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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded open set whose [[boundary|05D5EGMFYV]] $\\partial\\Omega$ is a $C^2$ hypersurface.\nLet $\\sigma$ be surface measure on $\\partial\\Omega$ and $\\nu$ the outward unit normal.\nLet $\\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] with $-\\Delta\\Phi = \\delta_0$, and let $\\varphi \\colon \\partial\\Omega \\to \\R$ be continuous.\nLet $S\\varphi$ be the [[single layer potential|9T6ZEV4125]] and $D\\varphi$ the [[double layer potential|EG8Z3FWV9A]] with density $\\varphi$.\nFor $x \\in \\partial\\Omega$ put\n$$K\\varphi(x) = \\int_{\\partial\\Omega} \\frac{\\partial \\Phi(x - y)}{\\partial \\nu(y)}\\, \\varphi(y)\\, d\\sigma(y), \\qquad K'\\varphi(x) = \\int_{\\partial\\Omega} \\frac{\\partial \\Phi(x - y)}{\\partial \\nu(x)}\\, \\varphi(y)\\, d\\sigma(y);$$\nboth integrals converge absolutely.\nThen:\n\n1. $S\\varphi$ is continuous on $\\R^n$.\n2. The restriction of $D\\varphi$ to $\\Omega$ extends continuously to $\\overline{\\Omega}$, with boundary values $D_-\\varphi = K\\varphi - \\tfrac12 \\varphi$ on $\\partial\\Omega$.\nThe restriction of $D\\varphi$ to $\\R^n \\setminus \\overline{\\Omega}$ extends continuously to $\\R^n \\setminus \\Omega$, with boundary values $D_+\\varphi = K\\varphi + \\tfrac12 \\varphi$ on $\\partial\\Omega$.\n3. For $x \\in \\partial\\Omega$ the limits\n$$\\partial_\\nu S_\\pm\\varphi(x) = \\lim_{h \\to 0^+} \\nu(x) \\cdot \\nabla S\\varphi\\big(x \\pm h\\,\\nu(x)\\big)$$\nexist uniformly on $\\partial\\Omega$, and $\\partial_\\nu S_\\pm \\varphi = K'\\varphi \\mp \\tfrac12 \\varphi$.\n\n## Remarks\n\nProof idea: for $\\varphi \\equiv 1$, the divergence theorem gives Gauss's integral, $D1 = -1$ in $\\Omega$, $D1 = 0$ outside $\\overline{\\Omega}$ and $K1 = -\\tfrac12$ on $\\partial\\Omega$.\nFor general $\\varphi$, write $D\\varphi(x) = D\\big(\\varphi - \\varphi(x_0)\\big)(x) + \\varphi(x_0)\\, D1(x)$ for $x$ near $x_0 \\in \\partial\\Omega$.\nThe first term is continuous across $\\partial\\Omega$ at $x_0$, because the $C^2$ boundary makes the kernel weakly singular and the density vanishes at $x_0$.\nSo the whole jump comes from $\\varphi(x_0)\\, D1$, and the normal derivative of $S\\varphi$ is handled in the same way.\n\nThe jumps can be read off: $D_+\\varphi - D_-\\varphi = \\varphi$ and $\\partial_\\nu S_-\\varphi - \\partial_\\nu S_+\\varphi = \\varphi$.\nExample: for the unit ball in $\\R^3$ and $\\varphi \\equiv 1$, $S1 = 1$ inside and $S1 = 1/\\abs{x}$ outside.\nSo $\\partial_\\nu S_-1 = 0$ and $\\partial_\\nu S_+1 = -1$ on the sphere, and both equal $K'1 \\mp \\tfrac12$ with $K'1 = -\\tfrac12$.\n\n## Explanations\n\n### L1 human\n\nA layer of charge on a surface gives a potential that is continuous across the surface, but whose slope jumps by the charge density.\nA layer of dipoles gives a potential that itself jumps across the surface, by the dipole density.\nThe jump is split evenly: half of it on each side of the value on the surface.\n\n### L2 human\n\nThese formulas are the bridge between the Laplace equation in a domain and integral equations on its boundary.\nIf one looks for the solution of the [[Dirichlet problem|8D56GQ4AG8]] as $D\\varphi$, part 2 turns the boundary condition into the equation $K\\varphi - \\tfrac12\\varphi = f$; this is the boundary integral method.\nFor the Neumann problem one uses $S\\varphi$ and part 3 in the same way.\nBecause $K$ and $K'$ are compact operators on $C(\\partial\\Omega)$ and adjoint to each other for the pairing of $L^2(\\partial\\Omega)$, the [[Fredholm alternative|4HH8C9V8JQ]] then governs solvability.\n",
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