Green representation for classical Poisson data

Fix $x\in \mathrm{\Omega }$ and choose ${\epsilon }_0>0$ with $B(x,{\epsilon }_0)\subset \mathrm{\Omega }$; for $0<\epsilon <{\epsilon }_0$ put $D_{\epsilon }:=\mathrm{\Omega }\setminus \bar{B}(x,\epsilon )$. Since $\bar{B}(x,\epsilon )\subset B(x,{\epsilon }_0)\subset \mathrm{\Omega }$, the set $D_{\epsilon }$ is open, bounded, contained in $\mathrm{\Omega }$, and nonempty (any point at distance $(\epsilon +{\epsilon }_0)\mathrm{/}2$ from $x$ lies in it), and its boundary is the disjoint union of $\mathrm{\partial }\mathrm{\Omega }$ and $S(x,\epsilon )$: every point of $S(x,\epsilon )$ lies in the open set $\mathrm{\Omega }$, so near such a point $D_{\epsilon }$ coincides with a ball minus the closed Euclidean ball, whose boundary is the regular level set $S(x,\epsilon )$ of [F8], and near a point of $\mathrm{\partial }\mathrm{\Omega }$ it coincides with $\mathrm{\Omega }$ because the removed ball is at positive distance from that point. Hence $D_{\epsilon }$ is a bounded $C^1$ domain in the sense of [F7], with outward normal ${\nu }_{\mathrm{\Omega }}$ on $\mathrm{\partial }\mathrm{\Omega }$ and outward normal $\sigma (y):=-(y-x)\mathrm{/}\mathrm{\mid }y-x\mathrm{\mid }$ on $S(x,\epsilon )$, since moving from $y\in S(x,\epsilon )$ in direction $\sigma$ enters the excluded ball while moving in direction $-\sigma$ leaves it.

Estimating on the small sphere: differentiating the two profiles of [F3] gives $D\mathrm{\Phi }(w)=-\frac{w}{{\omega }_{n-1}\mathrm{\mid }w{\mathrm{\mid }}^n}$ for $w\mathrm{\ne }0$, in both the power and the logarithmic case, because the radial profile $q$ with $\mathrm{\Phi }(w)=q(\mathrm{\mid }w\mathrm{\mid })$ has $q^{\mathrm{\prime }}(s)=-\frac{1}{{\omega }_{n-1}{s}^{n-1}}$ for $s>0$, and $\mathrm{\nabla }\mathrm{\mid }\cdot \mathrm{\mid }(w)=w\mathrm{/}\mathrm{\mid }w\mathrm{\mid }$ [F12]; consequently $D\mathrm{\Phi }(w)\cdot \sigma (w)=\frac{1}{{\omega }_{n-1}\mathrm{\mid }w{\mathrm{\mid }}^{n-1}}$ for $w\mathrm{\ne }0$. On $S(x,\epsilon )$ this gives $D\mathrm{\Phi }(y-x)\cdot \sigma (y)=\frac{1}{{\omega }_{n-1}{\epsilon }^{n-1}}$, while $\mathrm{\Phi }(y-x)$ has modulus $\frac{{\epsilon }^{2-n}}{(n-2){\omega }_{n-1}}$ for $n\ge 3$ and $\frac{\mathrm{\mid }\log \epsilon \mathrm{\mid }}{2\pi }$ for $n=2$ [F3]; and $\mathrm{\mid }D{H}_x(y)\mathrm{\mid }\le b_x$ for $y\in \bar{\mathrm{\Omega }}$ with $b_x:=\mathrm{\parallel }D{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}<\mathrm{\infty }$ by [F14]. The sphere has surface measure ${\omega }_{n-1}{\epsilon }^{n-1}$ by [F9], and $\mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}<\mathrm{\infty }$ by [F14].

