‘High spots’ theorems for sloshing problems

Kulczycki, Tadeusz; Kuznetsov, Nikolay

doi:10.1112/blms/bdp021

Cited by 20 publications

(24 citation statements)

References 22 publications

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“…[3,Section 3], [26], [30]) one obtains the following domain monotonicity results for eigenvalues for both the Dirichlet-Steklov problem (2.1)-(2.3) and the sloshing problem (1.1)-(1.4). We omit the proofs of these results because they are very similar to the proofs of Propositions 3.…”

Section: Fundamental Eigenvaluementioning

confidence: 99%

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On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

Kulczycki

Kwaśnicki

2012

Proceedings of the London Mathematical Society

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Abstract. We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W ⊂ R 3 . We study the case when W is an axially symmetric, convex, bounded domain satisfying the John condition. The Cartesian coordinates (x, y, z) are chosen so that the mean free surface of the liquid lies in (x, z)-plane and y-axis is directed upwards (y-axis is the axis of symmetry). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction ϕ there is a change of x, z coordinates by a rotation around y-axis so that ϕ is odd in x-variable.The second result of the paper gives the following monotonicity property of the fundamental eigenfunction ϕ. If ϕ is odd in x-variable then it is strictly monotonic in x-variable. This property has the following hydrodynamical meaning. If liquid oscillates freely with fundamental frequency according to ϕ then the free surface elevation of liquid is increasing along each line parallel to x-axis during one period of time and decreasing during the other half period. The proof of the second result is based on the method developed by D. Jerison and N. Nadirashvili for the hot spots problem for Neumann Laplacian.

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Section: Fundamental Eigenvaluementioning

confidence: 99%

“…The other one treat fundamental eigenfunctions in troughs (their cross-sections are subject to the same condition), and some vertical axisymmetric containers. Moreover, it was shown in [26] that for vertical-walled containers with horizontal bottom the question about high spots is equivalent to the hot spots conjecture.…”

Section: Introductionmentioning

confidence: 99%

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

Kulczycki

Kwaśnicki

2012

Proceedings of the London Mathematical Society

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“…According to formula (2.1), the corresponding eigenfunction is f 1 (x, y, z) = u 0,1 (x, y), and it solves problem (2.2)-(2.5) for m = 0 and n = n 0,1 . Theorem 2.1 proved by Kulczycki & Kuznetsov (2009) where the trace u 0,1 (x, 0) attains its extremum values, say, (0, 0) and (a, 0) are the minimum and maximum points, respectively. Then, f 1 (x, 0, z) is equal to u 0,1 (0, 0) and u 0,1 (a, 0) on the whole edges…”

Section: High-spots Theorem For Troughsmentioning

confidence: 99%

“…First results about high spots for fundamental eigenfunctions of the sloshing problem were obtained by Kulczycki & Kuznetsov (2009), who proved some theorems about the location of these points in the two-dimensional case and for some vertical axisymmetric containers. Moreover, it was shown that for verticalwalled containers with a horizontal bottom, the question of whether high spots is equivalent to the 'hot spots' conjecture of J. Rauch (see Burdzy (2006) for a survey of results about the latter conjecture).…”

Section: Introductionmentioning

confidence: 99%

On the ‘high spots’ of fundamental sloshing modes in a trough

Kulczycki

Kuznetsov

2010

Proc. R. Soc. A.

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We study an eigenvalue problem with a spectral parameter in a boundary condition. The problem describes sloshing of a heavy liquid in a container, which means that the unknowns are the frequencies and modes of the liquid's free oscillations. The question of 'high spots' (the points on the mean free surface, where its elevation attains the maximum and minimum values) is considered for fundamental sloshing modes in troughs of uniform cross section. For troughs, whose cross sections are such that the horizontal, top interval is the one-to-one orthogonal projection of the bottom, the following result is obtained: any fundamental eigenfunction attains its maximum and minimum values only on the boundary of the rectangular free surface of the trough.

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“…The Steklov eigenvalues have a number of domain monotonicity properties that bear directly on the present discussion (cf., e. g., Kulczycki and Kuznetsov (2009)). We recall that the Steklov eigenvalues admit the variational characterization (cf., e. g., Lamberti and Provenzano (2013); Bogosel et al (2017); Girouard and Polterovich (2017)) (30) σ n = inf…”

Section: Monotonicity With Respect To Cuttingmentioning

confidence: 99%

Functional optimality of the sulcus pattern of the human brain

Kröger

Ortiz

2018

Mathematical Medicine and Biology: A Journal of the IMA

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We develop a mathematical model of information transmission across the biological neural network of the human brain. The overall function of the brain consists of the emergent processes resulting from the spread of information through the neural network. The capacity of the brain is therefore related to the rate at which it can transmit information through the neural network. The particular transmission model under consideration allows for information to be transmitted along multiple paths between points of the cortex. The resulting transmission rates are governed by potential theory. According to this theory, the brain has preferred and quantized transmission modes that correspond to eigenfunctions of the classical Steklov eigenvalue problem, with the reciprocal eigenvalues quantifying the corresponding transmission rates. We take the model as a basis for testing the hypothesis that the sulcus pattern of the human brain has evolved to maximize the rate of transmission of information between points in the cerebral cortex. We show that the introduction of sulci, or cuts, in an otherwise smooth domain indeed increases the overall transmission rate. We demonstrate this result by means of numerical experiments concerned with a spherical domain with a varying number of slits on its surface.

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‘High spots’ theorems for sloshing problems

Cited by 20 publications

References 22 publications

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

On the ‘high spots’ of fundamental sloshing modes in a trough

Functional optimality of the sulcus pattern of the human brain

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