The Newtonian-potential convolution in $\mathbb{R}^3$: given $f : \mathbb{R}^3 \to \mathbb{R}$,
phiConvolution f x is $\int_{\mathbb{R}^3} \Phi(x - y) f(y) \, d^3 y$, where $\Phi$ is the
3D fundamental solution of the Laplacian.
Instances For
Leibniz/differentiation-under-the-integral for the Newtonian potential: if $f$ is smooth and compactly supported, then $\Delta(\Phi * f)(x) = \int \Phi(x-y) \Delta f(y) \, d^3y$.
Newtonian-potential inversion: convolution of $\Delta f$ with the fundamental solution $\Phi$ recovers $f$, i.e. $\int \Phi(x-y) \Delta f(y) \, d^3 y = f(x)$ for smooth compactly supported $f$.
Solution of Poisson's equation on $\mathbb{R}^3$: for smooth compactly supported $f$, $u = \Phi * f$ satisfies $\Delta u = f$ pointwise.
Far-field decay of the Newtonian potential: there exist constants $C_1, R_0 > 0$ such that $|(\Phi * f)(x)| \le C_1 / |x|$ whenever $|x| > R_0$, reflecting the $1/r$ decay of $\Phi$ in $\mathbb{R}^3$.
Boundedness of the Newtonian potential away from the origin: the convolution $\Phi * f$ is uniformly bounded on $\{|x| > 1\}$.
Combined decay estimate for the Poisson solution $u = \Phi * f$: there is a constant $C > 0$ such that $|u(x)| \le C / |x|$ for all $|x| > 1$, obtained by combining the far-field and global boundedness estimates.
On the boundary of the (Euclidean) ball $\{y : |y - p| < R\}$, the outward unit normal at $\sigma$ is the radial unit vector $(\sigma - p)/R$.
The surface (Hausdorff) measure on $\partial\Omega$, declared opaque since its precise construction is not used here.
The surface area $\omega_n$ of the unit sphere $S^{n-1} \subset \mathbb{R}^n$, given by $\omega_n = 2 \pi^{n/2} / \Gamma(n/2)$ for $n \ge 1$.
Instances For
The surface area of the unit sphere $S^2 \subset \mathbb{R}^3$ equals $4\pi$.
Green's second identity on a Lipschitz domain $\Omega \subset \mathbb{R}^n$: for $u, v \in C^2(\bar\Omega)$, $$\int_\Omega \bigl(v \Delta u - u \Delta v\bigr)\, dy = \int_{\partial\Omega} \bigl(v \, \nabla_{\hat N} u - u \, \nabla_{\hat N} v\bigr)\, d\sigma.$$
Removing a small closed ball from a Lipschitz domain $\Omega$ centred at an interior point keeps the resulting set $\Omega \setminus \overline{B(x,\varepsilon)}$ a Lipschitz domain.
The map $y \mapsto \Phi(x - y)$ is $C^2$ on $\overline{\Omega \setminus \overline{B(x,\varepsilon)}}$, since the singularity at $y = x$ has been removed.
A $C^2$ function on $\overline\Omega$ remains $C^2$ when restricted to the closure of the punctured domain $\Omega \setminus \overline{B(x,\varepsilon)}$.
On the punctured domain $\Omega \setminus \overline{B(x,\varepsilon)}$, since $\Delta_y \Phi(x - y) = 0$, the volume term $u(y)\, \Delta_y \Phi(x-y)$ vanishes and the integrand reduces to $\Phi(x - y) \Delta u(y)$.
Decomposition of a surface integral over $\partial(\Omega \setminus \overline{B(x,\varepsilon)})$ into the contribution from $\partial\Omega$ minus the contribution from the inner sphere $\partial B(x,\varepsilon)$ (with the normal pointing inward toward $x$ contributing the sign flip).
Specialisation of surfaceIntegral_punctured_normalDeriv_decomp to the integrand
$\Phi(x - \sigma) \nabla_{\hat N} u - u(\sigma) \nabla_{\hat N} \Phi$ appearing in the
representation formula.
Green's second identity applied on the punctured domain $\Omega \setminus B(x,\varepsilon)$ with $v(y) = \Phi(x - y)$: the volume integral equals the $\partial\Omega$ surface terms minus the sphere terms at $|y - x| = \varepsilon$.
