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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set with smooth boundary, with $n \\ge 2$, and let $m \\ge 1$.\nLet $A$ be a linear differential operator of order $2m$ with smooth coefficients on $\\overline{\\Omega}$.\nLet $B_1, \\ldots, B_m$ be linear differential operators with smooth coefficients on $\\partial \\Omega$, of orders $m_1, \\ldots, m_m$ less than $2m$.\nConsider the [[boundary value problem|TM8VQJWGHZ]]\n$$A u = f \\ \\text{in } \\Omega, \\qquad B_j u = g_j \\ \\text{on } \\partial \\Omega \\quad (j = 1, \\ldots, m).$$\nIt is *elliptic* if:\n\n1. $A$ is properly [[elliptic|XY8GEQSWPR]] on $\\overline{\\Omega}$;\n2. $B_1, \\ldots, B_m$ satisfy the [[Lopatinskii–Shapiro condition|48MEJEMEXM]] with respect to $A$ at every point of $\\partial \\Omega$.\n\n## Remarks\n\nExample: the Dirichlet problem $-\\Delta u = f$ in $\\Omega$, $u = g$ on $\\partial \\Omega$ has $m = 1$, the Laplacian is properly elliptic, and the Dirichlet condition satisfies the Lopatinskii–Shapiro condition.\nNon-example: $-\\Delta u = f$ with the two conditions $u = 0$ and $\\partial_\\nu u = 0$ on $\\partial \\Omega$ is not elliptic, since an operator of order $2$ takes exactly one boundary condition.\nAgmon, Douglis and Nirenberg proved an a priori estimate for elliptic problems.\nFor each integer $s \\ge 0$ there is $C \u003e 0$ with\n$$\\norm{u}_{H^{2m+s}(\\Omega)} \\le C \\Big( \\norm{Au}_{H^s(\\Omega)} + \\sum_{j=1}^m \\norm{B_j u}_{H^{2m+s-m_j-1/2}(\\partial \\Omega)} + \\norm{u}_{L^2(\\Omega)} \\Big),$$\nwhere $H^k$ denotes Sobolev spaces.\nFor a properly elliptic operator of order $2$, the Dirichlet condition always satisfies the Lopatinskii–Shapiro condition.\nIndeed, the bounded solutions of the ordinary differential equation form a line spanned by a function that does not vanish at $0$.\n",
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