Degenerate elliptic operators in one dimension

Details Degenerate elliptic operators in one dimension Degenerate elliptic operators in one dimension

2009-09-03

arxiv.org/abs/0909.0567v1

Mathematics

Analysis of PDEs

28 pages

Scientific paper

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H\varphi=-(c \varphi')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative $ c\in W^{1,\infty}_{\rm loc}(\Ri)$ the corresponding operator $H$ has a unique submarkovian extension.

Affiliated with

Also associated with

No associations

Rating

Degenerate elliptic operators in one dimension 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 Degenerate elliptic operators in one dimension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Degenerate elliptic operators in one dimension will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-663436