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  "body": "\n## Summary\n\nThe Lopatinskii–Shapiro condition, also called the complementing condition, is an algebraic condition on boundary operators.\nIt is what a boundary value problem for an [[elliptic operator|XY8GEQSWPR]] needs to be well behaved.\nIt is checked at each boundary point by freezing coefficients and looking at an ordinary differential equation in the normal direction.\nWhen it holds, the problem has a priori estimates in Sobolev spaces and is a Fredholm problem.\nIt plays for boundary conditions the role that ellipticity plays for the operator itself.\n\n## Explanations\n\n### L1 human\n\nAn elliptic equation inside a domain needs the right number and the right kind of conditions on the boundary.\nTo test them, zoom in on one boundary point until the boundary looks flat and the coefficients look constant.\nThen look for solutions that oscillate along the boundary and decay into the interior.\nThe condition says that the boundary conditions kill all such solutions.\nFor the Laplacian, prescribing the values $u$ on the boundary passes this test, and so does prescribing the normal derivative.\n",
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