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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $U \\subset \\R^n$ be a bounded open set.\n\n1. Let $1 \\le p \u003c \\infty$. There is a constant $C = C(n, p, U)$ such that $\\norm{u}_{L^p(U)} \\le C \\norm{\\nabla u}_{L^p(U)}$ for all $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}_0(U)$.\n2. Assume moreover that $U$ is connected and has $C^1$ boundary, and let $1 \\le p \\le \\infty$. There is a constant $C = C(n, p, U)$ such that\n$$\\norm{u - u_U}_{L^p(U)} \\le C \\norm{\\nabla u}_{L^p(U)} \\quad \\text{for all } u \\in W^{1,p}(U),$$\nwhere $u_U = \\frac{1}{\\abs{U}} \\int_U u \\, dx$ is the mean value of $u$ on $U$.\n\n## Remarks\n\nFor part 1 one can take $U$ inside a slab $\\set{0 \u003c x_1 \u003c d}$.\nFor smooth $u$ with compact support, $u(x) = \\int_0^{x_1} \\partial_1 u \\, dt$, and [[Hölder's inequality|JQ9WXC729D]] gives $\\norm{u}_{L^p} \\le d \\norm{\\partial_1 u}_{L^p}$; density extends this to $W^{1,p}_0(U)$.\nPart 2 is proved by contradiction.\nIf it failed, there would be functions with mean $0$, $L^p$ norm $1$ and gradients tending to $0$.\nThe [[Rellich–Kondrachov theorem|YM3JZ34PME]] would give a limit with zero gradient, hence constant on the connected set $U$.\nThis limit has mean $0$, so it is $0$, which contradicts the norm $1$.\n\nThe hypotheses are needed.\nIn part 2 one must subtract the mean, since constant functions have zero gradient.\nConnectedness is needed: on two disjoint intervals, a function equal to $1$ on one and $-1$ on the other has mean $0$, zero gradient and nonzero norm.\n\n## Explanations\n\n### L1 human\n\nA function that vanishes on the boundary of a bounded region cannot be large inside unless its slope is large somewhere.\nThe Poincaré inequality says that the size of the function is bounded by the size of its slope, up to a constant that depends on the region.\nFor functions without boundary values, the same holds after subtracting the average value.\n\n### L2 human\n\nThe inequality shows that $\\norm{\\nabla u}_{L^2}$ is a norm on $H^1_0(U)$ equivalent to the full $H^1$ norm.\nThis is exactly the coercivity needed to apply the Lax–Milgram theorem to the Dirichlet problem for $-\\Delta u = f$.\nThe best constant for $p = 2$ in part 1 is $1 / \\sqrt{\\lambda_1}$, where $\\lambda_1$ is the first Dirichlet eigenvalue of $-\\Delta$ on $U$.\n",
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