Laplacian, on the Arrowhead Curve
| dc.contributor.author | Claire David | |
| dc.date.accessioned | 2021-03-04T13:56:16Z | |
| dc.date.available | 2021-03-04T13:56:16Z | |
| dc.date.issued | 2020 | |
| dc.description.abstract | 
 In terms of analysis on fractals, the Sierpinski gasket stands out as one of the most studied example.
 The underlying aim of those studies is to determine a differential operator equivalent to the classic Laplacian.
 The classically adopted approach is a bidimensional one, through a sequence of so-called prefractals, i.e. a sequence of graphs that converges towards the considered domain.
 The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on the prefractals.
 
 It turns out that the gasket is also the image of a Peano curve, the so-called Arrowhead one, obtained by means of similarities from a starting point which is the unit line.
 This raises a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry.
 Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account its specific geometry.
 
 Another difference due to the geometry, is encountered may one want to build a specific measure.
 For memory, the sub-cells of the Kigami and Strichartz approach are triangular and closed: the similarities at stake in the building of the Curve called for semi-closed trapezoids.
 As far as we know, and until now, such an approach is not a common one, and does not appear in such a context.
 
 It intererestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve.
 Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach.
 In doing so, we make the comparison -- and the link -- between three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach, using decimation; the Mosco approach.
 
 
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| dc.identifier.issn | 2409-8906 | |
| dc.identifier.uri | https://card-file.ontu.edu.ua/handle/123456789/16666 | |
| dc.source | Proceedings of the International Geometry Center | |
| dc.title | Laplacian, on the Arrowhead Curve |