Existence and regularity for the electrostatic boundary value problem

Dirichlet branch. By [F8] the bounded smooth domain ${\mathrm{\Omega }}_v$ carries the boundary-chart presentation required by the solvability theorem [F1]. With $\epsilon :={\epsilon }_{0}I$, smooth and uniformly positive on $\overline{{\mathrm{\Omega }}_v}$, and smooth $\rho$, F1 gives exactly one $u\in H^1({\mathrm{\Omega }}_v)$ with trace $g$, $C^{\mathrm{\infty }}$ on $\overline{{\mathrm{\Omega }}_v}$, satisfying $-\mathrm{div}({\epsilon }_0\mathrm{\nabla }u)=\rho$ pointwise and $u{\mathrm{\mid }}_{\mathrm{\partial }{\mathrm{\Omega }}_v}=g$; the weak uniqueness in the $H^1$ class with trace $g$ is part of the same clause. Dividing by ${\epsilon }_0>0$ gives $-\mathrm{\Delta }u=\rho \mathrm{/}{\epsilon }_0$ and the equation and trace displayed in (a), so $\phi :=u$ is the asserted solution and it is unique in the stated class by F7 as well. The field $E:=-\mathrm{\nabla }\phi$ is $C^{\mathrm{\infty }}$ on $\overline{{\mathrm{\Omega }}_v}$ and satisfies $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{curl}E=0$ by [F2], which is the fixed-time reading of the pure electrostatic regime with source $\rho$; hence $(E,0)$ realises that regime on the vacuum region. This is (a).

Neumann branch. Let $g\in C^{\mathrm{\infty }}(\mathrm{\partial }{\mathrm{\Omega }}_v)$ satisfy the componentwise compatibility of (b); with $\epsilon ={\epsilon }_{0}I$ the conormal datum of F1 is ${\epsilon }_{0}g$ and its compatibility ${\int }_{{\mathrm{\Omega }}^{(j)}}\rho +{\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{(j)}}{\epsilon }_{0}g=0$ is exactly the displayed hypothesis, so F1 gives a weak solution, $C^{\mathrm{\infty }}$ on each closed component, unique up to one constant per component and normalised by zero componentwise means; after division by ${\epsilon }_0$ the equation is $-\mathrm{\Delta }\phi =\rho \mathrm{/}{\epsilon }_0$ and the conormal trace is ${\mathrm{\partial }}_{\nu }\phi =g$ by the definition of the conormal derivative and the classical normal derivative [F1, F2]. The necessity of the compatibility is F7, whose flux identity gives ${\int }_{{\mathrm{\Omega }}^{(j)}}\rho +{\epsilon }_0{\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{(j)}}gdS=0$ for any classical solution. This is (b).

Transmission branch. Let the finitely many disjoint closed smooth interfaces $\mathrm{\Gamma }$, sheet density ${\sigma }_{\mathrm{\Gamma }}$ and outer Dirichlet data $g$ be as in (d); with $\epsilon ={\epsilon }_{0}I$ on each side, piecewise smooth $\rho$ and smooth ${\sigma }_{\mathrm{\Gamma }}$, F1 gives a globally $H^1$ weak solution with outer Dirichlet data, $C^{\mathrm{\infty }}$ on the closure of each side, equal one-sided traces, unique by F7(iii), and flux jump $n\cdot [{\epsilon }_0\mathrm{\nabla }\phi ]=-{\sigma }_{\mathrm{\Gamma }}$. Since $E=-\mathrm{\nabla }\phi$ on each side, $[{\epsilon }_0\mathrm{\nabla }\phi ]={\epsilon }_0[\mathrm{\nabla }\phi ]=-{\epsilon }_0[E]$, so $n\cdot [E]={\sigma }_{\mathrm{\Gamma }}\mathrm{/}{\epsilon }_0$, the sign convention of [F6]; the weak right-hand side is ${\int }_{{\mathrm{\Omega }}_v}\rho v+{\sum }_{\mathrm{\Gamma }}{\int }_{\mathrm{\Gamma }}{\sigma }_{\mathrm{\Gamma }}Tv$ as stated. No dielectric coefficient is introduced because $\epsilon ={\epsilon }_{0}I$ on every side. This is (d).

