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  "body": "\n## Definition\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 $f \\colon \\partial\\Omega \\to \\R$ be continuous, and consider the interior [[Dirichlet problem|8D56GQ4AG8]] with data $f$.\nFor a continuous density $\\varphi$ on $\\partial\\Omega$, let $D\\varphi$ be its [[double layer potential|EG8Z3FWV9A]], and let $K\\varphi(x) = \\int_{\\partial\\Omega} \\frac{\\partial \\Phi(x - y)}{\\partial \\nu(y)} \\varphi(y)\\, d\\sigma(y)$ for $x \\in \\partial\\Omega$, as in the [[jump relations|J58YA8QNNX]].\nHere $\\Phi$ is the fundamental solution of the Laplacian with $-\\Delta\\Phi = \\delta_0$, $\\sigma$ is surface measure on $\\partial\\Omega$ and $\\nu$ is the outward unit normal.\nThe *boundary integral method* (double layer method) for this problem consists of the following steps.\n\n1. Seek the solution in the form $u = D\\varphi$ on $\\Omega$, with $\\varphi$ unknown.\n2. Solve the boundary integral equation\n$$-\\tfrac12 \\varphi(x) + K\\varphi(x) = f(x), \\qquad x \\in \\partial\\Omega,$$\nfor a continuous $\\varphi$.\n3. Take $u = D\\varphi$ on $\\Omega$ and $u = f$ on $\\partial\\Omega$.\n\n## Remarks\n\nBy the jump relations, the interior boundary values of $D\\varphi$ are $K\\varphi - \\tfrac12\\varphi$, so step 2 is exactly the condition that $u$ is continuous on $\\overline{\\Omega}$ with $u = f$ on $\\partial\\Omega$.\nSince $\\partial\\Omega$ is $C^2$, the kernel of $K$ is weakly singular and $K$ is a [[compact operator|M2NEP1K5AE]] on $C(\\partial\\Omega)$.\nSo the equation is a Fredholm equation of the second kind.\nBy the [[Fredholm alternative|4HH8C9V8JQ]], it is solvable for every $f$ once the homogeneous equation has only the solution $\\varphi = 0$.\nThis holds when $\\R^n \\setminus \\overline{\\Omega}$ is connected.\n\nExample: let $\\Omega$ be the unit ball in $\\R^3$ and $f \\equiv 1$.\nGauss's integral gives $K1 = -\\tfrac12$ on the sphere, so $\\varphi \\equiv -1$ solves $-\\tfrac12\\varphi + K\\varphi = \\tfrac12 + \\tfrac12 = 1$.\nThen $u = D(-1) = 1$ in the ball, the correct solution.\n\nNon-example: let $\\Omega$ be the shell $\\set{x \\in \\R^3 : 1 \u003c \\abs{x} \u003c 2}$, whose complement has the bounded component $\\set{\\abs{x} \\le 1}$.\nLet $\\varphi = 1$ on the inner sphere and $\\varphi = 0$ on the outer sphere.\nThe outward normal of $\\Omega$ on the inner sphere points toward the origin, so $D\\varphi = 1$ in the inner ball and $D\\varphi = 0$ outside it, in particular on $\\Omega$.\nThus $\\varphi \\ne 0$ solves the homogeneous equation, and the equation of step 2 is not uniquely solvable.\n",
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