Uniform-field dielectric sphere: candidate solution and its class

On $r\mathrm{\ne }0$ the functions $z$ and $z{r}^{-3}$ are smooth and $r^{-3}$ is smooth away from the origin; the sum and scalar rules of [L1] give $\mathrm{\Delta }z=0$ and $\mathrm{\Delta }(z{r}^{-3})=z\mathrm{\Delta }(r^{-3})+2\mathrm{\nabla }z\cdot \mathrm{\nabla }(r^{-3})$, with ${\mathrm{\partial }}_i{r}^{-3}=-3x_i{r}^{-5}$ and $\mathrm{\Delta }(r^{-3})={\sum }_i{\mathrm{\partial }}_i(-3x_i{r}^{-5})=-9r^{-5}+15r^{-5}=6r^{-5}$; hence $\mathrm{\Delta }(z{r}^{-3})=6z{r}^{-5}+2\cdot (-3z{r}^{-5})=0$, so $\mathrm{\Delta }{\phi }_{\mathrm{i}\mathrm{n}}=0$ on $r<a$ and $\mathrm{\Delta }{\phi }_{\mathrm{o}\mathrm{u}\mathrm{t}}=0$ on $r>a$; the interior potential is linear on the entire closed ball, including the origin, while the exterior expression and its gradient are smooth up to $S$ because $r\ge a>0$ on that closed exterior side.

Put $q:=z\mathrm{/}r$ for the fixed directional factor along a radial line. At $r=a$: ${\phi }_{\mathrm{i}\mathrm{n}}(a)=-C{E}_{0}aq$ and ${\phi }_{\mathrm{o}\mathrm{u}\mathrm{t}}(a)=(-E_{0}a+d\mathrm{/}{a}^2)q$, whose equality for every angle is the scalar equation $C=1-d\mathrm{/}(E_0{a}^3)$; the radial derivatives are ${\mathrm{\partial }}_r{\phi }_{\mathrm{i}\mathrm{n}}=-C{E}_{0}q$ and ${\mathrm{\partial }}_r{\phi }_{\mathrm{o}\mathrm{u}\mathrm{t}}=(-E_0-2d\mathrm{/}{a}^3)q$, whose equality after the weights ${\epsilon }_{\mathrm{i}\mathrm{n}},{\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is ${\epsilon }_{\mathrm{i}\mathrm{n}}C={\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}(1+2d\mathrm{/}(E_0{a}^3))$. Substituting the displayed $C$ and $d$ satisfies both equations (the determinant of the pair is ${\epsilon }_{\mathrm{i}\mathrm{n}}+2{\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}>0$), so the two matching identities hold. By the chain rule along any smooth surface chart, equal traces have equal tangential derivatives and hence equal tangential electric fields; conversely, equality of those derivatives makes the trace difference constant along every piecewise smooth path on the sphere, whose points are joined by great-circle arcs, without fixing its value. The displayed trace computation fixes it to zero. The weighted radial identity is exactly normal-$D$ continuity because $E=-\mathrm{\nabla }\phi$ and $D=\epsilon E$. The asymptotics (c) follow from $\mathrm{\mid }z\mathrm{\mid }\le r$: $\mathrm{\mid }{\phi }_{\mathrm{o}\mathrm{u}\mathrm{t}}+E_{0}z\mathrm{\mid }=\mathrm{\mid }d\mathrm{\mid }\mathrm{\mid }z\mathrm{\mid }{r}^{-3}\le \mathrm{\mid }d\mathrm{\mid }{r}^{-2}$ and $\mathrm{\nabla }(dz{r}^{-3})=d(r^{-3}{e}_3-3z{r}^{-5}x)$ has squared norm $d^2{r}^{-6}(1+3q^2)\le 4d^2{r}^{-6}$ because $\mathrm{\mid }q\mathrm{\mid }\le 1$, hence norm at most $2\mathrm{\mid }d\mathrm{\mid }{r}^{-3}$.

For the declared static completion, [L3] gives $\mathrm{div}D=\{\mathrm{div}D\}+n\cdot [D]{\delta }_S$ and $\mathrm{curl}E=\{\mathrm{curl}E\}+n\times [E]{\delta }_S$, with analogous formulas for $B$ and $H$; by step 1.1 the piecewise classical divergences and curls of $D$ and $E$ vanish (each side is a constant multiple of the gradient of a harmonic function, so $\mathrm{div}E=0$ and $\mathrm{curl}E=0$ there, the latter by the mixed-partial commutation of [L1]), and by step 1.2 the normal jump $n\cdot [D]$ and tangential jump $n\times [E]$ vanish; since $B=H=0$ and ${\rho }_f=J_f=0$ with ${\sigma }_f=K_f=0$, all four distributional macroscopic equations hold on the local region and the hypotheses of Stationary macroscopic interface jump conditions for D, B, E and H ($S=\{r=a\}$ a compact embedded $C^2$ hypersurface in $\mathrm{\Omega }$ because ${\bar{B}}_a\subseteq \mathrm{\Omega }$, locally integrable fields, $C^1$ off $S$, joint one-sided $C^1$ extensions, stationary free sheet data) are verified.

