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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
to be invariant under the
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psych ...
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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
s.
The theorem had in fact been previously discovered by
Mitio Nagumo
Mitio (Michio) Nagumo ( ja, 南雲 道夫; May 7, 1905 – February 6, 1995) was a Japanese mathematician, who specialized in the theory of differential equations. He gave the first necessary and sufficient condition for positive invaria ...
in 1942 and is also known as the Nagumo theorem.
Statement
Let ''F'' be closed subset of a C
2 manifold ''M'' and let ''X'' be a
vector field on ''M'' which is
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ...
. The following conditions are equivalent:
* Every
integral curve of ''X'' starting in ''F'' remains in ''F''.
* (''X''(''m''),''v'') ≤ 0 for every exterior normal vector ''v'' at a point ''m'' in ''F''.
Proof
Following , 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
''n''. In that case ''X'' locally satisfies a Lipschitz condition
:
If ''F'' is closed, the distance function ''D''(''x'') = ''d''(''x'',''F'')
2 has the following differentiability property:
:
where the minimum is taken over the closest points ''z'' to ''x'' in ''F''.
:To check this, let
::
:where the minimum is taken over ''z'' in ''F'' such that ''d''(''x'',''z'') ≤ ''d''(''x'',''F'') + ε.
:Since ''f''
ε is homogeneous in ''h'' and increases uniformly to ''f''
0 on any sphere,
::
:with a constant ''C''(ε) tending to 0 as ε tends to 0.
:This differentiability property follows from this because
::
:and similarly if , ''h'', ≤ ε
::
The differentiability property implies that
:
minimized over closest points ''z'' to ''c''(''t''). For any such ''z''
:
Since −, ''y'' − ''c''(''t''),
2 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
derivative from the right of
:
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
Literature
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*, Theorem 8.5.11
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{{DEFAULTSORT:Bony-Brezis theorem
Ordinary differential equations
Dynamical systems
Manifolds