Geometrical logarithmic capacitance

Abstract

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.

Introduction

We accelerate our exploitation of the $2\le n$-dimensional log-capacity made in [77], [75]. Specifically, we are interested in a more systematic study of the logarithm-involved set function performing on an arbitrarily compact set $K\subseteq {\mathbb{R}}^n$ – i.e. – the log-capacity of K:

$${\mathrm{C}}_{\log }(K)=\exp (\underset{|x|\to \infty }{\lim }(\log |x|-u_K(x)\left)\right)$$

whose existence is guaranteed by [36, Theorem 1.1 & Remarks 1.4-1.5] & [14, Remark 2.4] and where $u_K$ is the unique weak solution – i.e. – the n-equilibrium potential of K (based on the natural logarithm log which naturally appears in not only computing the voltage/capacitance difference between the coaxial cylindrical diodes/capacitors but also conducting a renormalization group investigation within the condensed-matter & high-energy physics [27], [60]) of the exterior boundary value problem:

$$\{\begin{array}{cc}-\mathrm{div}(|\mathrm{\nabla }u{|}^{n-2}\mathrm{\nabla }u)=0\hfill & \text{in}{\mathbb{R}}^n\setminus K;\hfill \\ u=0\hfill & \text{on}K;\hfill \\ 0<{\mathrm{lim\hspace{0.17em}inf}}_{|x|\to \infty }\frac{u(x)}{\log |x|}\le {\mathrm{lim\hspace{0.17em}sup}}_{|x|\to \infty }\frac{u(x)}{\log |x|}<\infty .\hfill \end{array}$$

Evidently, (1.1) is nonlinear unless $n=2$. Though it is really difficult to find ${\mathrm{C}}_{\log }(K)$, we always have

$$\{\begin{array}{c}{\mathrm{C}}_{\log }\left(\overline{{\mathbb{B}}^n}\right)={\mathrm{C}}_{\log }\left({\mathbb{S}}^{n-1}\right)=1;\hfill \\ {\mathrm{C}}_{\log }(x_0+rK)=r{\mathrm{C}}_{\log }(K),\hfill \end{array}$$

provided

$$\{\begin{array}{cc}\overline{{\mathbb{B}}^n}=\{x\in {\mathbb{R}}^n:|x|\le 1\}\hfill & \mathrm{\&}{\omega }_n=V(\overline{{\mathbb{B}}^n})=\text{volume of}\overline{{\mathbb{B}}^n};\hfill \\ {\mathbb{S}}^{n-1}=\{x\in {\mathbb{R}}^n:|x|=1\}\hfill & \mathrm{\&}{\sigma }_{n-1}=S({\mathbb{S}}^{n-1})=\text{surface-area of}{\mathbb{S}}^{n-1};\hfill \\ x_0+rK=\{x_0+rx:x\in K\}\hfill & \forall (x_0,r)\in {\mathbb{R}}^n\times (0,\infty ).\hfill \end{array}$$

As usual, the first equation in (1.1) says that u is n-harmonic in ${\mathbb{R}}^n\setminus K$. Meanwhile, K is said to be the polar whenever ${\mathrm{C}}_{\log }(K)=0$. However, it is perhaps appropriate to emphasize that the capacitary identification

$${\mathrm{C}}_{\log }(K)={\mathrm{C}}_{\log }(\partial K)$$

reveals physically that the net electric charge of a conductor K resides entirely on its boundary ∂K (existing as the surface of K).

