An approach to symmetrization via polarization

Brock, Friedemann; Solynin, Alexander Yu.

doi:10.1090/s0002-9947-99-02558-1

Cited by 161 publications

(162 citation statements)

References 32 publications

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“…Since for , we have If we are in the singular range , we have for some . Since the continuous Steiner symmetrisation decreases the modulus of continuity (see [ 8 , Theorem 3.3] and [ 8 , Corollary 3.1]), we also have . Further, Lemma 1 and the arguments of [ 17 , Proposition 2.7] guarantee that the expressions can be controlled by and the -Hölder seminorm of .…”

Section: Stationary Statesmentioning

confidence: 93%

Ground states in the diffusion-dominated regime

et al. 2018

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We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

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Section: Stationary Statesmentioning

confidence: 93%

Ground states in the diffusion-dominated regime

et al. 2018

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“…We first show that the rearrangement • is smoothing in the sense of [21] (see also [4]). Given r > 0 write E r for the r -neighbourhood of an L 1 -measurable set E in (B, d); by convention, ∅ r = ∅.…”

Section: A Pólya-szegö Inequalitymentioning

confidence: 91%

“…We first discuss the operation of polarisation for integrable functions on B (see [4] and references therein). For ν ∈ S n−1 the closed half-space H = H ν is defined by…”

Section: Spherical Cap Symmetrymentioning

confidence: 99%

An Unstable Two-Phase Membrane Problem and Maximum Flux Exchange Flow

McGillivray

2016

Appl Math Optim

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Let U be a bounded open connected set in R n (n ≥ 1). We refer to the unique weak solution of the Poisson problem − u = χ A on U with Dirichlet boundary conditions as u A for any measurable set A in U. The function ψ := u U is the torsion function of U. Let V be the measure V := ψ L n on U where L n stands for ndimensional Lebesgue measure. We study the variational problem I (U, p) := sup J (A) − V (U) p 2 with p ∈ (0, 1) where J (A) := A u A dx and the supremum is taken over measurable sets A ⊂ U subject to the constraint V (A) = pV (U). We relate the above problem to an unstable two-phase membrane problem. We characterise optimsers in the case n = 1. The proof makes use of weighted isoperimetric and Pólya-Szegö inequalities.

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“…First we recall the notion of polarization (or, equivalently, two-point rearrangement) of sets and functions, see, e.g., [8,4] We define two polarizations of a measurable set Ω with respect to H a as follows:…”

Section: Auxiliary Factsmentioning

confidence: 99%

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

Bobkov

Kolonitskii

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation −∆pu = f (u) in bounded Steiner symmetric domains Ω ⊂ R N under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

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An approach to symmetrization via polarization

Cited by 161 publications

References 32 publications

Ground states in the diffusion-dominated regime

Ground states in the diffusion-dominated regime

An Unstable Two-Phase Membrane Problem and Maximum Flux Exchange Flow

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

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