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.

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 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 Dirichlet Problem for a Second Order Elliptic Equation Degenerating on the Whole Boundary When the Boundary Condition Is Satisfied in the Presence of Some Weight

2006 ◽  
Vol 13 (3) ◽  
pp. 573-579
Author(s):  
Mikheil Usanetashvili
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
General Type ◽  
Type Equation ◽  
Second Order ◽  
Elliptic Type ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation

Abstract The first boundary value problem with weight is investigated for a general-type second order elliptic type equation degenerating on the whole boundary.

scholarly journals Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1381
Author(s):  
Jinru Wang ◽  
Wenhui Shi ◽  
Lin Hu
Keyword(s):  
Elliptic Equations ◽  
Numerical Solutions ◽  
Homogeneous Boundary Condition ◽  
Riesz Bases ◽  
Condition Numbers ◽  
Order Elliptic Equation ◽  
Approximation Properties ◽  
Numerical Examples ◽  
Stiffness Matrices ◽  
Second Order Elliptic Equation

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

BLOW-UP ANALYSIS FOR LIOUVILLE TYPE EQUATION IN SELF-DUAL GAUGE FIELD THEORIES

2005 ◽  
Vol 07 (02) ◽  
pp. 177-205 ◽  
Author(s):  
HIROSHI OHTSUKA ◽  
TAKASHI SUZUKI
Keyword(s):  
Gauge Field ◽  
Blow Up ◽  
Type Equation ◽  
Singular Part ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Liouville Type ◽  
Blow Up Analysis ◽  
The Green Function ◽  
Dirac Measures

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

scholarly journals Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

2020 ◽  
Vol 20 (2) ◽  
pp. 373-384
Author(s):  
Quoc-Hung Nguyen ◽  
Nguyen Cong Phuc
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Linear Range ◽  
Measure Data ◽  
Regularity Conditions ◽  
Compactly Supported ◽  
Singular Data ◽  
Quasilinear Elliptic

AbstractWe characterize the existence of solutions to the quasilinear Riccati-type equation\left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.with a distributional or measure datum σ. Here {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ({p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that {p>1} and {q>p}. For measure data, we assume that they are compactly supported in Ω, {p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range {p-1<q<1}. We also assume more regularity conditions on {\mathcal{A}} and on {\partial\Omega\Omega} in this case.

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

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