c****n
发帖数: 2031 | 1 Given a smooth closed 2-manifold S in R^3. Define the function d(x), x\in R^
3,
to be the unsigned distance function to the surface S. （For example in 1D,
if S is a point x0, then d(x)=|x-x0|.）The function d(x) is only continuous
but not smooth, e.g. it's not differentiable on S. Now the question is: is
there a sequence (countable or uncountable) d_\epsilon(x)->d(x) uniformly (i
.e. under supremum norm), where d_\epsilon are smooth and {x:d_\epsilon(x)=0
}=S, for all \epsilon(i.e. preserve zero | G******i
发帖数: 163 | 2 Dentoe by N(epsilon) the epsilon neighborhood of S.
For each small \epsilon>0, choose a smooth
f_\epsilon : R^3 -N(1/2*epsilon) -> (1/2*epsilon,\infty)
and uniformly approximates d_\epsilon(x) on R^3 -N(1/2*epsilon)
Define d_\epsilon(x) on R^3 as follows:
d_\epsilon(x) =d(x)^2 in N(epsilon);
d_\epsilon(x) =f_\epsilon in R^3 -N(2*epsilon);
in between, make a smooth and positive transition.
R^
,
continuous
(i
=0
【在 c****n 的大作中提到】
: Given a smooth closed 2-manifold S in R^3. Define the function d(x), x\in R^
: 3,
: to be the unsigned distance function to the surface S. （For example in 1D,
: if S is a point x0, then d(x)=|x-x0|.）The function d(x) is only continuous
: but not smooth, e.g. it's not differentiable on S. Now the question is: is
: there a sequence (countable or uncountable) d_\epsilon(x)->d(x) uniformly (i
: .e. under supremum norm), where d_\epsilon are smooth and {x:d_\epsilon(x)=0
: }=S, for all \epsilon(i.e. preserve zero
| c****n
发帖数: 2031 | 3 Thanks!
~~~~~~~~Here you mean f_4\epsilon, right?
【在 G******i 的大作中提到】
: Dentoe by N(epsilon) the epsilon neighborhood of S.
: For each small \epsilon>0, choose a smooth
: f_\epsilon : R^3 -N(1/2*epsilon) -> (1/2*epsilon,\infty)
: and uniformly approximates d_\epsilon(x) on R^3 -N(1/2*epsilon)
: Define d_\epsilon(x) on R^3 as follows:
: d_\epsilon(x) =d(x)^2 in N(epsilon);
: d_\epsilon(x) =f_\epsilon in R^3 -N(2*epsilon);
: in between, make a smooth and positive transition.
:
: R^
|
|
