View source for (13E)-labda-7,13-dien-15-ol synthase

{{Orphan|date=August 2013}}

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 (mathematics)|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 equation]]s.

==Statement==
Let ''F'' be closed subset of a C<sup>2</sup> 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''.

==Proof==
Following {{harvtxt|Hörmander|1983}}, to prove that the first condition implies the second, let ''c''(''t'') be an integral curve with
''c''(0) = ''x'' in ''F'' and ''dc/dt''= ''X''(''c''). Let ''g'' have a local maximum on ''F'' at ''x''. Then ''g''(''c''(''t'')) ≤ ''g'' (''c''(0)) for ''t'' small and positive. Differentiating, this implies that ''g'' '(''x'')⋅''X''(''x'') ≤ 0.

To prove the reverse implication, since the result is local. it enough to check it in '''R'''<sup>''n''</sup>. In that case ''X'' locally satisfies a Lipschitz condition

:<math>\displaystyle{|X(a)-X(b)|\le C|a-b|.}</math>

If ''F'' is closed, the distance function ''D''(''x'') = ''d''(''x'',''F'')<sup>2</sup> has the following differentiability property:

:<math>\displaystyle{D(x+h)=D(x) + \min_{z} 2h\cdot(x-z)  + o(|h|),}</math>

where the minimum is taken over the closest points ''z'' to ''x'' in ''F''.

:To check this, let

::<math>\displaystyle{f_\varepsilon(h)=\min_{z} 2h\cdot(x- z),}</math>

:where the minimum is taken over ''z'' in ''F'' such that ''d''(''x'',''z'') ≤ ''d''(''x'',''F'') + ε.

:Since ''f''<sub>ε</sub> is homogeneous in ''h'' and increases uniformly to ''f''<sub>0</sub> on any sphere,

::<math>\displaystyle{f_0(h) \ge f_\varepsilon(h) \ge f_0(h) - C(\varepsilon) |h|,}</math>

:with a constant ''C''(ε) tending to 0 as ε tends to 0.

:This differentiability property follows from this because

::<math>\displaystyle{D(x+h) \le |x+h-z|^2 \le |z-x|^2 +2h\cdot (x-z) + |h|^2 =D(x) +f_0(h) +|h|^2}</math>

:and similarly if |''h''| ≤ ε

::<math>\displaystyle{D(x+h) \ge D(x) + f_\varepsilon(h) + |h|^2.}</math>

The differentiability property implies that

:<math>\displaystyle{\lim{\delta\downarrow 0} {D(c(t+\delta)) - D(c(t))\over \delta} =2\min_z X(c(t))\cdot (c(t)-z),}</math>

minimized over closest points ''z'' to ''c''(''t''). For any such ''z''

:<math>\displaystyle{2X(c(t))\cdot (c(t)-z)= 2 X(z)\cdot (c(t)-z) - 2(X(z)-X(c(t)))\cdot (c(t)-z).}</math>

Since −|''y'' − ''c''(''t'')|<sup>2</sup> has a local maximum on ''F'' at ''y'' = ''z'', ''c''(''t'') −  ''z'' is an exterior normal vector at ''z''. So the first term on the right hand side is non-negative. The Lipschitz condition for ''X'' implies the second term is bounded above by 2''C''⋅''D''(''c''(''t'')). Thus the [[one-sided derivatives|derivative from the right]] of

:<math>\displaystyle{e^{-2Ct}D(c(t))}</math>

is non-positive, so it is a non-increasing function of ''t''. Thus if ''c''(0) lies in ''F'', ''D''(''c''(0))=0 and hence ''D''(''c''(''t'')) = 0 for ''t'' > 0, i.e. ''c''(''t'') lies in ''F'' for ''t'' > 0.

==References==
*{{citation|first=Jean-Michel|last= Bony|title= Principe du Maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénerés|journal= Ann. Inst. Fourier|volume= 19|year= 1969|pages= 277–304}}
*{{citation|first=Haim|last= Brezis|title= On a characterization of flow-invariant sets|journal= Comm. Pure App. Math.|volume= 223|year=1970|pages= 261–263}}
*{{citation|first=Lars|last=Hörmander|title=Analysis of Partial Differential Operators I|publisher=Springer-Verlag|year=1983|isbn=3-540-12104-8|pages=300–305}}, Theorem 8.5.11
*{{citation|title=The Theorems of Bony and Brezis on Flow-Invariant Sets|first=R. M.|last= Redheffer|
journal=The American Mathematical Monthly |volume= 79|year= 1972|pages= 740–747}}

{{DEFAULTSORT:Bony-Brezis theorem}}
[[Category:Ordinary differential equations]]
[[Category:Dynamical systems]]
[[Category:Manifolds]]