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  "body": "\n## Summary\n\nA viscosity solution is a notion of weak solution for fully nonlinear first-order and second-order equations.\nSuch equations, for example Hamilton–Jacobi equations, need not have classical solutions.\nIt replaces derivatives of the solution, which may not exist, by derivatives of smooth test functions that touch the solution from above or from below.\nThe notion was introduced by Crandall and Lions, and for many equations it gives existence, uniqueness and stability under uniform limits.\nIts name comes from the vanishing viscosity method.\nIn this method a small multiple of the Laplacian is added to the equation and then sent to zero.\n\n## Explanations\n\n### L1 human\n\nThink of a function with a sharp corner, like a roof.\nIt has no derivative at the ridge, so it cannot satisfy a differential equation there in the usual sense.\nThe viscosity approach tests the function with smooth curves that touch it from above or from below at a point.\nIt asks the equation to hold as an inequality for the slopes of those curves.\nFor the equation $\\abs{u'} = 1$ on $(-1, 1)$ with $u = 0$ at both ends, this singles out the roof $u(x) = 1 - \\abs{x}$ and rejects the valley $\\abs{x} - 1$.\n",
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