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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^n$, $n \\ge 2$, be a bounded connected open set with smooth boundary $\\partial\\Omega$ and outward unit normal $\\nu$.\nLet $H^{1/2}(\\partial\\Omega)$ and $H^{-1/2}(\\partial\\Omega)$ be the boundary [[Sobolev spaces|M4CEZ5K9HP]] of order $\\tfrac12$ and $-\\tfrac12$.\nFor $f \\in H^{1/2}(\\partial\\Omega)$, let $u_f \\in H^1(\\Omega)$ be the weak solution of the [[Dirichlet problem|8D56GQ4AG8]]\n$$\\Delta u = 0 \\text{ in } \\Omega, \\qquad u = f \\text{ on } \\partial\\Omega.$$\nThe boundary condition is taken in the sense of the [[trace operator|Z2T5R88QDB]].\nThe *Dirichlet-to-Neumann operator* is the map\n$$\\Lambda \\colon H^{1/2}(\\partial\\Omega) \\to H^{-1/2}(\\partial\\Omega), \\qquad \\Lambda f = \\partial_\\nu u_f|_{\\partial\\Omega}.$$\nMore generally, let $\\gamma$ be measurable on $\\Omega$ with $c \\le \\gamma \\le c^{-1}$ for some $c \u003e 0$.\nThe operator $\\Lambda_\\gamma$ uses the equation $\\nabla \\cdot (\\gamma \\nabla u) = 0$ and the weakly defined [[Neumann data|GS3KT2H645]] $\\gamma \\, \\partial_\\nu u|_{\\partial\\Omega}$.\n\n## Remarks\n\nExample: let $\\Omega$ be the unit disk and $f(\\theta) = e^{ik\\theta}$ with $k \\in \\Z$.\nThen $u_f = r^{\\abs{k}} e^{ik\\theta}$ is harmonic, and $\\partial_r u_f = \\abs{k}$ at $r = 1$, so $\\Lambda f = \\abs{k} f$.\nThus on the circle $\\Lambda = (-\\partial_\\theta^2)^{1/2}$.\nNon-example: replacing $\\Delta u = 0$ by $\\Delta u + \\lambda u = 0$, where $\\lambda$ is a Dirichlet eigenvalue of $-\\Delta$ on $\\Omega$, does not define a map, because the boundary value problem has no unique solution.\nIn general $\\Lambda$ is self-adjoint and nonnegative on $L^2(\\partial\\Omega)$, with kernel the constants, and it is a first-order elliptic [[pseudo-differential operator|FF27N0BWTP]] with principal symbol $\\abs{\\xi}$.\nIt is also called the Poincaré–Steklov operator, and the weak form is $\\int_{\\partial\\Omega} (\\Lambda f) \\, g = \\int_\\Omega \\nabla u_f \\cdot \\nabla u_g$.\n",
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