On a Two-Dimensional Elliptic Problem with Large Exponent in Nonlinearity

Ren, Xiaofeng; Wei, Juncheng

doi:10.2307/2154740

Cited by 22 publications

(6 citation statements)

References 5 publications

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“…In this section, we will prove some identities for the Green function of (− Δ) p under the Navier boundary conditions. Part of these formulas were former proved by Brezis and Peletier 1 when p = 1, N > 2, Ren and Wei 10 when p = 1, N = 2, Chou and Geng 4 when p = 2, N > 4, and Takahashi 11 when p = 2, N = 4.…”

Section: Integral Identities For Green's Function Of \Documentclass{amentioning

confidence: 90%

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Some identities of Green's function for the polyharmonic operator with the Navier boundary conditions and its applications

Takahashi

2012

Mathematische Nachrichten

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Section: Integral Identities For Green's Function Of \Documentclass{amentioning

confidence: 90%

“…The argument is almost the same as before, so we should be brief. Again we only treat the case p ≥ 2, since the formula was proved in 10 when p = 1. Recall Γ( r ) = − C p log r and Δ l Γ = B l r −2 l for l = 1, 2, …, on ∂ B r , here B l is defined in (3.11).…”

Section: Integral Identities For Green's Function Of \Documentclass{amentioning

confidence: 99%

Some identities of Green's function for the polyharmonic operator with the Navier boundary conditions and its applications

Takahashi

2012

Mathematische Nachrichten

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“…Finally, by combing the results above, we can prove the optimal approximation capabilities for the proposed IFE space through the following error estimation for the global interpolation operator.Theorem There exists a constant C independent of the interface location such that the following estimate holds for every

u \in {boldPH}_{italicint}^{2} (T)

{(‖‖, I_{h} boldu - boldu)}_{0, Ω} + h {(||, I_{h} boldu - boldu)}_{1, h, Ω} + h^{2} {(||, I_{h} boldu - boldu)}_{2, h, Ω} \leq {italicCh}^{2} {(‖‖, u)}_{2, Ω},

where | · | 1, h , Ω and | · | 2, h , Ω are the usual discrete semi‐norms defined according to the mesh

{scriptT}_{h}

. Proof By putting and together over all the elements T , we have

{(‖‖, I_{h} boldu - boldu)}_{0, Ω} + h {(||, I_{h} boldu - boldu)}_{1, h, Ω} + h^{2} {(||, I_{h} boldu - boldu)}_{2, h, Ω} \leq {italicCh}^{2} ({(‖‖, u)}_{2, Ω} + {(‖‖, u)}_{1, 6, Ω}) .

Then using the inequality

{(‖‖, w)}_{1, p, Ω}^{2} \leq C {(‖‖, w)}_{2, Ω}^{2}

for any w ∈ W 1, p (Ω) from , we have .…”

Section: Construction Of Ife Spacesmentioning

confidence: 93%

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

Guo

Lin

2019

Numerical Methods Partial

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We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear, and rotated Q1 polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q1 IFE shape functions are always guaranteed regardless of the Lamé parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi‐point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.

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“…Now,

\begin{array}{l} {‖v‖}_{W^{1, p} ((), S_{K})}^{2} & = \sum_{K \in S_{K}} {‖v‖}_{W^{1, p} ((), K)}^{2} \\ = \sum_{K \in S_{K}} ({}, {‖v‖}_{W^{1, p} ((), K \cap Ω_{1})}^{2} + {‖v‖}_{W^{1, p} ((), K \cap Ω_{2})}^{2}) \\ = \sum_{K \in S_{K}} ({}, {‖{\overset{true˜}{v}}_{1}‖}_{W^{1, p} ((), K \cap Ω_{1})}^{2} + {‖{\overset{true˜}{v}}_{2}‖}_{W^{1, p} ((), K \cap Ω_{2})}^{2}) \\ \leq \sum_{K \in S_{K}} ({}, {‖{\overset{true˜}{v}}_{1}‖}_{W^{1, p} ((), K)}^{2} + {‖{\overset{true˜}{v}}_{2}‖}_{W^{1, p} ((), K)}^{2}) . \end{array}

We now recall Sobolev embedding inequality for two dimensions (cf. Ren and Wei )

{(‖‖, w)}_{L^{p} (K)} \leq {italicCp}^{\frac{1}{2}} {(‖‖, w)}_{H^{1} (K)} \forall w \in H^{1} (K), p > 2 .

…”

Section: Preliminariesmentioning

confidence: 99%

A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

Deka

2019

Numerical Methods Partial

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We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L∞(L2) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L∞(L2) norm.

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On a Two-Dimensional Elliptic Problem with Large Exponent in Nonlinearity

Cited by 22 publications

References 5 publications

Some identities of Green's function for the polyharmonic operator with the Navier boundary conditions and its applications

Some identities of Green's function for the polyharmonic operator with the Navier boundary conditions and its applications

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

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