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  "body": "\n## Definition\n\nLet $a \u003c b$, and consider a second-order ordinary differential equation for $y$ on $[a, b]$.\nA *Dirichlet boundary condition* is a pair of conditions\n$$y(a) = \\alpha, \\qquad y(b) = \\beta,$$\nwith given real numbers $\\alpha, \\beta$.\nMore generally, let $\\Omega \\subseteq \\R^d$ be a bounded domain, that is, a connected open set, with [[boundary|05D5EGMFYV]] $\\partial \\Omega$, and let $g \\colon \\partial \\Omega \\to \\R$ be a given function.\nFor an equation for $u$ on $\\Omega$, the Dirichlet boundary condition is $u(x) = g(x)$ for all $x \\in \\partial \\Omega$.\nThe condition is *homogeneous* if $\\alpha = \\beta = 0$, respectively $g = 0$.\n\n## Remarks\n\nA differential equation together with a Dirichlet condition is a Dirichlet problem, a kind of [[boundary value problem|TM8VQJWGHZ]].\n\nExample: $y'' = 0$ on $[0, 1]$ with $y(0) = 1$ and $y(1) = 3$.\nThe function $y = 1 + 2x$ has $y'' = 0$, $y(0) = 1$ and $y(1) = 3$, so it solves this Dirichlet problem.\n\nNon-example: the conditions $y(0) = 1$ and $y'(1) = 3$ do not form a Dirichlet condition, because the second one prescribes a derivative and not a value.\n\nFor the Laplace equation $\\Delta u = 0$ on a bounded domain with continuous $g$, a solution that is continuous up to the boundary is unique when it exists, by the maximum principle.\n",
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