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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set and let $k \\ge 1$ be an integer.\nFor multi-indices $\\alpha, \\beta$ with $\\abs{\\alpha}, \\abs{\\beta} \\le k$, let $A_{\\alpha\\beta} \\colon \\Omega \\to \\C$ be bounded measurable functions.\nAssume that $A_{\\alpha\\beta}$ is uniformly continuous on $\\Omega$ when $\\abs{\\alpha} = \\abs{\\beta} = k$.\nAssume the uniform ellipticity condition: there is $\\theta \u003e 0$ with\n$$\\operatorname{Re} \\sum_{\\abs{\\alpha} = \\abs{\\beta} = k} A_{\\alpha\\beta}(x)\\xi^\\alpha\\xi^\\beta \\ge \\theta\\abs{\\xi}^{2k} \\quad \\text{for all } x \\in \\Omega,\\ \\xi \\in \\R^n.$$\nFor $u, v$ in the Sobolev space $H^k_0(\\Omega)$, let\n$$B(u, v) = \\sum_{\\abs{\\alpha}, \\abs{\\beta} \\le k} \\int_\\Omega A_{\\alpha\\beta}(x)\\, \\partial^\\beta u(x)\\, \\overline{\\partial^\\alpha v(x)} \\, dx.$$\nThen there are constants $c \u003e 0$ and $G \\ge 0$ such that\n$$\\operatorname{Re} B(u, u) \\ge c\\norm{u}_{H^k(\\Omega)}^2 - G\\norm{u}_{L^2(\\Omega)}^2 \\quad \\text{for all } u \\in H^k_0(\\Omega).$$\n\n## Remarks\n\nThe form $B$ is the sesquilinear form of the [[elliptic operator|XY8GEQSWPR]] $Lu = \\sum (-1)^{\\abs{\\alpha}} \\partial^\\alpha(A_{\\alpha\\beta}\\partial^\\beta u)$ in divergence form.\nFor constant coefficients and only top-order terms, the inequality follows from Plancherel's theorem: the Fourier transform turns $\\operatorname{Re} B(u, u)$ into $\\int \\operatorname{Re}\\sigma(\\xi)\\abs{\\hat u(\\xi)}^2 \\, d\\xi \\ge \\theta \\int \\abs{\\xi}^{2k}\\abs{\\hat u(\\xi)}^2 \\, d\\xi$.\nFor variable coefficients, one freezes the coefficients on small pieces of $\\Omega$, using uniform continuity, and glues with a partition of unity.\nLower-order terms are absorbed with the interpolation inequality $\\norm{u}_{H^j} \\le \\varepsilon\\norm{u}_{H^k} + C_\\varepsilon\\norm{u}_{L^2}$ for $j \u003c k$.\n\nExample: for $k = 1$ and $B(u, u) = \\int_\\Omega \\abs{\\nabla u}^2 \\, dx$, the form of $-\\Delta$, one has $B(u, u) = \\norm{u}_{H^1}^2 - \\norm{u}_{L^2}^2$, so $c = 1$ and $G = 1$ work.\n\nEllipticity cannot be dropped.\nIn $\\R^2$, the form $\\int_\\Omega \\abs{\\partial_1 u}^2 \\, dx$ is not elliptic, and functions that oscillate fast in the $x_2$ direction make $\\norm{u}_{H^1}$ large while the form and $\\norm{u}_{L^2}$ stay bounded.\n\n## Explanations\n\n### L1 human\n\nAn elliptic operator controls all derivatives of order $k$, not only some of them.\nGårding's inequality makes this precise in terms of energy: the energy $B(u, u)$ bounds the full Sobolev norm of $u$, up to a harmless multiple of $\\norm{u}_{L^2}^2$.\nIt is the basic estimate behind the existence of weak solutions to elliptic boundary problems.\n\n### L2 human\n\nAdding $G$ to the operator, the form $B(u, v) + G\\ip{u}{v}$ becomes coercive on $H^k_0(\\Omega)$.\nThe Lax–Milgram theorem then gives a unique weak solution of $(L + \\lambda)u = f$ for every $f$ and every $\\lambda \\ge G$.\nCombined with the compact embedding of $H^k_0(\\Omega)$ in $L^2(\\Omega)$, this yields the Fredholm alternative and a discrete spectrum for the Dirichlet problem.\nThe constant $G$ cannot in general be taken to be $0$, because $L$ may have eigenvalues with negative real part.\n",
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