Absolute finiteness and measurability. The volume integrand $y\mapsto G_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)$ is defined for $y\mathrm{\ne }x$ and bounded in modulus by $(\mathrm{\mid }\mathrm{\Phi }(y-x)\mathrm{\mid }+\mathrm{\parallel }{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})})\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}$, and $\mathrm{\Phi }(\cdot -x)$ is locally integrable on the bounded set $\mathrm{\Omega }$ by [F13]; hence an integrable dominating function exists, and the integrand is Borel by [F1, F17] and $\mathrm{\Delta }u\in C(\bar{\mathrm{\Omega }})$. The boundary integrand $u(y)P_{\mathrm{\Omega }}(x,y)$ is bounded in modulus by $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}({\sup }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }D\mathrm{\Phi }(\cdot -x)\mathrm{\mid }+b_x)$, a finite bound because $\mathrm{\partial }\mathrm{\Omega }$ is compact and at positive distance from $x$ [F3, F14]; the boundary has finite surface measure [F10], so the boundary integral is absolutely finite as well.

Nonnegativity of the Poisson kernel. First fix $x\in \mathrm{\Omega }$. By [F2], for $y\mathrm{\ne }x$ we have $G_{\mathrm{\Omega }}(x,y)=\mathrm{\Phi }(y-x)-H_x(y)$. The corrector $H_x$ is bounded on $\bar{\mathrm{\Omega }}$ by [F14], whereas $\mathrm{\Phi }(y-x)$ tends to $+\mathrm{\infty }$ as $y\to x$ by [F3]. Thus, for every sufficiently small $\epsilon >0$ as in step 1.1, $G_{\mathrm{\Omega }}(x,\cdot )>0$ on the inner sphere $S(x,\epsilon )$ and is zero on $\mathrm{\partial }\mathrm{\Omega }$ by [F1]. It is harmonic on $D_{\epsilon }$ and continuous on $\overline{D_{\epsilon }}$ by [F1]–[F3]. The weak minimum principle [F4], applicable to the bounded nonempty open set $D_{\epsilon }$ without a connectedness hypothesis, gives $G_{\mathrm{\Omega }}(x,y)\ge 0$ for every $y\in D_{\epsilon }$. Given any $y\in \mathrm{\Omega }\setminus \{x\}$, choose such an $\epsilon <\mathrm{\mid }y-x\mathrm{\mid }$; hence $G_{\mathrm{\Omega }}(x,y)\ge 0$ throughout $\mathrm{\Omega }\setminus \{x\}$. Now fix $y_0\in \mathrm{\partial }\mathrm{\Omega }$ and, for small $t>0$, put $y_t:=y_0-t{\nu }_{\mathrm{\Omega }}(y_0)$. By the local subgraph property in [F7], $y_t\in \mathrm{\Omega }$ for all sufficiently small $t>0$, and $y_t\mathrm{\ne }x$ for those $t$; define $g(t):=G_{\mathrm{\Omega }}(x,y_t)$ for such $t$ and $g(0):=G_{\mathrm{\Omega }}(x,y_0)=0$. Symmetry and the corrector regularity give a $C^2$ expression for $G_{\mathrm{\Omega }}(x,y)$ near $y_0$ by [F2, F3, F7]; the chain rule with the classical normal derivative [F5, F11, F12] gives $g^{\mathrm{\prime }}(0^{+})=P_{\mathrm{\Omega }}(x,y_0)$. Since $g(t)\ge 0$, its one-sided difference quotients $g(t)\mathrm{/}t$ are nonnegative, and so $P_{\mathrm{\Omega }}(x,y_0)\ge 0$.

