Bony–Brezis theorem

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: • 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 is enough to check it in Rn. In that case X locally satisfies a Lipschitz condition :\displaystyle{|X(a)-X(b)|\le C|a-b|.} If F is closed, the distance function D(x) = d(x,F)2 has the following differentiability property: :\displaystyle{D(x+h)=D(x) + \min_{z} 2h\cdot(x-z) + o(|h|),} where the minimum is taken over the closest points z to x in F. :To check this, let ::\displaystyle{f_\varepsilon(h)=\min_{z} 2h\cdot(x- z),} :where the minimum is taken over z in F such that d(x,z) ≤ d(x,F) + ε. :Since fε is homogeneous in h and increases uniformly to f0 on any sphere, ::\displaystyle{f_0(h) \ge f_\varepsilon(h) \ge f_0(h) - C(\varepsilon) |h|,} :with a constant C(ε) tending to 0 as ε tends to 0. :This differentiability property follows from this because ::\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} :and similarly if |h| ≤ ε ::\displaystyle{D(x+h) \ge D(x) + f_\varepsilon(h) + |h|^2.} The differentiability property implies that :\displaystyle{\lim{\delta\downarrow 0} {D(c(t+\delta)) - D(c(t))\over \delta} =2\min_z X(c(t))\cdot (c(t)-z),} minimized over closest points z to c(t). For any such z :\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).} 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 2C⋅D(c(t)). Thus the derivative from the right of :\displaystyle{e^{-2Ct}D(c(t))} 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==

• • • • • • • • , Theorem 8.5.11 • • == See also ==