Towards a reversed Faber–Krahn inequality for the truncated Laplacian

Birindelli, Isabeau; Galise, Giulio; Ishii, Hitoshi

doi:10.4171/rmi/1146

Cited by 12 publications

(23 citation statements)

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“…We finally remark that sort of reversed Faber-Krahn and Lieb inequality have been recently proved for the first eigenvalue of a degenerate operator called truncated Laplacian, see [2].…”

Section: Giovanni Cupini and Eugenio Vecchimentioning

confidence: 81%

“…then Ω must be a ball. We refer to [ We end this section with a brief recap on the optimization eigenvalue problem (2). We refer to [3]…”

mentioning

confidence: 99%

“…Let Ω ⊂ R n be an open and bounded connected set with Lipschitz boundary. Let α ∈ (0, α Ω (A)), and let (u, D) be an optimal pair which realizes the double infimum in (2). To simplify the notation, in this section we will denote Λ Ω (α, A) by Λ.…”

mentioning

confidence: 99%

“…Since we assume Λ Ω = Λ Ω , we can simply write Λ omitting the dependance on the set. Let us consider an optimal pair (u, D) of the double minimization problem (2) on Ω. Clearly, they satisfy the second order Euler-Lagrange equation associated to (2), i.e.…”

mentioning

confidence: 99%

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Faber-Krahn and Lieb-type inequalities for the composite membrane problem

Cupini¹,

Vecchi²

2019

Communications on Pure &Amp; Applied Analysis

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The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest first Dirichlet eigenvalue of the Laplacian. Another inequality related to the first eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the first Dirichlet eigenvalues of the Laplacian of two different domains with the first Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem.

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“…We finally remark that sort of reversed Faber-Krahn and Lieb inequality have been recently proved for the first eigenvalue of a degenerate operator called truncated Laplacian, see [2].…”

Section: Giovanni Cupini and Eugenio Vecchimentioning

confidence: 81%

“…then Ω must be a ball. We refer to [ We end this section with a brief recap on the optimization eigenvalue problem (2). We refer to [3]…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Faber-Krahn and Lieb-type inequalities for the composite membrane problem

Cupini¹,

Vecchi²

2019

Communications on Pure &Amp; Applied Analysis

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“…Problem () is the evolution version of the elliptic problem

({, \begin{matrix} λ_{j} false(D^{2} z (x) false) = 0, & in Ω, \\ z (x) = g (x), & on \partial Ω, \end{matrix})

which was extensively studied in [1, 3–7, 11, 12, 22, 23]. In particular, for

j = 1

and

j = N

, problem () is the equation for the convex and concave envelope of

g

normalΩ

, respectively, that is, the solution

z

is the biggest convex (smallest concave) function

u

, satisfying

u ⩽ g

(

u ⩾ g

) on

\partial Ω

, see [22, 23].…”

Section: Introductionmentioning

confidence: 99%

The evolution problem associated with eigenvalues of the Hessian

Blanc

Esteve

Rossi

2020

J. London Math. Soc.

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In this paper, we study the evolution problem ⎧ ⎪ ⎨ ⎪ ⎩ ut(x, t) − λj(D 2 u(x, t)) = 0, in Ω × (0, +∞), u(x, t) = g(x, t), on ∂Ω × (0, +∞), u(x, 0) = u0(x), in Ω, where Ω is a bounded domain in R N (which verifies a suitable geometric condition on its boundary) and λj(D 2 u) stands for the jth eigenvalue of the Hessian matrix D 2 u. We assume that u0 and g are continuous functions with the compatibility condition u0(x) = g(x, 0), x ∈ ∂Ω. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, g(x, t) = g(x), we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as t → ∞. For j = 1, the limit profile is just the convex envelope inside Ω of the boundary datum g, while for j = N , it is the concave envelope. We obtain this result with two different techniques: with partial differential equations (PDE) tools and with game-theoretical arguments. Moreover, in some special cases (for affine boundary data), we can show that solutions coincide with the stationary solution in finite time (which depends only on Ω and not on the initial condition u0).

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Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

Nguyen²,

Wang

2020

Geometric Analysis

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Dedicated to Gang Tian on his 60th birthday with friendship. AbstractWe study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is C 1,1 . We obtain as a consequence a Liouville theorem for entire solutions which are approximable by C 1,1 solutions on larger and larger compact domains, and, in particular, for entire C 1,1 loc solutions: they are either constants or standard bubbles.

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Towards a reversed Faber–Krahn inequality for the truncated Laplacian

Cited by 12 publications

References 0 publications

Faber-Krahn and Lieb-type inequalities for the composite membrane problem

Faber-Krahn and Lieb-type inequalities for the composite membrane problem

The evolution problem associated with eigenvalues of the Hessian

Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

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