Of basic importance in geometric function theory and mathematical physics is that if K is a compact subset of ${\mathbb{R}}^2\equiv \mathbb{C}$ (the finite complex plane) then ${\mathrm{C}}_{\log }(K)$ not only enjoys the strong log-subadditivity for any compact sets $K_1,K_2\subseteq {\mathbb{R}}^2$ (cf. [61]):

$$\log {\mathrm{C}}_{\log }(K_1\cup K_2)+\log {\mathrm{C}}_{\log }(K_1\cap K_2)\le \log {\mathrm{C}}_{\log }(K_1)+\log {\mathrm{C}}_{\log }(K_2),$$

but also equals the transfinite diameter $\mathrm{d}(K)$ or the Chebyshev constant $\mathrm{t}(K)$ or the Robin capacity $\exp (-\mathrm{r}(K))$ (cf. e.g. [2], [55], [69], [57], [62], [63], [43], [21], [23], [24], [18], [19], [72]):

• Geometrically speaking, the transfinite diameter of K is (cf. [21])

  $$\mathrm{d}(K)=\underset{k\to \infty }{\lim }{(\underset{z_1,\cdots ,z_k\in K}{\max }\prod _{1\le i<j\le k}|z_i-z_j|)}^{\frac{2}{(k-1)k}}.$$

• Functionally speaking, the Chebyshev constant of K is (cf. [21])

  $$\mathrm{t}(K)=\underset{k\to \infty }{\lim }(\underset{\mathrm{deg}(p)\le k-1}{\min }\underset{z\in K}{\max }|z^k-p(z)|),$$

  where $\mathbb{C}\ni z\mapsto p(z)$ is a polynomial with degree $\mathrm{deg}(p)$.

• Physically speaking, the Robin capacity of K is (cf. [19])

  $$\exp (-\mathrm{r}(K))=\exp (-\underset{\nu }{\inf }\underset{K}{\int }\underset{K}{\int }(\log |z-w{|}^{-1})d\nu (z)d\nu (w)),$$

  where $\mathrm{r}(K)$ is the Robin's minimum potential energy and ν runs over each unit charge distribution on K – actually – there exists a unique minimizing measure ${\nu }_0$ (called the 2-equilibrium distribution over K) such that its 2-conductor potential is

  $$U_0(z)=\underset{K}{\int }\log |z-w{|}^{-1}d{\nu }_0(w)=\log |z{|}^{-1}+\mathcal{O}(|z{|}^{-1})$$

  which equals $\mathrm{r}(K)$ quasi-everywhere on K (i.e., except for a polar set) and the Green function of $\mathbb{C}\setminus K$ with singularity at infinity (i.e., the solution to (1.1) under $n=2$) is

  $$G(z,\infty )=\mathrm{r}(K)-U_0(z).$$

Moreover, it is perhaps appropriate to recall that if $K\subseteq \mathbb{C}$ is simply connected then ${\mathrm{C}}_{\log }(K)$ coincides with the conformal radius $\mathrm{c}(K)$ of K since there is a unique holomorphic function

$$f(z)=\mathrm{c}(K)z+\sum _{j=0}^{\infty }{c}_j{z}^{-j}$$

mapping $\mathbb{C}\setminus \overline{{\mathbb{B}}^2}$ onto $\mathbb{C}\setminus K$ and sending ∞ to ∞, and so ${\mathrm{C}}_{\log }(\cdot )$ naturally appears in the calculus of variations for planar convex bodies (cf. [35], [9], [14]) and its extension to the setting for plurisubharmonic functions on the complex n-space ${\mathbb{C}}^n$ occurs in [37], [7].

As explicitly demonstrated in [77], [75], the discovered properties of ${\mathrm{C}}_{\log }(\cdot )$ shed light on a deep and wide connection between applied and pure mathematics which plays an increasing role in variational calculus, quasiconformal geometry and mathematical physics. Nevertheless, we feel that there is a highly strong need to explore more about the geometrical properties of the nonlinear set function ${\mathrm{C}}_{\log }(\cdot )$. In this paper we shall present eight novel issues involving such a geometric-nonlinear log-capacitary analysis in any dimension greater than one.

• In Section 2 we provide an equivalent definition of ${\mathrm{C}}_{\log }(\cdot )$ using the reduced conformal modulus (cf. Theorem 2.1), thereby showing that ${\mathrm{C}}_{\log }(\cdot )$ is not only a Choquet capacity but also satisfies an asymptotically strong log-subadditivity (cf. Theorem 2.2).