Integrability of $y \mapsto \Phi(x - y) f(y)$ on $\Omega$: the local singularity of the fundamental solution at $y = x$ is integrable, so the product is integrable on the domain.
As $\varepsilon \downarrow 0$, the punctured-domain integral $\int_{\Omega \setminus B(x,\varepsilon)} \Phi(x-y) \Delta u(y) \, dy$ converges to the full volume integral over $\Omega$.
For $n \ge 3$, on the sphere $|σ - x| = \varepsilon$ the fundamental solution satisfies $\|\Phi(x - \sigma)\| \le C_F / \varepsilon^{n-2}$ for some constant $C_F \ge 0$.
If $u \in C^2(\overline\Omega)$ then the normal derivative $\nabla_{\hat N} u$ is uniformly bounded on $\partial\Omega$ (and in fact globally, by some constant $M$).
Linear upper bound on the surface area of a sphere of radius $\varepsilon$ in $\mathbb{R}^n$: there exists $C_S \ge 0$ with $\mathrm{area}(\partial B(x, \varepsilon)) \le C_S \varepsilon^{n-1}$.
Pointwise-times-area bound for $n \ge 3$: there exists $K$ such that for every $\varepsilon > 0$ one can choose a pointwise bound $B$ for $\Phi(x-\sigma) \nabla_{\hat N}u(\sigma)$ on the sphere with $B \cdot \mathrm{area}(\partial B(x,\varepsilon)) \le K \varepsilon$.
Linear-in-$\varepsilon$ estimate on the sphere integral $\int_{|σ-x|=ε} \Phi(x-\sigma) \nabla_{\hat N} u(\sigma)\, d\sigma$ for $n \ge 3$: it is bounded by $K\varepsilon$ for some constant $K$.
Alias for sphere_integral_Phi_gradU_bound: for $n \ge 3$ the third sphere term
$R_3(\varepsilon)$ in the punctured-domain decomposition is $O(\varepsilon)$.
Refined sphere-integral bound in dimension $n = 2$, where $\Phi(x) \sim \log|x|$: the third sphere term is dominated by $K \varepsilon (|\log \varepsilon| + C)$ for some constants $K$ and $C \ge 0$.
In dimension $n = 2$, the sphere integral $R_3(\varepsilon) = \int_{|σ-x|=ε} \Phi(x-\sigma) \nabla_{\hat N} u(\sigma) \, d\sigma \to 0$ as $\varepsilon \downarrow 0$, using the $\varepsilon |\log \varepsilon|$ bound.
Combined result: in any dimension $n \ge 2$, the sphere integral $\int_{|σ-x|=ε} \Phi(x-\sigma) \nabla_{\hat N} u(\sigma) \, d\sigma$ vanishes as $\varepsilon \downarrow 0$.
Specification of the outward unit normal on a sphere of radius $\varepsilon$ centred at $x$: at $\sigma$ with $|\sigma - x| = \varepsilon$, the normal vector is the radial direction $(\sigma - x)/\varepsilon$.
On the sphere of radius $\varepsilon$ centred at $x$, the Euclidean distance $|x - \sigma|$ equals $\varepsilon$.
Explicit Fréchet derivative of $z \mapsto \Phi(x - z)$ away from the pole, given as a sum involving the radial direction $(\sigma - x)/|x - \sigma|$ and the coordinate projections.
On the sphere $|σ - x| = \varepsilon$, applying the Fréchet derivative of $z \mapsto \Phi(x - z)$ to the radial unit vector $(\sigma - x)/\varepsilon$ yields the constant $1/(\omega_n \varepsilon^{n-1})$.
On the sphere $\partial B(x, \varepsilon)$, the normal derivative of $z \mapsto \Phi(x-z)$ is the constant $1/\mathrm{area}(\partial B(x,\varepsilon))$, regardless of the surrounding domain $\Omega$.
Total flux of the fundamental solution through a sphere equals $1$: integrating the normal derivative $\nabla_{\hat N}\Phi(x - \sigma)$ over $\partial B(x,\varepsilon)$ gives exactly $1$, the source strength.
