One-radius results for supermedian functions on $\Bbb R^d$, $d\le 2$

Mathematics – Complex Variables

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

Scientific paper

A classical result states that every lower bounded superharmonic function on $\Bbb R^2$ is constant. In this paper the following (stronger) one-circle version is proven. If $f\colon \Bbb R^2\to (-\infty,\infty]$ is lower semicontinuous, $\liminf_{|x|\to\infty} f(x)/\ln|x|\ge 0$, and, for every $x\in\Bbb R^2$, $1/(2\pi) \int_0^{2\pi} f(x+r(x)e^{it}) dt\le f(x)$, where $r\colon \Bbb R^2\to (0,\infty)$ is continuous, $\sup_{x\in\Bbb R^2} (r(x)-|x|)<\infty$, and $\inf_{x\in\Bbb R^2} (r(x)-|x|)=-\infty$, then $f$ is constant. Moreover, it is shown that, with respect to the assumption $r\le c|\cdot|+M$ on $\Bbb R^d$, there is a striking difference between the restricted volume mean property for the cases $d=1$ and $d=2$.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

One-radius results for supermedian functions on $\Bbb R^d$, $d\le 2$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with One-radius results for supermedian functions on $\Bbb R^d$, $d\le 2$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and One-radius results for supermedian functions on $\Bbb R^d$, $d\le 2$ will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-120110

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.