3
$\begingroup$

Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit $$ \lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz) $$ where

  • $B(x,r)=\{y \in \mathbb{R}^d \mid |y-x|<r\}$ is the Euclidean ball centered at $x$ with radius $r$ and
  • $m$ is the Lebesgue measure on $\mathbb{R}^d$,

could exist (possibly in a almost-everywhere sense).

Two more or less elementary answers possibly are

  • $D$ is a Caccioppoli set and $x$ is a point of its reduced boundary and
  • when $D$ has a singular part with finite (electrostatic or Newtonian) capacity (see this Q&A for reference).

Are there any other cases? Also references are welcome.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.