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  "body": "\n## Problem\n\nLet $D \\subseteq \\R^d$ be a bounded open set, called the design region.\nLet $\\mathcal{O}$ be a class of open subsets $\\Omega \\subseteq D$, the admissible shapes, for example those with a given volume.\nFor each $\\Omega \\in \\mathcal{O}$ let $y_\\Omega$ be the solution of a boundary value problem for a [[partial differential equation|GC5BDZ6QBC]] on $\\Omega$.\nLet $J(\\Omega) = j(\\Omega, y_\\Omega)$ be a real-valued shape [[functional|1EQM0DJ4NC]].\nThe *shape optimization problem* is to find $\\Omega^* \\in \\mathcal{O}$ with $J(\\Omega^*) \\le J(\\Omega)$ for every $\\Omega \\in \\mathcal{O}$.\n\n## Remarks\n\nIt is a form of [[optimal control of partial differential equations|S2BBZPAFDR]] in which the control is the domain itself.\nExample: minimize the first Dirichlet eigenvalue $\\lambda_1(\\Omega)$ of the Laplacian among open sets $\\Omega \\subseteq \\R^d$ of a given volume.\nThe Rayleigh–Faber–Krahn inequality solves it: the minimizers are the balls.\nIn general minimizers need not exist in a given class of shapes, since minimizing sequences can develop holes or oscillating boundaries.\nOne then restricts the class, for example by a uniform cone condition or a perimeter bound, or relaxes the problem; relaxation leads to topology optimization.\nGradient methods use the [[shape derivative|9XFEG6ANGM]] of $J$, often computed with an adjoint equation.\n\n## Explanations\n\n### L1 human\n\nShape optimization asks for the best shape of an object: a wing with least drag, a beam with greatest stiffness, a drum with lowest tone.\nThe quality of a shape is measured through the solution of a physical equation on that shape.\nFor example, among all drums of the same area, the round drum has the lowest fundamental tone.\n",
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      "first_stated": "1907",
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      "posed_by": "Jacques Hadamard"
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