Single Layer Potentials and the Cauchy-Kowalewski Theorem

1973 ◽  
Vol 80 (1) ◽  
pp. 61
Author(s):  
P. A. Nickel
Keyword(s):  
Single Layer ◽  
Layer Potentials

On norm of single layer potentials on segments

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Seyed M. Zoalroshd
Keyword(s):  
Upper Bound ◽  
Single Layer ◽  
Exact Expression ◽  
Operator Norm ◽  
Equal Length ◽  
Layer Potentials ◽  
Schmidt Norm ◽  
Special Case

AbstractWe show that, for a special case, equality of the spectra of single layer potentials defined on two segments implies that these segments must have equal length. We also provide an upper bound for the operator norm and exact expression for the Hilbert–Schmidt norm of single layer potentials on segments.

scholarly journals Single layer potentials on surfaces with small Lipschitz constants

2014 ◽  
Vol 418 (2) ◽  
pp. 676-712 ◽  
Author(s):  
Vladimir Kozlov ◽  
Johan Thim ◽  
Bengt Ove Turesson
Keyword(s):  
Single Layer ◽  
Layer Potentials ◽  
Lipschitz Constants

Simply and doubly periodic arrangements of the equi-stress holes in a perforated elastic plane: The single-layer potential approach

2017 ◽  
Vol 23 (5) ◽  
pp. 805-819 ◽  
Author(s):  
Shmuel Vigdergauz
Keyword(s):  
Periodic Structures ◽  
Rotational Symmetry ◽  
Infinite Plate ◽  
Single Layer ◽  
Mapping Technique ◽  
Elastic Plane ◽  
Layer Potentials ◽  
Layer Potential ◽  
Linear Elastostatics ◽  
Wide Range

The layer potentials of two-dimensional linear elastostatics are applied as a novel building block for the semi-analytical design of non-standard arrangements of the equi-stress holes in an infinite plate under a given bulk-type loading. Thematically, this paper begins where our previous work left off with the main attention being focused on periodic structures of low rotational symmetry which are hard to tackle by the more customary conformal mapping technique. The analytical derivations are backed up by numerical simulations, in which a genetic algorithm is utilized to identify the optimal interface shapes for different geometries and a wide range of the governing parameters.

scholarly journals Modified Representations for the Close Evaluation Problem

2021 ◽  
Vol 26 (4) ◽  
pp. 69
Author(s):  
Camille Carvalho
Keyword(s):  
Single Layer ◽  
Linear Partial Differential Equation ◽  
Representation Formula ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Layer Potentials ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Integral Equation Methods ◽  
Evaluation Problem

When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem’s solution. Similar to Gauss’s law used to modify Laplace’s double-layer potential, we use modified representations of Laplace’s single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions.

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