Estimates of Dirichlet eigenvalues for degenerate $$\triangle _{\mu }$$-Laplace operator

Chen, Hua; Chen, Hongge; Li, Jin-Ning

doi:10.1007/s00526-020-01765-x

Cited by 6 publications

(2 citation statements)

References 26 publications

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“…In addition, for the homogeneous vector fields X defined on R n , Proposition 2.8 below shows an ingenious relationship between the homogeneous dimension Q and the pointwise dimension ν(x), which says Remark 1.3. Theorem 1.3 also improves our previous estimates of Dirichlet eigenvalues for homogeneous Hörmander operators in [24].…”

Section: Introduction and Main Resultssupporting

confidence: 79%

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Chen¹,

Chen²,

Li³

2022

Preprint

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We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators △ X = m j=1 X 2 j on a bounded connected open domain containing the origin, where X 1 , X 2 , . . . , X m are linearly independent smooth vector fields in R n satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that Ω is a bounded connected open domain in R n containing the origin, and its boundary ∂Ω is smooth and non-characteristic for X. Combining the subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry and some refined analysis involving convex geometry, we establish the explicit asymptotic behaviourwhere λ k denotes the k-th Dirichlet eigenvalue of △ X on Ω, Q 0 is a positive rational number, and d 0 is a non-negative integer. Furthermore, we also give the optimal bounds of index Q 0 , which depends on the homogeneous dimension associated with vector fields X 1 , X 2 , . . . , X m .

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Section: Introduction and Main Resultssupporting

confidence: 79%

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Chen¹,

Chen²,

Li³

2022

Preprint

Self Cite

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“…Here we mention, without the sake of completeness, Franchi and Lanconelli [15,16,17] for the Hölder regularity of weak solutions and for the embedding of the associated Sobolev spaces, Garofalo and Shen [19] for Carleman estimates and unique continuation results, D'Ambrosio [9] for Hardy inequalities, Thuy and Tri [31] and Kogoj and Lanconelli [24] for semilinear problems. Finally we mention Chen and Chen [4], Chen, Chen, Duan and Xu [5], Chen, Chen and Li [6] and Chen and Luo [7] for asymptotic bounds for eigenvalues.…”

mentioning

confidence: 99%

The first Grushin eigenvalue on cartesian product domains

Luzzini¹,

Provenzano²,

Stubbe³

2022

Preprint

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In this paper we consider the first eigenvalue λ 1 (Ω) of the Grushin operator ∆ G := ∆x 1 + |x 1 | 2s ∆x 2 with Dirichlet boundary conditions on a bounded domain Ω of R d = R d 1 +d 2 . We prove that λ 1 (Ω) admits a unique minimizer in the class of domains with prescribed finite volume which are the cartesian product of a set in R d 1 and a set in R d 2 , and that the minimizer is the product of two balls Ω * 1 ⊆ R d 1 and Ω * 2 ⊆ R d 2 . Moreover, we provide a lower bound for |Ω * 1 | and for λ 1 (Ω * 1 × Ω * 2 ). Finally, we consider the limiting problem as s tends to 0 and to +∞.

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Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

Luyen¹,

Cuong²

2021

Rend. Circ. Mat. Palermo, II. Ser

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Estimates of Dirichlet eigenvalues for degenerate $$\triangle _{\mu }$$-Laplace operator

Cited by 6 publications

References 26 publications

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach

The first Grushin eigenvalue on cartesian product domains

Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

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