An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

Chernyavskaya, Nina A. ; Shuster, Leonid A.

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 709-716.

Résumé

We consider the weighted space $W_1^{(2)}(\mathbb{R},q)$ of Sobolev type $$ W_1^{(2)}(\mathbb R,q)=\left \{y\in A_{\rm loc}^{(1)}(\mathbb R)\colon \|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}\infty \right \} $$ and the equation $$ - y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb R. \leqno (1) $$ Here $f\in L_1(\mathbb{R})$ and $0\le q\in L_1^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathbb{R}).$ \endgraf We prove the following: \item {1)} The problems of embedding $W_1^{(2)}(\mathbb{R},q)\hookrightarrow L_1(\mathbb{R})$ and of correct solvability of (1) in $L_1(\mathbb{R})$ are equivalent; \item {2)} an embedding $W_1^{(2)}(\mathbb{R},q)\hookrightarrow L_1(\mathbb{R})$ exists if and only if $$\exists a>0\colon \inf _{x\in \mathbb R}\int _{x-a}^{x+a} q(t) {\rm d} t>0.$$

Détail

DOI : 10.1007/s10587-012-0041-6

Classification : 34B24, 34B40, 46E35
Mots-clés : Sobolev space; embedding theorem; Sturm-Liouville equation