The 𝐴𝐵 program in geometric analysis: sharp Sobolev inequalities and related problems

Druet, Olivier; Hebey, Emmanuel

doi:10.1090/memo/0761

Cited by 67 publications

(79 citation statements)

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“…For the existence of extremal functions for the classical Moser-Trudinger inequality, we would like to mention Carleson and Chang [3], Flucher [8], Lin [12], Li [10] and Li-Liu [11]. About extremals for optimal Sobolev inequalities on Riemannian manifolds, we refer the reader to Druet and Hebey [6] and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

Yang

2007

Trans. Amer. Math. Soc.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

Yang

2007

Trans. Amer. Math. Soc.

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“…This can be considered as an important component of the A-B program in geometric analysis as proposed by Druet and Hebey [22]. For simplicity, we only discuss that case for q = 1.…”

Section: Linearly Perturbed Borderline Variational Problems With An Imentioning

confidence: 99%

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

Ghoussoub

Robert

2015

Bull. Math. Sci.

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In this survey paper, we consider variational problems involving the HardySchrödinger operator L γ := − − γ |x| 2 on a smooth domain of R n with 0 ∈ , and illustrate how the location of the singularity 0, be it in the interior of or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter can be stated as:where γ < n 2 4 , s ∈ [0, 2) and 2 (s) := 2(n−s) n−2 . We address questions regarding the explicit values of the optimal constant C := μ γ,s ( ), as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties often lead to situations where the best constants μ γ,s ( ) do not depend on the domain, and hence they are not attainable. We consider two different approaches for "breaking the homogeneity" of the problem, and restoring compactness. One approach was initiated by Brezis-Nirenberg, when γ = 0 and s = 0, and by Janelli, when γ > 0 and s = 0. It is suitable for the case where the singularity 0 is in the interior of , and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang for γ = 0, s > 0, and by C.S. Lin et al. and Ghoussoub-Robert, when γ = 0, s ≥ 0. It consists of considering domains, where the singularity 0 is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0 ∈ , then a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" theorem for the operator L γ is required in lower dimensions. If the singularity 0 belongs to the boundary ∂ , then the local geometry around 0 (convexity and mean curvature) plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" theorem is needed for the lower dimensions. Each case leads to a distinct notion of critical dimension for the operator L γ .

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“…It at least goes back to [3], for more references see [10]. Relevant for the study of boundary value problems for differential operators is the Sobolev trace inequality that has been intensively studied, see for example, [11,12,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

The Best Sobolev Trace Constant in Periodic Media for Critical and Subcritical Exponents

Bonder¹,

Orive²,

Rossi³

2009

Glasgow Math. J.

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Abstract. In this paper we study homogenization problems for the Sobolev trace embedding H 1 (Ω) → L q (∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, we consider H 1 and L q spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of an homogenized limit problem and the best trace constant converges to a homogenized best trace constant. Our results are in fact more general, we can also consider general operators of the form a ε (x, ∇u) with nonlinear Neumann boundary conditions. In particular, we can deal with the embedding W 1,p (Ω) → L q (∂Ω).

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The 𝐴𝐵 program in geometric analysis: sharp Sobolev inequalities and related problems

Cited by 67 publications

References 0 publications

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

The Best Sobolev Trace Constant in Periodic Media for Critical and Subcritical Exponents

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