Uniqueness of classical Dirichlet and compatible Neumann solutions

Put $w:=u_1-u_2$. By [F4, F7], $w$ is a real $C^2(\bar{\mathrm{\Omega }})$ function and $\mathrm{\Delta }w=\mathrm{\Delta }{u}_1-\mathrm{\Delta }{u}_2=0$, so $w$ is harmonic; also $w\in C^2(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})$. If $u_1=u_2$ on $\mathrm{\partial }\mathrm{\Omega }$ then $w=0$ on $\mathrm{\partial }\mathrm{\Omega }$, and if ${\mathrm{\partial }}_{\nu }{u}_1={\mathrm{\partial }}_{\nu }{u}_2$ on $\mathrm{\partial }\mathrm{\Omega }$ then, since ${\mathrm{\partial }}_{\nu }w=D{u}_1\cdot \nu -D{u}_2\cdot \nu$ by [F3, F4], also ${\mathrm{\partial }}_{\nu }w=0$ there. For every real constant $c$, [F4] gives $\mathrm{\Delta }(u_1+c)=\mathrm{\Delta }{u}_1$ and ${\mathrm{\partial }}_{\nu }(u_1+c)={\mathrm{\partial }}_{\nu }{u}_1$, so a constant shift changes neither datum.

Suppose $u\in C^2(\bar{\mathrm{\Omega }})$ is real, $-\mathrm{\Delta }u=f$ and ${\mathrm{\partial }}_{\nu }u=g$ on $\mathrm{\partial }\mathrm{\Omega }$. By [F7] the field $\mathrm{\nabla }u$ lies in $C^1(\bar{\mathrm{\Omega }};{\mathbb{R}}^n)$, so the divergence theorem [F3] applies to it: ${\int }_{\mathrm{\Omega }}\mathrm{\Delta }udx={\int }_{\mathrm{\Omega }}\mathrm{div}\mathrm{\nabla }udx={\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\nabla }u\cdot \nu dS={\int }_{\mathrm{\partial }\mathrm{\Omega }}gdS$, the last step by [F3] and the definition of $g$. Since $\mathrm{\Delta }u=-f$ pointwise, this is $-{\int }_{\mathrm{\Omega }}f={\int }_{\mathrm{\partial }\mathrm{\Omega }}g$, that is, ${\int }_{\mathrm{\Omega }}f=-{\int }_{\mathrm{\partial }\mathrm{\Omega }}gdS$, the compatibility equation of clause 3.

Dirichlet uniqueness. Assume $u_1=u_2$ on $\mathrm{\partial }\mathrm{\Omega }$, so that $\mathrm{\Delta }{u}_1=\mathrm{\Delta }{u}_2=\mathrm{\Delta }$ and $w$ has zero boundary trace by step 1.1. First route: the published uniqueness theorem [F1] applies to $u_1,u_2$, which lie in $C^2(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})$ by [F7] and have equal Laplacians and equal boundary values; hence $u_1=u_2$ on $\bar{\mathrm{\Omega }}$. Second route: if in addition a Dirichlet Green function with the regularity of [F8] exists, then all hypotheses of the zero-Dirichlet representation are met by $w$, which is real, $C^2(\bar{\mathrm{\Omega }})$, harmonic and has zero trace; the representation gives $w(x)={\int }_{\mathrm{\Omega }}{G}_{\mathrm{\Omega }}(x,y)\cdot 0dy=0$ for every $x\in \mathrm{\Omega }$, hence $u_1=u_2$ on $\mathrm{\Omega }$ and, by continuity [F7], on $\bar{\mathrm{\Omega }}$. Either route gives clause 1.

Neumann data determine exactly the affine family. Assume ${\mathrm{\partial }}_{\nu }{u}_1={\mathrm{\partial }}_{\nu }{u}_2$ on $\mathrm{\partial }\mathrm{\Omega }$. By step 1.1 the difference $w$ is real, harmonic and satisfies ${\mathrm{\partial }}_{\nu }w=0$ on $\mathrm{\partial }\mathrm{\Omega }$, and $w\in C^2(\bar{\mathrm{\Omega }})\subset C^1(\bar{\mathrm{\Omega }})$ by [F7]; so the first Green identity [F2] applies with both slots equal to $w$: ${\int }_{\mathrm{\Omega }}(w\mathrm{\Delta }w+\mathrm{\mid }Dw{\mathrm{\mid }}^2)dx={\int }_{\mathrm{\partial }\mathrm{\Omega }}w{\mathrm{\partial }}_{\nu }wdS$. The right side is $0$ because ${\mathrm{\partial }}_{\nu }w=0$, and $\mathrm{\Delta }w=0$, so ${\int }_{\mathrm{\Omega }}\mathrm{\mid }Dw{\mathrm{\mid }}^{2}dx=0$. The integrand $\mathrm{\mid }Dw{\mathrm{\mid }}^2$ is nonnegative with finite integral, so it vanishes almost everywhere by [F6]; if $\mathrm{\mid }Dw(y_0)\mathrm{\mid }>0$ at some $y_0\in \mathrm{\Omega }$, continuity of $Dw$ would give a ball $B(y_0,r)\subset \mathrm{\Omega }$ and a constant $c>0$ with $\mathrm{\mid }Dw{\mathrm{\mid }}^2\ge c$ on that ball, whence ${\int }_{\mathrm{\Omega }}\mathrm{\mid }Dw{\mathrm{\mid }}^2\ge c\lambda (B(y_0,r))>0$ by monotonicity and the positive ball measure of [F6], a contradiction. Hence $Dw=0$ on the nonempty open connected set $\mathrm{\Omega }$, and [F5] makes $w$ constant on $\mathrm{\Omega }$; by continuity [F7] that constant is the value on $\bar{\mathrm{\Omega }}$. Conversely, step 1.1 shows that $u_1+c$ has the same source and the same outward Neumann trace for every real $c$, so the solution set is exactly the affine family $u_1+\mathbb{R}$ whenever one solution exists.

The three clauses are independent statements: clause 1 uses only the Dirichlet data, clause 2 only the Neumann data, and clause 3 is the necessary equation of step 1.2. No existence is asserted, and no sufficiency of the compatibility equation is claimed; in particular the second clause says that the solution set is either empty or a full affine line in $C^2(\bar{\mathrm{\Omega }})$. The empty-support and zero-data cases are included: if $f=0$ and the traced data are zero, then $u\equiv 0$ is a solution and clauses 1–2 apply with no exception, while clause 3 reads $0=0$. Countable Choice is inherited from the Green, surface, Green-identity and divergence conventions of [F1], [F2], [F3] and [F8]; the pointwise differentiation, energy and limiting arguments add no further choice. For complex-valued $u_1,u_2$ the argument applies to $\mathrm{R}\mathrm{e}u$ and $\mathrm{I}\mathrm{m}u$ separately, since the Laplacian and the normal derivative are real-linear and the Green identity used is stated for real functions; the statement is formulated for real data. Dimension $n=1$ is excluded by [F1] and [F3].