Conformal invariants and partial differential equations

Chang, Shinan

doi:10.1090/s0273-0979-05-01058-x

Cited by 57 publications

(75 citation statements)

References 92 publications

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“…Bo Guan [9] and John Urbas [22] have also made important contributions to the equation. Some of the techniques in these works can be modified to study equations in conformal geometry (see [3]). …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Necessary and sufficient conditions on solvability for Hessian inequalities

Ji¹,

Bao²

2010

Proc. Amer. Math. Soc.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Necessary and sufficient conditions on solvability for Hessian inequalities

Ji¹,

Bao²

2010

Proc. Amer. Math. Soc.

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“…This is a research direction currently under intensive study. The interested reader is referred to the survey [4].…”

Section: Q-curvature and Geometric Analysismentioning

confidence: 99%

What is Q-Curvature?

et al. 2008

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“…For "naively" defined Sobolev spaces, however, it may be false. A Trudinger-type inequality in a geometric setting reminds one of the role that such inequalities play in conformal geometry (see, e.g., [2][3][4]). The situation studied here is not directly related, but nevertheless, questions that arise in one context may be interesting in the other as well.…”

Section: The Inequalitymentioning

confidence: 99%

A Trudinger type inequality for maps into a Riemannian manifold

Moser

2008

Ann Glob Anal Geom

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The tension field of a map into a Riemannian manifold is the equivalent to the Laplacian of a function. However in contrast to the latter, the tension field is given by a nonlinear differential operator. Nevertheless, it permits an extension of a well-known Trudinger inequality that involves an Orlicz space for a function with exponential growth. Keywords Trudinger inequality · Tension field The inequalityFor n ≥ 4, let ⊂ R n be a bounded, convex, open set with smooth boundary and suppose that N is a Riemannian manifold. For a smooth map u :→ N that is constant on ∂ , consider the pull-back vector bundle u −1 TN over with the induced Levi-Civita connection D. The tension field of u is the section τ (u) = trace Ddu of this bundle. This defines a nonlinear differential operator τ of second order (for a representation in local coordinates, see, e.g., Jost [6]) that can be regarded as the Laplacian for a map into N . Therefore, for 1 ≤ p < ∞, the quantityis reminiscent of the norm of the Sobolev space W 2, p 0 ( ). This analogy leads to the question whether certain inequalities for Sobolev functions can be extended to spaces of maps into N . In this note, we examine the case p = n 2 , and we study the analog of an inequality proved by Adams [1] that generalizes results of Trudinger [14] and J. Moser [8].

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Conformal invariants and partial differential equations

Cited by 57 publications

References 92 publications

Necessary and sufficient conditions on solvability for Hessian inequalities

Necessary and sufficient conditions on solvability for Hessian inequalities

What is Q-Curvature?

A Trudinger type inequality for maps into a Riemannian manifold

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