Apply the second Green identity [F6] on the bounded $C^1$ domain $D_{\epsilon }$ of step 1.1 to the real functions $U:=u$ and $V:=G_{\mathrm{\Omega }}(x,\cdot )$, the latter being $C^2$ on $\overline{D_{\epsilon }}$ by [F2, F3, F7] and harmonic on $D_{\epsilon }$ by [F1, F3, F18]. Both volume integrals are finite because $D_{\epsilon }\subset \mathrm{\Omega }$ and both functions are bounded on $\overline{D_{\epsilon }}$ [F14], so ${\int }_{D_{\epsilon }}{G}_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)dy={\int }_{\mathrm{\partial }{D}_{\epsilon }}{G}_{\mathrm{\Omega }}(x,y){\mathrm{\partial }}_{\nu }u(y)dS(y)-{\int }_{\mathrm{\partial }{D}_{\epsilon }}u(y){\mathrm{\partial }}_{\nu }{G}_{\mathrm{\Omega }}(x,\cdot )(y)dS(y)\mathrm{.}$ On $\mathrm{\partial }\mathrm{\Omega }$ the trace $G_{\mathrm{\Omega }}(x,\cdot )=0$ by [F1], and the outward normal of $D_{\epsilon }$ there is ${\nu }_{\mathrm{\Omega }}$ by step 1.1, so the first boundary term vanishes and the second equals $-{\int }_{\mathrm{\partial }\mathrm{\Omega }}u(y){\mathrm{\partial }}_{{\nu }_{\mathrm{\Omega }}(y)}{G}_{\mathrm{\Omega }}(x,y)dS(y)={\int }_{\mathrm{\partial }\mathrm{\Omega }}u(y)P_{\mathrm{\Omega }}(x,y)dS(y)$ by [F5]. On $S(x,\epsilon )$ the outward normal is $\sigma$ by step 1.1, so the two boundary terms are ${\int }_{S(x,\epsilon )}{G}_{\mathrm{\Omega }}(x,y){\mathrm{\partial }}_{\sigma }u(y)dS(y)$ and $-{\int }_{S(x,\epsilon )}u(y){\mathrm{\partial }}_{\sigma }{G}_{\mathrm{\Omega }}(x,\cdot )(y)dS(y)$.

Limits on the small sphere. First term: by steps 1.1 and 1.2, $\mid G_{\mathrm{\Omega }}(x,y)\mid \le \mathrm{\mid }\mathrm{\Phi }(y-x)\mathrm{\mid }+\mathrm{\parallel }{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}$ and $\mathrm{\mid }{\mathrm{\partial }}_{\sigma }u(y)\mathrm{\mid }\le \mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}$ on $S(x,\epsilon )$, so the triangle inequality for integrals [F14] and the surface measure ${\omega }_{n-1}{\epsilon }^{n-1}$ of [F9] bound this term by $\mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}(\frac{{\epsilon }^{2-n}}{(n-2){\omega }_{n-1}}+\mathrm{\parallel }{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}){\omega }_{n-1}{\epsilon }^{n-1}=\mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}(\frac{\epsilon }{n-2}+\mathrm{\parallel }{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}{\omega }_{n-1}{\epsilon }^{n-1})$ for $n\ge 3$, and by $\mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}(\frac{\mathrm{\mid }\log \epsilon \mathrm{\mid }}{2\pi }+\mathrm{\parallel }{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})})2\pi \epsilon$ for $n=2$; both tend to $0$ as $\epsilon \downarrow 0$, using $\epsilon \mathrm{\mid }\log \epsilon \mathrm{\mid }\to 0$ from [F13]. Second term: by steps 1.1 and 1.2, ${\mathrm{\partial }}_{\sigma }{G}_{\mathrm{\Omega }}(x,\cdot )(y)=D\mathrm{\Phi }(y-x)\cdot \sigma (y)-D{H}_x(y)\cdot \sigma (y)=\frac{1}{{\omega }_{n-1}{\epsilon }^{n-1}}-D{H}_x(y)\cdot \sigma (y)$ on $S(x,\epsilon )$, so ${\int }_{S(x,\epsilon )}{\mathrm{\partial }}_{\sigma }{G}_{\mathrm{\Omega }}(x,\cdot )dS=1+R_{\epsilon }$ with $\mathrm{\mid }{R}_{\epsilon }\mathrm{\mid }\le b_x{\omega }_{n-1}{\epsilon }^{n-1}\to 0$; hence $-u(x){\int }_{S(x,\epsilon )}{\mathrm{\partial }}_{\sigma }{G}_{\mathrm{\Omega }}(x,\cdot )dS\to -u(x)$. For the remaining piece, the mean value theorem [F15] applied along segments from $x$ to $y\in S(x,\epsilon )$ gives $\mathrm{\mid }u(y)-u(x)\mathrm{\mid }\le \mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}\epsilon$, so $\mid {\int }_{S(x,\epsilon )}(u(y)-u(x)){\mathrm{\partial }}_{\sigma }{G}_{\mathrm{\Omega }}(x,\cdot )(y)dS(y)\mid \le \mathrm{\parallel }Du{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}\epsilon (\frac{1}{{\omega }_{n-1}{\epsilon }^{n-1}}+b_x){\omega }_{n-1}{\epsilon }^{n-1}⟶0.$

