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  "body": "\n## Counterexample\n\nConsider the [[integral functional|JRQZ803Q5Y]]\n$$J(u) = \\int_{-1}^{1} x^2 \\, u'(x)^2 \\, dx$$\non the set $\\mathcal{A}$ of functions $u \\in C^1([-1, 1])$ with $u(-1) = -1$ and $u(1) = 1$.\nThen $\\inf_{\\mathcal{A}} J = 0$, but no $u \\in \\mathcal{A}$ has $J(u) = 0$, so $J$ has no minimiser on $\\mathcal{A}$.\n\nCheck that the infimum is $0$.\nFor $\\varepsilon \u003e 0$, put $u_\\varepsilon(x) = \\arctan(x / \\varepsilon) / \\arctan(1 / \\varepsilon)$, which lies in $\\mathcal{A}$.\nWith $c_\\varepsilon = \\arctan(1 / \\varepsilon)$, one has $u_\\varepsilon'(x) = \\varepsilon / \\big(c_\\varepsilon (\\varepsilon^2 + x^2)\\big)$.\nSince $x^2 \\le \\varepsilon^2 + x^2$,\n$$J(u_\\varepsilon) \\le \\frac{1}{c_\\varepsilon^2} \\int_{-1}^{1} \\frac{\\varepsilon^2}{\\varepsilon^2 + x^2} \\, dx = \\frac{2 \\varepsilon}{c_\\varepsilon},$$\nwhich tends to $0$ as $\\varepsilon \\to 0$, because $c_\\varepsilon \\to \\pi / 2$.\n\nCheck that $0$ is not attained.\nIf $J(u) = 0$, then $x u'(x) = 0$ for all $x$, by continuity of the integrand.\nSo $u' = 0$ for $x \\ne 0$, hence also at $0$, and $u$ is constant.\nThis contradicts $u(-1) = -1 \\ne 1 = u(1)$.\n\n## Remarks\n\nRiemann had used the Dirichlet principle, the claim that a nonnegative energy with given boundary values attains its infimum.\nWeierstrass's example shows that this claim can fail, even for a nonnegative functional with a smooth integrand.\nThe minimising functions $u_\\varepsilon$ tend to the sign function, which jumps at $0$ and does not belong to the admissible class.\nThe integrand $x^2 \\xi^2$ is convex in $\\xi$, but its coefficient vanishes at $x = 0$.\nSo $J$ gives no bound on $u'$ near $0$, and minimising sequences can develop a jump there.\nThis is the failure of the compactness step of the [[direct method|TE379HBTY0]].\nHilbert later justified the Dirichlet principle for the Dirichlet energy by the direct method.\n\n## Explanations\n\n### L1 human\n\nThe problem asks for a curve from height $-1$ to height $1$ that makes a weighted energy as small as possible.\nThe weight $x^2$ is zero at the middle, so steep slopes there are almost free.\nCurves that jump more and more steeply at $0$ get energies closer and closer to $0$.\nBut a curve with energy exactly $0$ would have to be flat everywhere, and then it could not climb from $-1$ to $1$.\nSo the infimum is never reached: the best candidates converge to a jump, which is not an allowed curve.\n",
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