Example 4.1.21FC1E9D9B42F0

Example 4.1.21.FC1E9D9B42F0 Let $0<r<R$ and $K=\overline{B(0,r)}$. The function $x\mapsto G_ B(x,0)$ is radial and decreasing in $|x|$ by lemma 4.1.14. We claim that $u_ K=f_ K=w$, where

The function $w$ equals $-1$ on $K$ and $-G_ B(x,0)/G_ B(re_1,0)$ on the annulus $r<|x|<R$. It is subharmonic, because $-G_ B(\cdot ,0)/G_ B(re_1,0)$ is a positive multiple of $E_ d$ plus a constant on $B$, and it is nonpositive; hence it is admissible and $w\leq f_ K$. Conversely, let $v$ be admissible. On the annulus, $w$ is harmonic and continuous up to the boundary, with boundary values $-1$ on $|x|=r$ and $0$ on $|x|=R$. Since $v\leq -1$ on the inner sphere and $v\leq 0$ on $B$, upper semicontinuity shows that the boundary upper limits of $v$ are at most these values. The comparison principle (theorem 3.4.5) on the annulus gives $v\leq w$ there, and $v\leq -1=w$ on $K$. Hence $f_ K=w$, and since $w$ is continuous, $u_ K=w$. Near $\partial B$, $u_ K$ is $-1/G_ B(re_1,0)$ times $G_ B(\cdot ,0)$, whose outward normal derivative on $\partial B$ is $-(s_{d-1}R^{d-1})^{-1}$. Therefore the flux formula of proposition 4.1.16 gives

These are relative capacities. To compute the kernel capacities of the same ball, let $\sigma _ r$ be the normalized surface measure on the sphere $|x|=r$ and $p=E_ d*\sigma _ r$. If $|x|>r$, then $p(x)$ is the spherical mean over $\partial B(0,r)$ of the function $y\mapsto E_ d(x-y)$, which is harmonic on a neighborhood of $\overline{B(0,r)}$; hence $p(x)=E_ d(x)$ by the mean-value property (theorem 3.1.7). Surface measure is invariant under orthogonal transformations (subsection 3.1.1), which act transitively on spheres centered at the origin, so $p$ is radial. On the ball $|x|<r$, the function $p$ is harmonic: by proposition 3.4.11 and theorem 3.1.11 it agrees almost everywhere with a harmonic function there, hence everywhere by theorem 3.4.12. Since $p$ is radial, for $0<|x|<r$ the value $p(x)$ is the spherical mean of $p$ over $\partial B(0,|x|)$, which equals $p(0)$ by the mean-value property; moreover, $p(0)=E_ d(re_1)$ because $E_ d$ is radial. Finally, $p$ is constant on the sphere $|x|=r$, which is the support of $\sigma _ r$, so part (1) of theorem 4.1.10 shows that $p$ is continuous at the points of the sphere. Hence $p=E_ d(re_1)$ on $K$.

Consequently the Newtonian potential of $\sigma _ r$ equals $\Gamma _ d(re_1)$ on $K$ and $\Gamma _ d(x)$ for $|x|>r$, and the logarithmic potential with a scale $D>2r$ equals $(2\pi )^{-1}\log (D/r)$ on $K$ and $(2\pi )^{-1}\log (D/|x|)$ for $|x|>r$. In particular $\sigma _ r$ has finite energy, and its potential $U^{\sigma _ r}$ has a constant value $V$ on $K$. Since $\sigma _ r$ is a nonzero measure with finite energy carried by the sphere $|x|=r$, theorem 4.1.17 shows that this sphere is not polar, although it has Lebesgue measure zero: for $0<\varepsilon <r$ it is contained in $B(0,r+\varepsilon )\setminus B(0,r-\varepsilon )$, whose measure $s_{d-1}\bigl ((r+\varepsilon )^ d-(r-\varepsilon )^ d\bigr )/d$ tends to zero as $\varepsilon \downarrow 0$.

Let $\nu $ be a probability measure on $K$ with finite energy. Then $\nu -\sigma _ r$ has total mass zero, and its positive and negative parts are dominated by $\nu $ and $\sigma _ r$, so they have finite energy because the kernel is nonnegative on $K\times K$. Moreover, $\int U^{\sigma _ r}\, \mathrm d(\nu -\sigma _ r)=V-V=0$. Hence proposition 4.1.5 gives

Thus $\sigma _ r$ is the equilibrium measure of $K$, the Newtonian capacity of $K$ is $(d-2)s_{d-1}r^{d-2}$ if $d\geq 3$, and its logarithmic capacity is $r$ if $d=2$. A probability measure on a compact set is also one on every larger compact set, so the kernel capacities are increasing; when $d=2$, we compute both logarithmic capacities with a scale exceeding the diameter of the larger set, as theorem 4.1.7 allows. Letting $r\downarrow 0$, we recover the fact that a point has kernel capacity zero.