On the minimal solution for quasilinear degenerate elliptic equation and its blow-up

Horiuchi, Toshio; Kumlin, Peter

doi:10.1215/kjm/1250283558

Cited by 7 publications

(3 citation statements)

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“…For the proof see [15,Lemma 2.3] for example. REMARK 5.1 The proof of Theorem 5.1 is organized in the following way.…”

Section: Lemma 52mentioning

confidence: 99%

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Missing terms in the weighted Hardy–Sobolev inequalities and its application

Ando¹,

Horiuchi²

2012

Kyoto J. Math.

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Let Ω be a bounded domain of R n (n ≥ 1) containing the origin. In the present paper we establish the weighted Hardy-Sobolev inequalities with sharp remainders. For example, when α = 1 − n/p and 1 < p < +∞ hold, we establish the following inequality.There exist positive numbers Λn,p,α, C, and R such that we have. This is called the critical Hardy-Sobolev inequality with a sharp remainder involving a singular weight A 1 (|x|) −p A 2 (|x|) −2 , in the sense that the improved inequality holds for this weight but fails for any weight more singular than this one. Here Λn,p,α is a sharp constant independent of each function u. Further we establish the Hardy-Sobolev inequalities in the subcritical case (α > 1 − n/p) and the supercritical case (α < 1 − n/p).As an application, we use our improved inequality to determine exactly when the first eigenvalues of the weighted eigenvalue problems for the operators represented by − div(|x| αp |∇u| p−2 ∇u) − μ/|x| n A 1 (|x|) −p |u| p−2 u (the critical case) will tend to zero as μ increases to Λn,p,α. This also gives us sufficient conditions for the operators to have the positive first eigenvalue in a certain nontrivial functional framework, and we study the eigenvalue problem in the borderline case.

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“…For the proof see [15,Lemma 2.3] for example. REMARK 5.1 The proof of Theorem 5.1 is organized in the following way.…”

Section: Lemma 52mentioning

confidence: 99%

“…For the detailed proof, see [12, Theorem 1 and the proof of compactness p. 383] for example (cf. [15,Proposition 4.2], [13,Proposition 1.2]). …”

Section: Proof Of Theorem 51mentioning

confidence: 99%

Missing terms in the weighted Hardy–Sobolev inequalities and its application

Ando¹,

Horiuchi²

2012

Kyoto J. Math.

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“…Besides, the diversification of materials for the membrane leads to different dielectric profiles f and corresponding solutions. We refer the readers to [9], [11], [12], [17], [19], [20], [23], [27] and references therein for more details, including the evaluation of λ * , discussions on the touchdown phenomenon and the existence and properties of the global solutions to both the elliptic and parabolic problems, as well as the study of the equations with varying dielectric properties from a theoretical perspective. Furthermore, we refer the readers to [22] for modified MEMS problems with a non-local term, to [29] for an advection term and to [8] and references therein for fourth-order problems.…”

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confidence: 99%

Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems

Cherfils¹,

Miranville²,

Peng³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - S

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Our aim in this paper is to study discretized parabolic problems modeling electrostatic micro-electromechanical systems (MEMS). In particular, we prove, both for semi-implicit and implicit semi-discrete schemes, that, under proper assumptions, the solutions are monotonically and pointwise convergent to the minimal solution to the corresponding elliptic partial differential equation. We also study the fully discretized semi-implicit scheme in one space dimension. We finally give numerical simulations which illustrate the behavior of the solutions both in one and two space dimensions.

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On the Caffarelli-Kohn-Niremberg type inequalities and related topics

Horiuchi¹

2019

Sugaku Expositions

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On the minimal solution for quasilinear degenerate elliptic equation and its blow-up

Cited by 7 publications

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Missing terms in the weighted Hardy–Sobolev inequalities and its application

Missing terms in the weighted Hardy–Sobolev inequalities and its application

Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems

On the Caffarelli-Kohn-Niremberg type inequalities and related topics

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