Hadamard variational formulas for p-torsion and p-eigenvalue with applications

Huang, Yong; Song, Changzhen; Xu, Lu

doi:10.1007/s10711-018-0318-5

Cited by 15 publications

(9 citation statements)

References 27 publications

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“…A related Hadamard formula was proved in [18] in the case where Ω is convex with C 2 boundary having positive Gauss curvature. The proof in [18] depends on the convexity but, as we show here, the validity of the result does not.…”

Section: Outline Of Proofmentioning

confidence: 99%

An inequality for the normal derivative of the Lane–Emden ground state

Frank

Larson

2022

Advances in Calculus of Variations

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We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 0\leq q-1\leq 1 , that is, minimizers of the Dirichlet integral among L q L^{q} -normalized functions. Our main result is a sharp lower bound on the L 2 L^{2} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ R d \Omega\subset\mathbb{R}^{d} , without assuming convexity.

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Section: Outline Of Proofmentioning

confidence: 99%

An inequality for the normal derivative of the Lane–Emden ground state

Frank

Larson

2022

Advances in Calculus of Variations

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“…The following basic facts and geometric inequalities about q-torsional rigidity can be referred to [20] or [36].…”

Section: Variational Formula For Q-torsional Rigidity Of General Conv...mentioning

confidence: 99%

The Lp Minkowski problem for q-torsional rigidity

Chen¹,

Zhao²,

Wang³

et al. 2022

Preprint

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In this paper, we introduce the so-called Lp q-torsional measure for p ∈ R and q > 1 by establishing the Lp variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the Lp Minkowski problem w.r.t. the q-torsional rigidity for discrete measure and general measure when 0 < p < 1 and q > 1.2010 Mathematics Subject Classification. 52A20 52A40.

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“…Notice that there are only few results of the Minkowski type problems on p-torsion measure with regard to p-Laplacian equation (p 2), the main reason forming this phenomenon may be that there is no weak convergence result of p-torsion measure. Huang-Song-Xu [18] established the Hadamard variational formula of p-torsional rigidity in the smooth category, further forming a frame for dealing with the associated Minkowski type problems on p-torsion measure.…”

Section: Introductionmentioning

confidence: 99%