  
Best Constant ◽  
Homogeneous Dirichlet Problem ◽  
Second Order Elliptic Equation ◽  
Bounded Open Subset ◽  
The Right

Let u be a week solution of homogeneous Dirichlet problem for a second order elliptic equation of divergence form, in a bounded open subset of ℝn. We prove, that if the right hand side of the equation is an element of H−1, n(Ω), then u belongs to the Orlicz space LΦ where Φ(t) = exp(|t|n/(n−1)) − 1. We employ the properties of the Schwartz symmetrization thus obtaining the “best” constant of the estimate.

The maximal operator on weighted variable Lebesgue spaces

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
David Cruz-Uribe ◽  
Lars Diening ◽  
Peter Hästö
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lebesgue Spaces ◽  
Hölder Continuous ◽  
Variable Lebesgue Spaces ◽  
Muckenhoupt Condition ◽  
Necessary And Sufficient ◽  
Weighted Variable

AbstractWe study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces L ωp(·) (Ω). For a given log-Hölder continuous exponent p with 1 < inf p ⩽ supp < ∞ we present a necessary and sufficient condition on the weight ω for the boundedness of M. This condition is a generalization of the classical Muckenhoupt condition.

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