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  "body": "\n## Summary\n\nA Dirichlet eigenvalue of a bounded domain $\\Omega$ is a number $\\lambda$ such that $-\\Delta u = \\lambda u$ has a nonzero solution vanishing on the boundary.\nThese numbers form an increasing sequence of positive numbers that tends to infinity.\nThey are the squared frequencies of a vibrating membrane fixed along its edge.\nThey also control the decay of heat in $\\Omega$ with zero boundary temperature.\nHow they depend on the shape of $\\Omega$ is the subject of spectral geometry, with results such as the [[Faber–Krahn inequality|E7TY2TPXX7]].\n\n## Explanations\n\n### L1 human\n\nFix the edge of a drum and strike it.\nThe drum can vibrate only in certain patterns, and each pattern has its own pitch.\nThe Dirichlet eigenvalues are the squares of these pitches, up to a physical constant.\nFor the interval $(0, \\pi)$, the patterns are $\\sin(kx)$ and the eigenvalues are $1, 4, 9, 16, \\dots$.\n",
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