Saturday, 23 June 2007

ca.analysis and odes - Lipschitz functions in mathbbRnmathbbRn

Let f=(f1,ldots,fn):[a,b]rightarrowmathbbRnf=(f1,ldots,fn):[a,b]rightarrowmathbbRn be a continuously differentiable function. (See the comments above for an explanation as to why the hypotheses have been strengthened.)



For 1leqileqn1leqileqn, let



Li=maxxin[a,b]|fi(x)|Li=maxxin[a,b]|fi(x)|,



so that, by the Mean Value Theorem, for x,yin[a,b]x,yin[a,b],



|fi(x)fi(y)|=|fi(c)||xy|leqLi|xy||fi(x)fi(y)|=|fi(c)||xy|leqLi|xy|.



Then, taking the standard Euclidean norm on mathbbRnmathbbRn,



|f(x)f(y)|2=sumni=1|fi(x)fi(y)|2leq(sumni=1L2i)|xy|2|f(x)f(y)|2=sumni=1|fi(x)fi(y)|2leq(sumni=1L2i)|xy|2,



so



|f(x)f(y)|leqsqrt(sumni=1L2i)|xy||f(x)f(y)|leqsqrt(sumni=1L2i)|xy|.



Thus we can take



L=sqrtsumni=1L2iL=sqrtsumni=1L2i.



Since all norms on mathbbRnmathbbRn are equivalent -- i.e., differ at most by a multiplicative constant -- the choice of norm on mathbbRnmathbbRn will change the expression of the Lipschitz constant LL in terms of the Lipschitz constants LiLi of the components, but not whether ff is Lipschitz.

No comments:

Post a Comment