On interrelation between resolvabilities of some boundary value problems for $L$-harmonic functions on unbounded open subsets of Riemannian manifolds

S. A. Korolkov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2014), pp. 17-26

Résumé

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 $\Omega$ $$ \left\{\begin{array}{c} Lu=0 \rm{\; in \;}\Omega,\\ u|_{\partial\Omega}=\varphi,\\ u\stackrel{\Omega}{\simeq}f. \end{array}\right. $$

Détail

Keywords: boundary problems, $L$-harmonic functions, Riemannian manifolds, solutions of the stationary Shrodinger equation, equivalent functions.