The volume term converges. For each $y\in \mathrm{\Omega }\setminus \{x\}$ one has $y\in D_{\epsilon }$ as soon as $0<\epsilon <\mathrm{\mid }y-x\mathrm{\mid }$, so ${\mathbf{1}}_{D_{\epsilon }}(y)G_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)\to G_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)$ pointwise on $\mathrm{\Omega }\setminus \{x\}$, a set of full measure by [F16]; the dominating function $M(y):=(\mathrm{\mid }\mathrm{\Phi }(y-x)\mathrm{\mid }+\mathrm{\parallel }{H}_x{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})})\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})}$ is integrable by [F13, F14] and bounds every term. Step 1.3 supplies measurability, so [F16] gives ${\int }_{D_{\epsilon }}{G}_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)dy\to {\int }_{\mathrm{\Omega }}{G}_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)dy$; the integral on the right is absolutely finite by step 1.3 and [F14].

Passing to the limit. Take a sequence ${\epsilon }_k\downarrow 0$ with $0<{\epsilon }_k<{\epsilon }_0$ and use the identity of step 2.2 for each $k$. Step 2.4 gives the limit of the left side, the boundary integral over $\mathrm{\partial }\mathrm{\Omega }$ is independent of $k$, the first sphere term tends to $0$ and the second to $-u(x)$ by step 2.3. Therefore ${\int }_{\mathrm{\Omega }}{G}_{\mathrm{\Omega }}(x,y)\mathrm{\Delta }u(y)dy={\int }_{\mathrm{\partial }\mathrm{\Omega }}u(y)P_{\mathrm{\Omega }}(x,y)dS(y)-u(x)\mathrm{.}$ Rearranging and using $\mathrm{\Delta }u=-(-\mathrm{\Delta }u)$ yields the displayed representation formula, with both integrals absolutely finite by step 1.3. Since $x\in \mathrm{\Omega }$ was arbitrary, the formula holds for every $x\in \mathrm{\Omega }$.

Substituting the constant function $u\equiv 1$, which lies in $C^2(\bar{\mathrm{\Omega }})$ with $\mathrm{\Delta }u=0$ by [F18], the formula of step 3.1 collapses to $1={\int }_{\mathrm{\partial }\mathrm{\Omega }}{P}_{\mathrm{\Omega }}(x,y)dS(y)$, which is the asserted normalization; combined with step 2.1 this proves $P_{\mathrm{\Omega }}\ge 0$ and unit boundary mass. The theorem makes no existence claim for Green functions, only uses the one supplied; the dimension $n=1$ is excluded by [F3] and no case of [F1] is left out. Countable Choice is inherited from the Green, Poisson-kernel, surface and Green-identity conventions cited in [F1], [F5], [F6] and [F9]; the excision, chain rule, mean value and limiting arguments above invoke no further choice. Complex-valued $u$ are handled by applying the real result to $\mathrm{R}\mathrm{e}u$ and $\mathrm{I}\mathrm{m}u$; the statement is formulated for real $u$.