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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be open, let $m \\ge 1$, and let\n$$P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\, \\partial^\\alpha$$\nbe a linear differential operator on $U$ with coefficients $a_\\alpha \\in C^\\infty(U)$, where $\\alpha$ runs over multi-indices.\nIts formal transpose is the operator $P^t \\varphi = \\sum_{\\abs{\\alpha} \\le m} (-1)^{\\abs{\\alpha}} \\partial^\\alpha (a_\\alpha \\varphi)$.\nLet $f$ be a locally integrable function on $U$.\nA locally integrable function $u$ on $U$ is a *weak solution* of $P u = f$ if\n$$\\int_U u \\, P^t \\varphi \\, dx = \\int_U f \\, \\varphi \\, dx \\quad \\text{for all } \\varphi \\in C_c^\\infty(U),$$\nwhere $C_c^\\infty(U)$ is the space of test functions on $U$.\n\n## Remarks\n\nThe identity is what integration by parts gives when $u$ is of class $C^m$ and $P u = f$ holds pointwise.\nSo every classical solution is a weak solution.\nConversely, a weak solution of class $C^m$ satisfies $P u = f$ almost everywhere, and everywhere if $f$ is continuous.\nIf $u$ has [[weak derivatives|C3EC148VZD]] up to order $m$, the condition says that $\\sum a_\\alpha D^\\alpha u = f$ almost everywhere.\n\nExample: on $\\R^2$ with variables $(x, t)$, let $u(x, t) = 1$ for $x \u003e t$ and $u(x, t) = 0$ for $x \\le t$, and let $P = \\partial_t + \\partial_x$, so $P^t \\varphi = -(\\varphi_t + \\varphi_x)$.\nPut $\\psi(y, t) = \\varphi(y + t, t)$; then $\\partial_t \\psi = \\varphi_x + \\varphi_t$ at the point $(y + t, t)$.\nHence $\\int\\!\\!\\int u \\, (\\varphi_t + \\varphi_x) \\, dx \\, dt = \\int_0^\\infty \\int_{\\R} \\partial_t \\psi(y, t) \\, dt \\, dy = 0$, because $\\psi(y, \\cdot)$ has compact support.\nSo $u$ is a weak solution of $u_t + u_x = 0$, though it is discontinuous.\n\nNon-example: the step $w(x, t) = 1$ for $x \u003e 0$ and $0$ for $x \\le 0$ is not a weak solution of the same equation.\nHere $\\int\\!\\!\\int w \\, (\\varphi_t + \\varphi_x) \\, dx \\, dt = -\\int_{\\R} \\varphi(0, t) \\, dt$, which is not zero for a test function with $\\varphi(0, t) \u003e 0$ near some $t$.\n\nFor second-order equations in divergence form, such as $-\\sum_{i,j} \\partial_j (a_{ij} \\partial_i u) = f$ with bounded coefficients, one often asks instead that $u$ lie in a Sobolev space $H^1_0(U)$ and that $\\int_U \\sum_{i,j} a_{ij} \\, \\partial_i u \\, \\partial_j v \\, dx = \\int_U f v \\, dx$ for all $v \\in H^1_0(U)$.\nThis is the variational form of the definition.\n",
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