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PreprintCC BY 4.0arXiv

Logarithmic capacity and torsion in planar shape optimization

  • David Ziener
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Abstract

We study a family of functionals involving a product of logarithmic capacity and torsional rigidity. Using several geometric inequalities, we identify the discs as the only maximizers among planar convex sets for certain values of the parameters. Finally, we prove regularity of maximizers in the non-critical case. This is done by establishing a shape-differentiability theorem for the logarithmic capacity in the class of planar convex bodies.

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Cite

BibTeXRIS
BibTeX
@misc{ziener2026logarithmic,
  title = {{Logarithmic capacity and torsion in planar shape optimization}},
  author = {David Ziener},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06205},
  eprint = {2609.06205},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06205},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Logarithmic capacity and torsion in planar shape optimization
AU  - David Ziener
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06205
UR  - https://arxiv.org/abs/2609.06205
AB  - We study a family of functionals involving a product of logarithmic capacity and torsional rigidity. Using several geometric inequalities, we identify the discs as the only maximizers among planar convex sets for certain values of the parameters. Finally, we prove regularity of maximizers in the non-critical case. This is done by establishing a shape-differentiability theorem for the logarithmic capacity in the class of planar convex bodies.
ER  -