On Monday, 13 November, at 16:20 in room 5239 NSU the next meeting of the seminar on geometric analysis will take place.
Speaker: Vladimir Nikolaevich Dubinin, Doctor of Physics and Mathematics, Corresponding Member. RAS
Title of the report: Variational formulas for conformal capacity.
Annotation:
The topic of this lecture is closely related to the variational principles of conformal mappings. These principles, rich in both mathematical and mechanical applications, show how conformal (quasi-conformal) mappings change with small changes in the imaged regions. The lecture consists of two parts. The first part provides a brief overview of some approaches to obtaining variational formulas. We will focus on the use of conformal mappings and integral representations, the use of Hadamard's variational formula for the Green's function, and the use of the method of extremal metrics. Then we will give a direct proof of the analogue of the classical Hadamard variational formula for the conformal capacitance of a generalized capacitor under deformation of equipotential lines. The second part provides a variational formula for the conformal capacitance in the case of changing current lines. Both formulas can be considered as variational formulas for the energy of the electrostatic field. Finally, a proof is given of variational formulas for quadratic forms with coefficients depending on internal radii, Robin radii, Green's functions, or Robin functions. Special cases of these formulas are considered as examples.