Conductor reading. Assume the bounded $C^{\mathrm{\infty }}$ ambient branch with outer data $g_0$ and conductor data $V_j$. The boundary of ${\mathrm{\Omega }}_v$ is the disjoint union of the compact boundary pieces $\mathrm{\partial }\mathrm{\Omega },\mathrm{\partial }{D}_1,\dots ,\mathrm{\partial }{D}_m$ [F3], and the function equal to $g_0$ on $\mathrm{\partial }\mathrm{\Omega }$ and to the constant $V_j$ on $\mathrm{\partial }{D}_j$ is smooth on each piece, hence admissible in (a) [step 1.1]; let $\phi$ be the resulting solution and $E=-\mathrm{\nabla }\phi$ on ${\mathrm{\Omega }}_v$, extended by $E:=0$ on the conductor. Then $\phi$ is continuous on $\overline{{\mathrm{\Omega }}_v}$ with the constant trace $V_j$ on each $\mathrm{\partial }{D}_j$ [F3], and $E$ has the joint one-sided $C^1$ regularity across $S=\mathrm{\partial }D$ because $E$ is $C^{\mathrm{\infty }}$ on $\overline{{\mathrm{\Omega }}_v}$, [F10] supplies its smooth ambient extension, and $0$ is smooth on the conductor side. Define $\sigma :={\epsilon }_{0}n\cdot E_{+}\in C^{\mathrm{\infty }}(\mathrm{\partial }D)$ directly; its pairing is a stationary-sheet source [F6]. In every boundary chart $X$, $\phi \circ X=V_j$, so the chain rule gives $\mathrm{\nabla }{\phi }_{+}\cdot {\mathrm{\partial }}_{a}X=0$ for both tangent directions, hence $n\times E_{+}=0$. Also $\nu =-n$ on the conductor boundary, so ${\mathrm{\partial }}_{\nu }\phi =\mathrm{\nabla }{\phi }_{+}\cdot (-n)=n\cdot E_{+}$ and $Q_j:={\int }_{\mathrm{\partial }{D}_j}\sigma dS={\epsilon }_0{\int }_{\mathrm{\partial }{D}_j}{\mathrm{\partial }}_{\nu }\phi dS$. For the distributional equations: [F5] applied on the compact embedded interface $S$ with ${\mathrm{\Omega }}_{-}$ the conductor, ${\mathrm{\Omega }}_{+}$ the vacuum and the normal $n$ from conductor to vacuum gives $\mathrm{div}E=\{\mathrm{div}E{\}}_{\text{piecewise}}+(n\cdot [E]){\delta }_S$ and $\mathrm{curl}E=\{\mathrm{curl}E{\}}_{\text{piecewise}}+(n\times [E]){\delta }_S$; the piecewise parts are ${\rho }_v\mathrm{/}{\epsilon }_0$ on the vacuum side and $0$ on the conductor side, and $[E]=E_{+}$ with $n\cdot E_{+}=\sigma \mathrm{/}{\epsilon }_0$, $n\times E_{+}=0$ by the preceding chart calculation, so the identities are exactly the adopted distributional vacuum Maxwell equations with the source pair $({\rho }_v+\sigma {\delta }_{\mathrm{\partial }D},0)$ [F6]. Extending $\phi$ by $V_j$ on each body gives the continuous model potential. These verified identities establish the conductor model, so [F4] now applies to it and confirms its surface-charge reading without being used to assume the equations just constructed. This is (c).

Scope. All existence and regularity statements above are exactly the clauses of [F1] for the smooth bounded domains of the statement, the constant coefficient ${\epsilon }_{0}I$, smooth data and finitely many disjoint smooth interfaces; the conductor clause additionally uses only the bounded smooth ambient branch of [F3] and the constant data there. No nonsmooth boundary, corner, edge, coefficient irregularity, prescribed-total-charge problem, unbounded exterior region without a decay class, dielectric response, measurement, apparatus or empirical claim is asserted; existence of the potential is not asserted to be a physical equilibrium, which is the adopted model of The ideal electrostatic conductor model [F3]. This is (e).