The Problem of Prescribed Critical Functions

Humbert, Emmanuel; Vaugon, Michel

doi:10.1007/s10455-005-1583-8

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…This theorem was proved by E. Humbert and M. Vaugon in the case f = cst = 1 and M not conformally diffeomorphic to the sphere, [24]. Their method works in the case of a non constant function f and an arbitrary manifold once it is proved that we can suppose the existence of positive critical functions satisfying the strict inequality (*) in theorem 2, result we included in this theorem (note that, as Supf > 0, we can always find a metric g' conformal to g such that M f dv g ′ > 0).…”

Section: Statement Of the Resultsmentioning

confidence: 88%

“…n−2 has a solution u > 0, and 2/: (h, f, g) is critical. Indeed, in this case h = F h ′ (u) and h ′ is critical for f and g ′ = u 4 n−2 g. E. Humbert et M. Vaugon proved this theorem in the case f = cste and for a manifold not conformaly diffeomorphic to the sphere [24]. Their method lies on the fact that for such a manifold , after a first conformal change of metric, B 0 (g)K(n, 2) −2 is a critical function, (we will denote these two constants K et B 0 ).…”

Section: Critical Triplementioning

confidence: 94%

“…as soon as i is large enough. We therefore fix i and s large enough to have (24) et ( 25) and we consider…”

Section: Fourth Stepmentioning

confidence: 99%

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Critical functions and elliptic PDE on compact riemannian manifolds

Collion¹

2010

Preprint

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We study in this work the existence of minimizing solutions to the critical-power type equation △gu + h.u = f.u n+2 n−2 on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was originally introduced by E. Hebey and M. Vaugon for the study of second best constant in the Sobolev embeddings. Along the way, we prove an important estimate concerning concentration phenomena's when f is a non-constant function. We give here intuitive details. 1

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Section: Statement Of the Resultsmentioning

confidence: 88%

Section: Critical Triplementioning

confidence: 94%

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Critical functions and elliptic PDE on compact riemannian manifolds

Collion¹

2010

Preprint

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“…Developments on the notions of weakly critical and critical functions may also be found in the papers Humbert and Vaugon [17], and Robert [19].…”

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confidence: 99%

Sharp Sobolev inequalities in the presence of a twist

Collion¹,

Hebey²,

Vaugon³

2007

Trans. Amer. Math. Soc.

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Abstract. Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Let also A be a smooth symmetrical positive (0, 2)-tensor field in M . By the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where H 2 1 (M ) is the standard Sobolev space of functions in L 2 with one derivative in L 2 . We investigate in this paper the value of the sharp K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Also let A be a smooth symmetrical (0, 2)-tensor field in M . In a local chart, A = (A ij ), i, j = 1, . . . , n. We assume that A is positive when acting on 1-forms in the sense that for any x ∈ M , and any η in the cotangent spaceThen, by the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where ∇u = (∂ i u) is the 1-form consisting (in local charts) of the first derivatives of u, dv g is the Riemannian volume element of g, and H 2 1 (M ) is the standard Sobolev space consisting of functions in L 2 with one derivative in L 2 . Clearly, the sharp constant B in (0.1) is V −2/n g , where V g is the volume of (M, g), and the corresponding sharp inequality holds true since it holds true for the classical Sobolev inequality [and |∇u| 2 is controled by A x (∇u, ∇u)]. On the other hand, as is easily understood by the fact that A charges some parts of the space M more than others, it is expected that A will affect the sharp constant K in (0.1). Note (0.1) is associated to the operator ∆ g A u = −div g (A x ∇u) which appears in several places in mathematical and physics literature.The questions we ask in this note are: what is the value K s = K s (g) of the sharp K in (0.1), does the corresponding sharp inequality hold true, and if yes, does its saturated version (where B is lowered to its minimum value under the constraint K = K s ) possess extremal functions? When A = g −1 , we are back to the classical problem (dealing with the classical Sobolev inequality). Possible references in book

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Sharp Sobolev inequalities for vector valued maps

Hebey

2006

Math. Z.

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The Problem of Prescribed Critical Functions

Abstract: Let (M, g) be a compact Riemannian manifold on dimension n ≥ 4 not conformally diffeomorphic to the sphere S n . We prove that a smooth function f on M is a critical function for a metricg conformal to g if and only if there exists x ∈ M such that f (x) > 0.

Cited by 3 publications

References 6 publications

Critical functions and elliptic PDE on compact riemannian manifolds

Critical functions and elliptic PDE on compact riemannian manifolds

Sharp Sobolev inequalities in the presence of a twist

Sharp Sobolev inequalities for vector valued maps

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