Mathematics
On interrelation between resolvabilities of some boundary value problems for $L$-harmonic functions on unbounded open subsets of Riemannian manifolds
S. A. Korolkov Volgograd State University
Abstract:
We study
$L$-harmonic functions,
i.e. solutions of the stationary Shrodinger equation
$$Lu\equiv\Delta u-c(x)u=0$$
on unbounded open set of Riemannian manifold and
establish some existence results.
Let
$M$ be a smooth connected noncompact Riemannian manifold
without boundary and
$\Omega$ be a simply connected unbounded open
set of
$M$ with
$C^1$-smooth boundary
$\partial \Omega$. Let
$\{B_k\}_{k=1}^\infty$ be a smooth exhaustion of
$M$, i.e. sequence
of precompact open subsets of
$M$ with
$C^1$-smooth boundaries
$\partial B_k$ such that
$M=\bigcup_{k=1}^\infty B_k$,
$\overline
B_k\subset B_{k+1}$ for all
$k$. In what follows we assume
$B_k\cap \Omega\ne\emptyset$, sets
$B_k\cap\Omega$ are simply
connected,
$\partial B_k$ and
$\partial \Omega$ are transversal
for all
$k$.
Let
$B'_k=B_k \setminus\Omega$ and
$v_{M\setminus B'_k}$ be a
$L$-potential of
$B'_k$ relative to
$M$ (see, for example,
[10; 11]). By the maximum principle, the
sequence
$\{v_{M\setminus B'_k}\}_{k=1}^\infty$ is point-wise
increasing and converges to an
$L$-harmonic in
$\Omega$ function
$v_{\Omega}=\lim\limits_{k\to\infty}v_{M\setminus B'_k}$. It is
easy to see that
$0\leq v_{\Omega}\leq 1$,
$v_\Omega|_{\partial\Omega}=1$. The function
$v_{\Omega}$ is
called the
$L$-potential of the
$\Omega$.
Two continuous in
$\Omega$ (in
$\partial\Omega$, resp.) functions
$f_1$ and
$f_2$ are called
weak equivalent in
$\Omega$
(in
$\partial\Omega$, resp.)
relative to $v_{\Omega}$
(
$f_1\stackrel{\Omega}{\simeq} f_2$,
$f_1\stackrel{\partial\Omega}{\simeq} f_2$, resp.) if there exists
some constant
$C$, such that
$|f_1-f_2|\leq C v_{\Omega}\rm{\; in
\;} \Omega$ (
$|f_1-f_2|\leq C v_{\Omega}$ in
$\partial\Omega$,
resp.).
A continuous function
$f$ in
$\Omega$ is called weak admissible
relative to
$\Omega$ (
$f\in K^*_\Omega(\Omega)$) if there is an
compact
$B$ and
$L$-harmonic function
$u$ in
$\Omega\setminus B$
such that
$u\stackrel{\Omega}{\simeq} f$ in
$\Omega\setminus B$.
We have the following results.
Theorem 1. Let
$B$ be an compact,
$v_{M\setminus B}$ be a
$L$-potential of
$B$ relative to
$M$ and
$u(x)$ be an
$L$-harmonic
in
$\Omega\setminus B$ function. Then there exists a constant
$C$
and
$L$-harmonic in
$\Omega$ function
$f$ such that
$ |f-u|\leq Cv_{M\setminus B} \in{\; in \;} \Omega\setminus B.$
Theorem 2. Let
$f\in K^*_\Omega(\Omega)$. Then for any
continuous in
$\partial\Omega$ function
$\varphi$ such that
$\varphi\stackrel{\Omega}{\simeq}f$ in
$\partial\Omega$, there
exists solution of the following problem in
$\Omega$
$$
\left\{
\begin{array}{c} Lu=0 \rm{\; in \;}\Omega,\\
u|_{\partial\Omega}=\varphi,\\
u\stackrel{\Omega}{\simeq}f.
\end{array}
\right.
$$
Keywords:
boundary
problems, $L$-harmonic functions, Riemannian manifolds, solutions of the stationary Shrodinger equation, equivalent functions.
UDC:
517.954
BBK:
22.161