Green functions and the Dirichlet spectrum

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged L1-moment spectra $\left \{\frac {\mathcal {A}_{k}\left ({B_{R}^{M}}\right )}{\text {vol}\left ({S_{R}^{M}}\right )}\right \}_{k=1}^{\infty }$ A k B R M vol S R M k = 1 ∞ , and the torsional rigidity $\mathcal {A}_{1}({B^{M}_{R}})$ A 1 ( B R M ) of a geodesic ball ${B^{M}_{R}}$ B R M in a Riemannian manifold Mn which satisfies that the mean curvatures of the geodesic spheres ${S^{M}_{r}}$ S r M included in it, (up to the boundary ${S^{M}_{R}}$ S R M ), are controlled by the radial mean curvature of the geodesic spheres $S^{\omega }_{r}(o_{\omega })$ S r ω ( o ω ) with same radius centered at the center oω of a rotationally symmetric model space $M^{n}_{\omega }$ M ω n . As a consecuence, we prove a first Dirichlet eigenvalue $\lambda _{1}({B^{M}_{R}})$ λ 1 ( B R M ) comparison theorem and show that equality with the bound $\lambda _{1}(B^{\omega }_{R}(o_{\omega }))$ λ 1 ( B R ω ( o ω ) ) , (where $B^{\omega }_{r}(o_{\omega })$ B r ω ( o ω ) is the geodesic r-ball in $M^{n}_{\omega }$ M ω n ), characterizes the L1-moment spectrum $\left \{\mathcal {A}_{k}({B^{M}_{R}})\right \}_{k=1}^{\infty }$ A k ( B R M ) k = 1 ∞ as the sequence $\left \{\mathcal {A}_{k}(B^{\omega }_{R})\right \}_{k=1}^{\infty }$ A k ( B R ω ) k = 1 ∞ and vice-versa.

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“…, obtained in the paper [24]. This formula for λ 1 (B ω R (o ω )) was subsequently extended in the paper [7] to any precompact domain ⊆ M, namely…”

Section: A Glimpse At Our Resultsmentioning

confidence: 91%

“…, for all k ≥ 1 and for all r ∈]0, R], we have that, given B M r ⊆ M in a Riemannian manifold (M, g) with r ∈]0, R], (see [24] and [7]):…”

Section: 45)mentioning

confidence: 99%

First Dirichlet Eigenvalue and Exit Time Moment Spectra Comparisons

Palmer

Erik

2023

Potential Anal

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“…The family of functions (1.7) are related with the moment functions of the Poisson hierarchy { u k } associated to g. In fact, in [3], is proved that since g is a rotationally symmetric metric tensor…”

Section: The Schwarz Symmetrization Of a Compact Open Domainmentioning

confidence: 99%

“…Observe that the conclusion of an equality between the mean curvatures pointed inward of geodesic spheres is a weaker than the conclusion of an isometry between geodesic balls. Indeed, in example 5.3 of [3] is shown a 4-dimensional geodesic ball non-isometric to the geodesic ball of M 4 κ , but with H St(p) = H(t, κ) for all t ∈ (0, R) .…”

Section: Introductionmentioning

confidence: 99%

First Eigenvalue of the Laplacian of a Geodesic Ball and Area-Based Symmetrization of its Metric Tensor

Gimeno¹,

Erik²

2021

Preprint

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Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with our upper bound if and only if the mean curvature pointed inward of each geodesic sphere is a radial function.

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First Dirichlet Eigenvalue and Exit Time Moments: A Survey

Gimeno,

Hurtado

2023

RSME Springer Series

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Green functions and the Dirichlet spectrum

Cited by 5 publications

References 26 publications

First Dirichlet Eigenvalue and Exit Time Moment Spectra Comparisons

First Dirichlet Eigenvalue and Exit Time Moment Spectra Comparisons

First Eigenvalue of the Laplacian of a Geodesic Ball and Area-Based Symmetrization of its Metric Tensor

First Dirichlet Eigenvalue and Exit Time Moments: A Survey

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