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In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive inner product with any exterior normal vector to the set. A vector is an ''exterior normal'' at a point of the closed set if there is a real-valued continuously differentiable function maximized locally at the point with that vector as its derivative at the point. If the closed subset is a smooth submanifold with boundary, the condition states that the vector field should not point outside the subset at boundary points. The generalization to non-smooth subsets is important in the theory of partial differential equations. The theorem had in fact been previously discovered by Mitio Nagumo in 1942 and is also known as the Nagumo theorem. ==Statement== Let ''F'' be closed subset of a C2 manifold ''M'' and let ''X'' be a vector field on ''M'' which is Lipschitz continuous. The following conditions are equivalent: *Any integral curve of ''X'' starting in ''F'' remains in ''F''. * (''X''(''m''),''v'') ≤ 0 for any exterior normal vector ''v'' at a point ''m'' in ''F''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bony–Brezis theorem」の詳細全文を読む スポンサード リンク
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