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  "body": "\n## Summary\n\nA functional on a Banach space is weakly lower semicontinuous if it cannot jump up in a weak limit.\nThat is, $J(u) \\le \\liminf_k J(u_k)$ whenever $u_k \\to u$ weakly.\nIt is [[semicontinuity|RN4G8SGP6P]] with respect to the [[weak topology|P6KHVSTPZV]], usually in its sequential form.\nTogether with coercivity it is the main hypothesis of the [[direct method|TE379HBTY0]].\nFor integral functionals it is closely tied to convexity of the integrand in the gradient variable.\n\n## Explanations\n\n### L1 human\n\nWeak convergence allows a sequence to oscillate faster and faster while its average settles down.\nThe functions $\\sin(kx)$ on $[0, 2\\pi]$ converge weakly to $0$, even though their size does not shrink.\nA weakly lower semicontinuous energy may lose value in such a limit, but it cannot gain value.\nThe norm itself is a typical example: the limit $0$ has norm $0$, below the norms of $\\sin(kx)$.\n",
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  "name": "Weak lower semicontinuity",
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