The normal derivative $\nabla_{\hat N}\Phi(x - \cdot)$ is integrable on the sphere $\partial B(x, \varepsilon)$ with respect to the surface measure.
The product $u(\sigma) \nabla_{\hat N} \Phi(x - \sigma)$ is integrable on the sphere $\partial B(x, \varepsilon)$ with respect to the surface measure.
Splitting the sphere integral $\int_{|σ-x|=ε} u(\sigma) \nabla_{\hat N} \Phi(x-\sigma)$ as $u(x)$ plus an error term $\int (u(\sigma) - u(x)) \nabla_{\hat N} \Phi$, using the normalisation $\int \nabla_{\hat N} \Phi = 1$.
For any $\sigma$ on the sphere $\partial B(x,\varepsilon)$, $|u(\sigma) - u(x)|$ is bounded above by the corresponding indexed supremum over the sphere.
The absolute value of the deviation sphere integral $\int (u(\sigma) - u(x)) \nabla_{\hat N}\Phi$ is bounded by the maximal oscillation $\sup_\sigma |u(\sigma) - u(x)|$ over the sphere.
Continuity at $x$ implies that the supremum of the oscillation $|u(\sigma) - u(x)|$ over the shrinking sphere $\partial B(x,\varepsilon)$ tends to zero as $\varepsilon \downarrow 0$.
The deviation sphere integral $\int (u(\sigma) - u(x)) \nabla_{\hat N}\Phi(x-\sigma)$ tends to $0$ as $\varepsilon \downarrow 0$, obtained by combining the deviation bound with the oscillation $\to 0$ result.
Mean-value-type limit: as $\varepsilon \downarrow 0$, the sphere integral $\int_{|σ-x|=ε} u(\sigma) \nabla_{\hat N}\Phi(x - \sigma)\, d\sigma$ converges to $u(x)$.
Reformulation: the sphere integral $\int u(\sigma) \nabla_{\hat N}\Phi(x-\sigma)$ admits the decomposition $u(x) + \text{error}(\varepsilon)$ where the error vanishes as $\varepsilon \downarrow 0$.
Alias for sphere_integral_u_normalDeriv_Phi_decomp: the fourth sphere term
$R_4(\varepsilon)$ decomposes as $u(x)$ plus a vanishing error.
The limit of the fourth sphere term is $u(x)$, the value of $u$ at the centre — the mean-value property of the fundamental solution.
Representation formula obtained by taking $\varepsilon \downarrow 0$ in Green's identity on the punctured domain: for $u \in C^2(\bar\Omega)$ and $x \in \Omega$, $$u(x) = \int_\Omega \Phi(x-y) \Delta u(y), dy
- \int_{\partial\Omega} \Phi(x-\sigma) \nabla_{\hat N} u(\sigma), d\sigma
- \int_{\partial\Omega} u(\sigma) \nabla_{\hat N} \Phi(x-\sigma), d\sigma.$$
Proposition 2.0.3 in general dimension $n \ge 2$: the representation formula for $u \in C^2(\bar\Omega)$ at an interior point $x \in \Omega$ in terms of the fundamental solution $\Phi$, the volume integral of $\Delta u$, and the boundary terms involving $u$ and $\nabla_{\hat N} u$.
Proposition 2.0.3 specialised to $\mathbb{R}^3$ with the 3D fundamental solution $\Phi_3(x) = -1/(4\pi|x|)$: for $u \in C^2(\mathbb{R}^3)$, $\Omega$ a Lipschitz domain and $x \in \Omega$, $$u(x) = \int_\Omega \Phi_3(x-y) \Delta u(y), dy
- \int_{\partial\Omega} \Phi_3(x-\sigma) \nabla_{\hat N} u(\sigma), d\sigma
- \int_{\partial\Omega} u(\sigma) \nabla_{\hat N} \Phi_3(x-\sigma), d\sigma.$$
Green function for the domain $\Omega \subset \mathbb{R}^n$ (Definition 2.0.2 together with the decomposition in Proposition 2.0.2): a function $G(x, y)$ on $\Omega \times \Omega$ with $\Delta_y G(x, y) = \delta(x)$ and $G(x, \sigma) = 0$ for $\sigma \in \partial \Omega$. It is packaged as the decomposition $G(x, y) = \Phi(x - y) - \phi(x, y)$, where the corrector $\phi(x, \cdot)$ is harmonic in $\Omega$ with boundary data $\Phi(x - \cdot)|_{\partial\Omega}$.
