scholarly journals Concentration phenomena for the fractional Q‐curvature equation in dimension 3 and fractional Poisson formulas

2021 ◽  
Author(s):  
Azahara DelaTorre ◽  
María del Mar González ◽  
Ali Hyder ◽  
Luca Martinazzi
Keyword(s):  
Curvature Equation ◽  
Concentration Phenomena

Concentration Phenomena for the Paneitz Curvature Equation in ℝN

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Zhongyuan Liu
Keyword(s):  
Curvature Equation ◽  
Concentration Phenomena ◽  
Curvature Problem

AbstractIn this paper, we study the Paneitz curvature problemΔwhere

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

scholarly journals Rigidity and trace properties of divergence-measure vector fields

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Gian Paolo Leonardi ◽  
Giorgio Saracco
Keyword(s):  
Vector Fields ◽  
Extremal Solution ◽  
Divergence Measure ◽  
Higher Dimension ◽  
Regular Domain ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Divergence Free Vector ◽  
Prescribed Mean Curvature Equation ◽  
Rigidity Property

AbstractWe consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where {\varphi(t)} is a non-negative convex function vanishing only at {t=0}. We show that this property is always satisfied in dimension {n=2}, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when {\varphi(t)=ct^{2}}) in dimension {n\geq 4}. The validity of the quadratic rigidity, which we prove in dimension {n=2}, implies the existence of the trace of a divergence-measure vector field ξ on an {\mathcal{H}^{1}}-rectifiable set S, as soon as its weak normal trace {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

scholarly journals The blessing of dimensionality for the analysis of climate data

2021 ◽  
Author(s):  
Bo Christiansen
Keyword(s):  
Climate Data ◽  
High Dimensions ◽  
Sample Mean ◽  
Simple Description ◽  
Atmospheric Sciences ◽  
Large Samples ◽  
Concentration Phenomena ◽  
Spatially Extended ◽  
Long Time ◽  
Effective Dimensions

Abstract. We give a simple description of the blessing of dimensionality with the main focus on the concentration phenomena. These phenomena imply that in high dimensions the length of independent random vectors from the same distribution have almost the same length and that independent vectors are almost orthogonal. In climate and atmospheric sciences we rely increasingly on ensemble modelling and face the challenge of analysing large samples of long time-series and spatially extended fields. We show how the properties of high dimensions allow us to obtain analytical results for, e.g., correlations between sample members and the behaviour of the sample mean when the size of the sample grows. We find that the properties of high dimensionality with reasonable success can be applied to climate data. This is the case although most climate data show strong anisotropy and both spatial and temporal dependence resulting in effective dimensions around 25–100.

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