Functional Inequalities: New Perspectives and New Applications

Ghoussoub, Nassif; Moradifam, Amir

doi:10.1090/surv/187

Cited by 104 publications

(145 citation statements)

References 152 publications

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“…The story here is the link-discovered by Ghoussoub and Moradifam [26,27]-between various improvements of this inequality confined to bounded domains and Sturm's theory regarding the oscillatory behavior of certain linear ordinary equations.…”

Section: To See Thatmentioning

confidence: 93%

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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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Section: To See Thatmentioning

confidence: 93%

“…Following Ghoussoub and Moradifam [27], we say that a non-negative C 1 -function P defined on an interval (0, R) is a Hardy Improving Potential (abbreviated as HIpotential) if the following improved Hardy inequality holds on every domain contained in a ball of radius R:…”

Section: To See Thatmentioning

confidence: 99%

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

Ghoussoub

Robert

2015

Bull. Math. Sci.

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“…In [45], Ghigi provided a proof based on the Prékopa-Leindler inequality, which is also explained in full detail in the book [47, of Ghoussoub and Moradifam. Let us mention that the book contains much more material and tackles the issues of improved inequalities under additional constraints, a question that was raised in [4] and later studied in [24][25]46].…”

Section: A Review Of the Literaturementioning

confidence: 99%

“…60] and [51]. A detailed statement can be found in [47,Lemma 17.1.2]. Competing symmetries are another aspect of symmetrization that we will not study in this paper and for which we refer to [19].…”

Section: A Review Of the Literaturementioning

confidence: 99%

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The Moser-Trudinger-Onofri inequality

Dolbeault

Esteban

Jankowiak

2015

Chin. Ann. Math. Ser. B

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This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.

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Some Hardy identities on half‐spaces

Duy

Lam

Phi

2021

Mathematische Nachrichten

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We prove some Hardy identities on the half‐space double-struckR+N$\mathbb {R}_{+}^{N}$. Our equalities imply correponding versions of the Hardy type inequalities with exact remainder terms on double-struckR+N$\mathbb {R}_{+}^{N}$. These equalities give straightforward understandings of the optimal constants as well as the nonexistence of nontrivial optimizers for various Hardy type inequalities on half‐spaces. These identities also provide the “virtual” ground state in the sense of Frank and Seiringer [13] for several Hardy type inequalities on double-struckR+N$\mathbb {R}_{+}^{N}$.

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Functional Inequalities: New Perspectives and New Applications

Cited by 104 publications

References 152 publications

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

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

The Moser-Trudinger-Onofri inequality

Some Hardy identities on half‐spaces

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