Mobile disks in hyperbolic space and minimization of conformal capacity. ETNA - Electronic Transactions on Numerical Analysis
Our focus is to study constellations of disjoint disks in the hyperbolicspace, i.e., the unit disk equipped with the hyperbolic metric. Each constellationcorresponds to a set $E$ which is the union of $m>2$ disks withhyperbolic radii $r_j>0, j=1,\ldots,m$. The centers of the disks are notfixed...
| Published in: | ETNA - Electronic Transactions on Numerical Analysis |
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| Main Authors: | , , |
| Format: | Journal/Newspaper |
| Language: | English |
| Published: |
oeaw
2024
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| Subjects: | |
| Online Access: | http://epub.oeaw.ac.at/?arp=buecher/Organisationseinheiten/_id105092_/ETNA/etna_Vol_60/pp1-19.pdf https://doi.org/10.1553/etna_vol60s1 |
| Summary: | Our focus is to study constellations of disjoint disks in the hyperbolicspace, i.e., the unit disk equipped with the hyperbolic metric. Each constellationcorresponds to a set $E$ which is the union of $m>2$ disks withhyperbolic radii $r_j>0, j=1,\ldots,m$. The centers of the disks are notfixed, and hence individual disks of the constellation are allowed tomove under the constraints that they do not overlap and theirhyperbolic radii remain invariant. Our main objective is to findcomputational lower bounds for the conformal capacity of a givenconstellation. The capacity depends on the centers and radii in a verycomplicated way even in the simplest cases when $m=3$ or $m=4$. In theabsence of analytic methods, our work is based on numerical simulationsusing two different numerical methods, the boundary integral equationmethod and the $hp$-FEM method, respectively. Our simulations combine capacitycomputation with minimization methods and produce extremal cases wherethe disks of the constellation are grouped next to each other. Thisresembles the behavior of animal colonies minimizingheat flow in arctic areas. |
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