For uniqueness, let $w_{\mathrm{i}\mathrm{n}}={\tilde{\phi }}_{\mathrm{i}\mathrm{n}}-{\phi }_{\mathrm{i}\mathrm{n}}$ on $\overline{B_a}$ and $w_{\mathrm{o}\mathrm{u}\mathrm{t}}={\tilde{\phi }}_{\mathrm{o}\mathrm{u}\mathrm{t}}-{\phi }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ on $\{r\ge a\}$; both are harmonic with $C^2$ regularity on the closed regions, their traces and weighted radial derivatives agree on $S$ by the shared interface conditions, and $\mathrm{\mid }{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{\mid }\le (M+\mathrm{\mid }d\mathrm{\mid })r^{-1}$ for all sufficiently large $r\ge 1$. Apply the first Green identity of [L2] with $u=v=w_{\mathrm{i}\mathrm{n}}$ on the ball $B_a$ and with $u=v=w_{\mathrm{o}\mathrm{u}\mathrm{t}}$ on the annulus $\{a<r<R\}$: since the Laplacians vanish, ${\int }_{B_a}\mathrm{\mid }\mathrm{\nabla }{w}_{\mathrm{i}\mathrm{n}}{\mathrm{\mid }}^2={\oint }_S{w}_{\mathrm{i}\mathrm{n}}{\mathrm{\partial }}_r{w}_{\mathrm{i}\mathrm{n}}$ and ${\int }_{a<r<R}\mathrm{\mid }\mathrm{\nabla }{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}{\mathrm{\mid }}^2={\oint }_{\{r=R\}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}{\mathrm{\partial }}_r{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}-{\oint }_S{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}{\mathrm{\partial }}_r{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$; multiplying by ${\epsilon }_{\mathrm{i}\mathrm{n}}$ and ${\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and adding, the two surface terms on $S$ cancel by $w_{\mathrm{i}\mathrm{n}}=w_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and ${\epsilon }_{\mathrm{i}\mathrm{n}}{\mathrm{\partial }}_r{w}_{\mathrm{i}\mathrm{n}}={\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}{\mathrm{\partial }}_r{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$, while the term on $\{r=R\}$ is bounded by $(4\pi R^2)\cdot (M+\mathrm{\mid }d\mathrm{\mid })R^{-1}\cdot C{R}^{-2}$ using [L2]'s Cauchy estimate on balls of radius $R\mathrm{/}4$ centred at $\mathrm{\mid }x\mathrm{\mid }=R$, which lie in the exterior for large $R$ and carry the bound $\mathrm{\mid }{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{\mid }\le C\mathrm{/}R$, hence tends to $0$. For each fixed $L>a$, the nonnegative weighted energy on the ball and $a<r<L$ is at most that on the ball and $a<r<R$ for $R>L$, and the latter is bounded by the absolute outer boundary term, which tends to zero. Thus every fixed bounded-region energy is zero; continuity and the positive weights imply that both gradients vanish, so $w_{\mathrm{i}\mathrm{n}}$ is constant on the ball and $w_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is constant on $r>a$; the decay forces that constant to be $0$, hence the common trace is $0$, and then $w_{\mathrm{i}\mathrm{n}}=0$ by the Dirichlet uniqueness of [L2] applied on the ball to $w_{\mathrm{i}\mathrm{n}}$ and the zero function.

The fields follow by differentiation: $E_{\mathrm{i}\mathrm{n}}=-\mathrm{\nabla }{\phi }_{\mathrm{i}\mathrm{n}}=C{E}_0{e}_3$ with $D_{\mathrm{i}\mathrm{n}}={\epsilon }_{\mathrm{i}\mathrm{n}}C{E}_0{e}_3$ on $r<a$; outside, ${\phi }_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the sum of the applied uniform potential and the potential $dz{r}^{-3}$ of a dipole at the origin with moment $p=4\pi {\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}d{e}_3$, and direct differentiation of the defined dipole potential gives $E_p(x)=[3(p\cdot n)n-p]\mathrm{/}(4\pi {\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}{r}^3)$, so the exterior field has precisely that form; the interior polarisation is $P_{\mathrm{i}\mathrm{n}}=D_{\mathrm{i}\mathrm{n}}-{\epsilon }_0{E}_{\mathrm{i}\mathrm{n}}=({\epsilon }_{\mathrm{i}\mathrm{n}}-{\epsilon }_0)C{E}_0{e}_3$ and the exterior polarisation is $P_{\mathrm{o}\mathrm{u}\mathrm{t}}=({\epsilon }_{\mathrm{o}\mathrm{u}\mathrm{t}}-{\epsilon }_0)E_{\mathrm{o}\mathrm{u}\mathrm{t}}$, so the bound surface charge density is the singular coefficient of ${\rho }_b=-\mathrm{div}P$ at $S$, namely ${\sigma }_b=n\cdot (P_{\mathrm{i}\mathrm{n}}-P_{\mathrm{o}\mathrm{u}\mathrm{t}})$ in $\mathrm{C}{\mathrm{m}}^{-2}$, by applying [L3] to the piecewise smooth $P$ with jump $P_{\mathrm{o}\mathrm{u}\mathrm{t}}-P_{\mathrm{i}\mathrm{n}}$.

Steps 1.1-1.2 verify (a)-(c), step 2.1 verifies (b) and the distributional thin-interface reading, step 2.2 gives the uniqueness statement (d) with existence exhibited by the displayed pair, and step 2.3 gives the fields, polarisation, exterior dipole moment and bound surface charge of (e). The scope (f) holds: the computation is quasistatic, of the declared linear branch at ${\sigma }_f=0$, conditional on the stated constants and regularity; the identification of $p$ with a dynamic wave polarisability requires a separate quasistatic/dipole approximation, and no material data, empirical claim or general existence theorem is supplied beyond this explicit transmission solution.