Geometrical logarithmic capacitance. (English) Zbl 1436.31010
Summary: This paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance (a very key notion in mathematical physics, quasiconformal geometry and variational calculus) through: (1) identifying with the reduced conformal module; (2) evaluating the minimal log-potential energy; (3) relating to both the volume-radius and the surface-radius; (4) linking with the \(n\)-harmonic radius and the log-capacity of the Kevin image of a compact surface; (5) finding the Minkowski inequality and the general variational formula for the log-capacity; (6) pinching the log-isocapacitary inequality from left and solving the left-prescribed problem for the normalized log-capacitary curvature measure; (7) pinching the log-isocapacitary inequality from right and handling the right-prescribed problem for the normalized log-capacitary curvature measure; (8) handling an overdetermination of the \(n\)-equilibrium potential of a given convex body via the log-capacitary concavity.
MSC:
| 31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J93 | Quasilinear elliptic equations with mean curvature operator |
| 53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |
Keywords:
reduced conformal modulus; geometric radii; log-isocapacitary inequality; log-potential energy; log-Minkowski’s inequalityReferences:
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