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  "body": "\n## Definition\n\nLet $\\Omega$ be a connected open set in $\\R^n$.\nConsider a linear second-order partial differential equation in $\\Omega$ with real coefficients.\nIts type at each point is given by the [[classification of second-order PDEs|5FC6BMM1NA]].\nThe equation is of *mixed type* in $\\Omega$ if both conditions hold:\n\n1. it is [[elliptic|B4TTNHSTRX]] on some nonempty open subset of $\\Omega$;\n2. it is [[hyperbolic|92C3S03AQZ]] on some nonempty open subset of $\\Omega$.\n\n## Remarks\n\nExample: the Tricomi equation $u_{yy} + y \\, u_{xx} = 0$ on $\\Omega = \\R^2$, written as $A u_{xx} + 2B u_{xy} + C u_{yy}$, has $A = y$, $B = 0$ and $C = 1$, so $AC - B^2 = y$.\nIt is elliptic on the open set $y \u003e 0$ and hyperbolic on the open set $y \u003c 0$, so it is of mixed type.\nNon-example: the equation $u_{xx} + (1 + y^2) u_{yy} = 0$ has $AC - B^2 = 1 + y^2 \u003e 0$ everywhere, so it is elliptic on all of $\\R^2$ and not of mixed type.\nIf the coefficients are continuous, the elliptic and hyperbolic regions are open, and in a connected $\\Omega$ they are separated by a closed set where the coefficient matrix is degenerate.\n\n## Explanations\n\n### L1 human\n\nAn equation is of mixed type when it is elliptic in some region and hyperbolic in another region of the same domain.\n",
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