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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be a nonempty bounded open set and let $H^1_0(\\Omega)$ be the [[Sobolev space|M4CEZ5K9HP]] of functions with zero boundary values.\nLet $\\Delta$ be the [[Laplacian|0E0B6RA8K0]].\nA real number $\\lambda$ is a *Dirichlet eigenvalue* of $\\Omega$ if some $u \\in H^1_0(\\Omega)$ with $u \\ne 0$ is a [[weak solution|SJZX81A783]] of\n$$-\\Delta u = \\lambda u \\quad \\text{in } \\Omega.$$\nThat is, $\\int_\\Omega \\nabla u \\cdot \\nabla v \\, dx = \\lambda \\int_\\Omega u v \\, dx$ for every $v \\in H^1_0(\\Omega)$.\nSuch a $u$ is a *Dirichlet eigenfunction* for $\\lambda$.\n\n## Remarks\n\nThe Dirichlet eigenvalues form a sequence $0 \u003c \\lambda_1(\\Omega) \u003c \\lambda_2(\\Omega) \\le \\lambda_3(\\Omega) \\le \\dots \\to \\infty$, listed with multiplicity, when $\\Omega$ is connected.\nThey are the [[eigenvalues|0BQAA2BK3Q]] of the Dirichlet Laplacian, a self-adjoint operator with compact inverse.\nThe first one is the minimum of the Rayleigh quotient: $\\lambda_1(\\Omega) = \\min \\set{ \\int_\\Omega \\abs{\\nabla u}^2 \\, dx / \\int_\\Omega u^2 \\, dx : u \\in H^1_0(\\Omega), u \\ne 0 }$.\n\nExample: for $\\Omega = (0, \\pi)$, the function $u(x) = \\sin(kx)$ with $k \\ge 1$ an integer vanishes at $0$ and $\\pi$ and satisfies $-u'' = k^2 u$.\nSo $k^2$ is a Dirichlet eigenvalue, and these are all of them.\n\nNon-example: on $(0, \\pi)$, the number $0$ is not a Dirichlet eigenvalue.\nThe solutions of $-u'' = 0$ are $u = a + bx$, and the boundary conditions force $a = b = 0$.\nWith the Neumann condition $u' = 0$ instead, the constant function shows that $0$ is a Neumann eigenvalue.\n",
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