• In Section 3 as an n-dimensional analogue of the Robin capacity we give a physical interpretation of ${\mathrm{C}}_{\log }(K)$ of a compact $K\subseteq {\mathbb{R}}^n$ per minimizing the energy integral

  $$\underset{K\times K}{\iint }\log |x-y{|}^{-1}d\nu (x)d\nu (y)$$

  which characterizes the potential energy of the charge distribution $d\nu (x)d\nu (y)$ in the presence of $\log |x-y{|}^{-1}$ (cf. Theorem 3.2).

• In Section 4 we discuss the relationship of ${\mathrm{C}}_{\log }(\cdot )$ to the volume-radius and the surface-radius (cf. Theorems 4.1 & 4.3).

• In Section 5 we discover that not just the n-harmonic radius ${\rho }_D$ of a bounded regular domain D exists as a sharp lower bound of ${\mathrm{C}}_{\log }(D)$ and it is concave provided that D is convex (cf. Theorem 5.1), but also that if the compact $K\subseteq {\mathbb{R}}^n$ has a regular interior $K^{\circ }\ni o$ and $K^{⁎}$ is its Kelvin image $K^{⁎}$ under inversion

  $$K\ni x\mapsto x^{⁎}=|x{|}^{-2}x\in K^{⁎}$$

  then

  $${\mathrm{C}}_{\log }(\partial K){\mathrm{C}}_{\log }(\partial K^{⁎})\ge 1$$

  with the equality for K being a ball (cf. Theorem 5.3).

• In Section 6 we establish not only the optimal Minkowski inequality for ${\mathrm{C}}_{\log }(\cdot )$ for a convex body (cf. Theorem 6.1) but also the general variational formula for ${\mathrm{C}}_{\log }(\cdot )$ for an Alexandrov's region (cf. Theorem 6.3).

• In Section 7 we squeeze the log-isocapacitary inequality (a comparison for the volume-radius and the log-capacity; cf. (4.2))

  $${\left(\frac{V(K)}{{\omega }_n}\right)}^{\frac{1}{n}}\le {\mathrm{C}}_{\log }(K)$$

  from the left side of (1.3) through a Gauss-like probability integral over $\overline{{\mathbb{R}}^n}$, thereby solving the left-prescribed problem for the normalized log-capacitary curvature measure (cf. (6.1)-(6.2))

  $${\sigma }_{n-1}^{-1}{g}_{⁎}(|\mathrm{\nabla }{u}_K{|}^{n}dS)\text{on}{\mathbb{S}}^{n-1},$$

  the push-forward measure on ${\mathbb{S}}^{n-1}$ of the potential-weighted $(n-1)$-dimensional Hausdorff measure

  $${\sigma }_{n-1}^{-1}|\mathrm{\nabla }{u}_K{|}^{n}dS\text{on}\partial K,$$

  under the Gauss map $g:\partial K\to {\mathbb{S}}^{n-1}$ (cf. Theorem 7.2 which is also viewed as resolving a log-capacitary variant of the Yau's Problem 59 on [79, p. 683]).

• In Section 8 we squeeze (1.3) from the right side of (1.3) through the mean-width-like integral on ${\mathbb{S}}^{n-1}$ (as a very nice description of $\overline{{\mathbb{R}}^n}$ through the stereographic projection [42, p. 112]), thereby handling the right-prescribed problem for

  $${\sigma }_{n-1}^{-1}{g}_{⁎}(|\mathrm{\nabla }{u}_K{|}^{n}dS)\text{on}{\mathbb{S}}^{n-1}$$

  (cf. Theorem 8.2 which is also regarded as the appropriate resolution of a log-capacitary analogue of the classical Minkowski-type problem solved in [51], [54], [11], [34], [35], [15]).

• In Section 9 we are motivated by the last three sections and [64] to consider an overdetermined problem for (1.1), thereby utilizing the log-capacitary concavity to show in Theorem 9.2 that a given convex body K is homothetic to a sublevel set of the n-equilibrium potential $u_K$ if and only if $\exp u_K$ is convex – equivalently – K is a closed ball.