Instances For
Instance: $2 \le 3$, supplied so that FundSolN and related results specialise to the
three-dimensional setting.
Unfolding lemma for surfaceIntegral: definitionally equal to the integral
$\int_{\partial\Omega} f(\sigma) \, d\sigma$.
Regularity-extension axiom: a function that is $C^2$ on the interior $\Omega$ and continuous on the closure $\bar\Omega$ is in fact $C^2$ on $\bar\Omega$.
Textbook-sign variant of the punctured-domain surface decomposition: the surface integral over $\partial(\Omega \setminus \overline{B(x,\varepsilon)})$ equals the $\partial\Omega$ boundary contribution plus (rather than minus) the inner-sphere contribution, reflecting the choice of inward-pointing normal on $\partial B(x,\varepsilon)$.
Green's identity on the punctured domain with the textbook sign convention (inward normal on the inner sphere), giving the version of the identity matching equation (2.0.30) in the text.
Core equation (2.0.30) of the textbook: representation of $u(x)$ in terms of $\Delta u$, the trace of $u$ on $\partial\Omega$, and the normal derivative of $u$, all paired with the fundamental solution $\Phi$, expressed using the textbook sign convention.
Equation (2.0.30) for solutions of the Poisson boundary value problem $\Delta u = f$ in $\Omega$, $u = g$ on $\partial\Omega$: $u(x)$ is represented as $-\int_\Omega \Phi(x-y) f(y), dy
- \int_{\partial\Omega} \Phi(x-\sigma) \nabla_{\hat N} u(\sigma), d\sigma
- \int_{\partial\Omega} g(\sigma) \nabla_{\hat N} \Phi(x-\sigma), d\sigma$.
Duplicate of the regularity-extension axiom under a Lipschitz domain assumption: $C^2$ on $\Omega$ together with continuity on $\bar\Omega$ promotes to $C^2$ on $\bar\Omega$.
Integrability axiom for surface integrals: every function under consideration is integrable on $\partial\Omega$ with respect to the surface measure.
Identity (2.0.33) at the core level: applying Green's second identity to $u$ and the harmonic corrector $\phi(x, \cdot)$ in $\Omega$ produces the relation $0 = \int_\Omega \phi(x, y) \Delta u(y), dy
- \int_{\partial\Omega} \Phi(x-\sigma) \nabla_{\hat N} u(\sigma), d\sigma
- \int_{\partial\Omega} u(\sigma) \nabla_{\hat N} \phi(x, \sigma), d\sigma$.
Equation (2.0.33) for the Poisson boundary value problem: substituting $\Delta u = f$ in $\Omega$ and $u = g$ on $\partial\Omega$ into the corrector identity yields a relation among the source $f$, boundary data $g$, the corrector $\phi$, and the normal derivative of $u$.
Volume-integral form of the decomposition $G = \Phi - \phi$: under integrability, $\int_\Omega f(y) G(x, y)\, dy = \int_\Omega \Phi(x-y) f(y)\, dy - \int_\Omega \phi(x, y) f(y)\, dy$.
Differentiability of the corrector $\phi(x, \cdot)$ at any point $\sigma$.
Normal-derivative decomposition $\nabla_{\hat N} G = \nabla_{\hat N} \Phi - \nabla_{\hat N} \phi$ inherited from $G(x, y) = \Phi(x - y) - \phi(x, y)$.
Surface-integral form of the Green-function decomposition: $\int_{\partial\Omega} g(\sigma) \nabla_{\hat N} G(x, \sigma)\, d\sigma$ splits as the $\Phi$-part minus the corrector $\phi$-part.
Theorem 2.2 (Green-function representation for solutions of the boundary value Poisson equation): given a Green function $G$ for $\Omega$ and a solution $u$ of $\Delta u = f$ in $\Omega$ with $u = g$ on $\partial \Omega$, $$u(x) = -\int_\Omega f(y) G(x, y), dy
- \int_{\partial\Omega} g(\sigma) \nabla_{\hat N} G(x, \sigma), d\sigma.$$