| Abstract: |
| In its simplest form, Sturm's comparison result is:
Theorem.
Let $q_1(x) \leq q_2(x)$ and not identically equal on $[a,b]. $
Let $u$ satisfy $u'' + q_1(x)u = 0$ on $[a,b]$, and
$v$ satisfy $v''+ q_2(x)v = 0$ on $[a,b]$.
Suppose that $a < b$ are consecutive zeros of $u$.
Then $v$ has at least one zero in $(a,b)$.
There is a discrete version for symmetric tri-diagonal operators with positive off-diagonal terms. However, it does not hold for general symmetric penta-diagonal operators with positive off-diagonal terms.
The main point today, is to replace the second derivative operator by a bounded integral operator, resulting in a nonlocal diffusion-like operator. Is there a similar comparison theorem? First, when $u''$ is replaced by $$Lu := \int J(x-y)[u(y)-u(x)]dy,$$ where $J\geq 0$ is continuous, even, with compact support and $J(0)>0$.
Then with a scaled version in $\mathbb{R}^n$
$$L_\epsilon u := \int_\Omega \frac1{\epsilon^{n+2}}J(\frac{x-y}{\epsilon})[u(y)-u(x)]dy,$$ for $\epsilon>0$ and small, and where $\Omega \subset \mathbb{R}^n$ is smoothly bounded, and $J\geq 0$ is radially symmetric with $f(0)>0$.
This operator converges, in some sense,
to a multiple of the Laplacian, as $\epsilon\to 0$. Does a Sturm-like comparison theorem hold when $\epsilon$ is small?
This is joint work with Guangyu Zhao. |
|