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Results tagged with geometric-measure-theory
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user 528383
Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.
2
votes
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Is $f'_n$ assumed to be the distributional derivative of $f_n$ or just the (a.e.) pointwise one? In the latter case, I think a counterexamples is given by considering a piecewise constant approximati …
6
votes
Accepted
Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?
It follows from co-area formula:
$$
\int_A |Df|=\int_{\mathbb R} \mathcal H^{n-1} (\{f=t\}\cap A).
$$
By taking $A=\{|Df|= 0\}$ one gets that
$$
\int_{\mathbb R} \mathcal H^{n-1} (\{f=t\}\cap \{|Df|= …
3
votes
Accepted
Solution to the Eikonal equation with almost everywhere continuous derivative
If you take $u$ to be the distance from the boundary of $\Omega$ the two properties you look for are statisfied at every point of differentiability for $u$, indeed in any such point $x$ the gradient o …
1
vote
Accepted
On nowhere differentiability of functions that just barely fail to be Lipschitz
As pointed out, this is possible only for $p\le n$. Let $u$ be an unbounded function in $W_0^{1,n}(B_{1/2})$, for instance
$$
u(x)=\log |\log |x||-\log \log 2
$$
and let ${q_i}_i$ be a countable dens …
4
votes
Accepted
Metric currents on singular measures in $\mathbb R^d$
The implication does not work. What it is true is that if a measure admits $k$ Alberti representations, it is absolutely continuous with respect to $\mathcal H^k$. Actually, it is absolutely continuou …