Brownian motion and the fundamental frequency of a drum

Bañuelos, Rodrigo; Carroll, Tom

doi:10.1215/s0012-7094-94-07517-0

Cited by 97 publications

(95 citation statements)

References 16 publications

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“…This quantity is called the inradius of the domain. It is known (see [2]) that there are positive constants C 1 , C 2 , C 3 , and C 4 such that…”

Section: Introductionmentioning

confidence: 99%

“…There has been considerable interest in obtaining sharp versions of the right-hand side of (5) and the left-hand side of (6) and in identifying the extremal domains. We refer the reader to R. Bañuelos and T. Carroll [2] for some of the extensive literature on this subject and for connections to other problems. These problems seem to be very difficult for arbitrary simply connected domains and conjectures; on how the extremals domains should look is not even available.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Brascamp-Lieb-Luttinger–type inequality and applications to symmetric stable processes

Bañuelos

Latała²,

Méndez-Hernández³

2001

Proc. Amer. Math. Soc.

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Abstract. We derive an inequality for multiple integrals from which we conclude various generalized isoperimetric inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius. Our multiple integral inequality is a replacement for the classical inequality of H. J. Brascamp, E. H. Lieb and J. M. Luttinger, where instead of fixing the volume of the domain one fixes its inradius.

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“…This quantity is called the inradius of the domain. It is known (see [2]) that there are positive constants C 1 , C 2 , C 3 , and C 4 such that…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Brascamp-Lieb-Luttinger–type inequality and applications to symmetric stable processes

Bañuelos

Latała²,

Méndez-Hernández³

2001

Proc. Amer. Math. Soc.

Self Cite

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“…See [7] for a generalization to a certain pseudo-Laplacian. 1 The constant 1/4 in λ Ω > 1/4ρ 2 Ω has since been improved using ideas from probability and conformal mapping, the currently best value is 0.6197, see [4]. (12) follows from (13) h Ω ≥ 1 ρ Ω and Cheeger's inequality.…”

Section: Cheeger's Constant and Inradiusmentioning

confidence: 99%

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

Grieser

2006

Arch. Math.

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Abstract. We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger's constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger's constant in terms of the inradius of a plane domain.

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“…Let R D be the supremum of the radii of all disks contained in D. This geometric quantity is called the inradius of the domain D. Many interesting and challenging extremal problems arise when one fixes R D and asks for those domains which maximize or minimize various quantities which depend on D. For example, one such problem which has been extensively studied and which remains open is the following: amongst all simply connected domains with inradius 1, find the one(s) which has the lowest fundamental frequency. It is shown in [2] that this problem is closely related to problems concerning the maximum expected lifetime of Brownian motion in the domain, the schlicht Bloch-Landau constant, and integral means for the derivative of the conformal mapping from the disk to the domain.…”

Section: Introductionmentioning

confidence: 99%

“…For arbitrary simply connected domains this problem (as well as the other problems mentioned above) are open even for ϕ(x) = x, which is the case of the lifetime of Brownian motion in D. We refer the reader to [2] for more on this and related problems.…”

Section: Introductionmentioning

confidence: 99%

Inradius and integral means for Green’s functions and conformal mappings

Bañuelos

Carroll

Housworth

1998

Proc. Amer. Math. Soc.

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Abstract. Let D be a convex planar domain of finite inradius R D . Fix the point 0 ∈ D and suppose the disk centered at 0 and radius R D is contained in D. Under these assumptions we prove that the symmetric decreasing rearrangement in θ of the Green's function G D (0, ρe iθ ), for fixed ρ, is dominated by the corresponding quantity for the strip of width 2R D . From this, sharp integral mean inequalities for the Green's function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison estimates for the hyperbolic metric of D with that of the strip and a careful analysis of geodesics.

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Brownian motion and the fundamental frequency of a drum

Cited by 97 publications

References 16 publications

A Brascamp-Lieb-Luttinger–type inequality and applications to symmetric stable processes

A Brascamp-Lieb-Luttinger–type inequality and applications to symmetric stable processes

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

Inradius and integral means for Green’s functions and conformal mappings

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